Recent zbMATH articles in MSC 42https://zbmath.org/atom/cc/422021-01-08T12:24:00+00:00WerkzeugSome equalities and inequalities for Parseval \(K\)-frames.https://zbmath.org/1449.420562021-01-08T12:24:00+00:00"Fu, Yuankang"https://zbmath.org/authors/?q=ai:fu.yuankang"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: We give the equalities and inequalities for Parseval \(K\)-frames by using a conclusion about the equalities and inequalities for \(K\)-frames. Then we introduce the concept of upper and lower indexes of Parseval \(K\)-frames and discuss some properties of upper and lower indexes of Parseval \(K\)-frames. The conclusions given in this paper generalize the corresponding conclusions.Constructions of \(K\)-frames in Hilbert spaces.https://zbmath.org/1449.420542021-01-08T12:24:00+00:00"Du, Dandan"https://zbmath.org/authors/?q=ai:du.dandan"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: A \(K\)-frame is a generalization of a frame in a Hilbert space. In this paper we use two Bessel sequences to construct a \(K\)-frame, a \({T_1}\)-frame or \({T_2}\)-frame in a Hilbert space. We also construct a \(P\)-frame or \(Q\)-frame by two \(K\)-frames in a Hilbert space. Our results generalize and improve the existing remarkable results.Sharp bound for the weighted bilinear Hardy operator on the \({L^p}\) space with power weight.https://zbmath.org/1449.420222021-01-08T12:24:00+00:00"Xiao, Fuyu"https://zbmath.org/authors/?q=ai:xiao.fuyuSummary: We study the boundedness of the weighted bilinear Hardy operator and the weighted bilinear Cesàro operator on the \({L^p}\) space with power weight and obtain norms of these operators on the \({L^p}\) space with power weight. As applications, we also calculate the sharp bounds of the bilinear Riemann-Liouville operator and the bilinear Weyl operator on the \({L^p}\) space with power weight.Boundedness for commutators of Marcinkiewicz integrals with variable kernel on Herz-Hardy spaces with variable exponents.https://zbmath.org/1449.420192021-01-08T12:24:00+00:00"Shao, Xukui"https://zbmath.org/authors/?q=ai:shao.xukuiSummary: With the help of the boundedness of commutators of the Marcinkiewicz integral on Lebesgue spaces with variable exponents and the theory of atomic decomposition on Herz-Hardy space with variable exponents, the boundedness of commutators of the Marcinkiewicz integral with variable kernels \(\mu_\Omega^b\) on homogeneous and nonhomogeneous Herz-Hardy spaces with variable exponents was given.The boundedness of maximal dyadic derivative operator on dyadic martingale Hardy space with variable exponents.https://zbmath.org/1449.420362021-01-08T12:24:00+00:00"Zhang, Chuanzhou"https://zbmath.org/authors/?q=ai:zhang.chuanzhou"Xia, Qi"https://zbmath.org/authors/?q=ai:xia.qi"Zhang, Xueying"https://zbmath.org/authors/?q=ai:zhang.xueyingSummary: In this paper, we research dyadic martingale Hardy spaces with variable exponents. By the characterization of log-Hölder continuity, the Doob's inequality is derived. Moreover, we prove the boundedness of the maximal dyadic derivative operator by the atomic decomposition of the variable exponent martingale space, which generalizes the conclusion in the classical case.On approximation of functions belonging to some classes of functions by \((N,p_n,q_n)(E,\theta )\) means of conjugate series of its Fourier series.https://zbmath.org/1449.420042021-01-08T12:24:00+00:00"Krasniqi, Xhevat Zahir"https://zbmath.org/authors/?q=ai:krasniqi.xhevat-zahir"Deepmala"https://zbmath.org/authors/?q=ai:deepmala.|deepmala.vandanaSummary: We obtain some new results on the approximation of functions belonging to some classes of functions by \((N,p_n,q_n)(E,\theta )\) means of conjugate series of its Fourier series. These results, under conditions assumed here, are better than those obtained previously by others. In addition, several particular results are derived from our results as corollaries.\(Tb\) criteria for Calderón-Zygmund operators on Lipschitz spaces with para-accretive functions.https://zbmath.org/1449.420282021-01-08T12:24:00+00:00"Zheng, Taotao"https://zbmath.org/authors/?q=ai:zheng.taotao"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingThe goal of this paper is to give a Tb criteria for the boundedness of Calderón-Zygmund operators on the Lipschitz spaces \[\mathrm{Lip}_b(\alpha)(\mathbb R^n).\] The main machine is to develop the Littlewood-Paley characterization for Lipschitz spaces \(\mathrm{Lip}_b(\alpha)(\mathbb R^n)\) and \(\mathrm{Lip}(\alpha)(\mathbb R^n)\), which also has its own value and significance.
Then, the authors prove that the Calderón-Zygmund operators \(T\) are bounded from \(\mathrm{Lip}_b(\alpha)(\mathbb R^n)\) to \(\mathrm{Lip}(\alpha)(\mathbb R^n)\) if and only if \[T(b) \,= \,0.\] For this purpose, the authors recall the useful tools that are the para-accretive function, test function spaces, an approximation to the identity and some discrete versions of the Calderón type reproducing formula.
Finally, it is pointed out that to prove the main result in the paper, the authors need the boundedness of Calderón-Zygmund operator on Hardy spaces.
Reviewer: Maria Alessandra Ragusa (Catania)Weighted estimates of fractional maximal operator and its commutator on weighted \(\lambda\)-central Morrey spaces.https://zbmath.org/1449.420342021-01-08T12:24:00+00:00"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Yang, Yuhe"https://zbmath.org/authors/?q=ai:yang.yuheSummary: By applying weighted inequalities and real variable methods, the boundedness of the fractional maximal operator with rough kernel is obtained in the weighted \(\lambda\)-central Morrey spaces is obtained. The boundedness of its commutator generated by a \(\lambda\)-central mean oscillation function is also proved.The compactness of commutators of bilinear fractional maximal operators on Multi-Morrey spaces.https://zbmath.org/1449.420312021-01-08T12:24:00+00:00"Guo, Qingdong"https://zbmath.org/authors/?q=ai:guo.qingdong"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangSummary: Let \({\mathcal{M}_\alpha}\) be the bilinear fractional maximal operators and let \(\vec{b} = ({b_1}, {b_2})\) be a collection of locally integrable functions. In this paper, we obtain that the commutators generated by the bilinear fractional maximal operators and the CMO (\(C_c^\infty\) closure under the BMO norm) functions are compact operators from the Morrey spaces to the Multi-Morrey spaces, where the commutators include the fractional maximal linear commutators \({\mathcal{M}_{\alpha, \sum \vec{b}}}\) and fractional maximal iterator commutators \({\mathcal{M}_{\alpha, \prod \vec{b}}}\). The conclusion of this paper is also new when the operators are linear.Weighted estimates of Marcinkiewicz integral with variable kernel on variable exponent Herz-Morrey spaces.https://zbmath.org/1449.420212021-01-08T12:24:00+00:00"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Shi, Jinli"https://zbmath.org/authors/?q=ai:shi.jinliSummary: By using hierarchical decomposition of functions and weighted inequalities, and by means of the weight boundedness on variable exponent Lebesgue spaces, we proved the boundedness of Marcinkiewicz integral operators with variable kernel on weighted variable exponent Herz-Morrey spaces.Some properties of frame for translate operator and modulation operator.https://zbmath.org/1449.420582021-01-08T12:24:00+00:00"Liu, Chuntai"https://zbmath.org/authors/?q=ai:liu.chuntaiSummary: A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. The author discussed some properties of some families of functions induced by translate operator or modulation operator, and gave some conditions for these families being not a continuous frame.\(BL_{p,\nu}^{m}\) estimates for the Riesz transforms associated with Laplace-Bessel operator.https://zbmath.org/1449.420122021-01-08T12:24:00+00:00"Ekincioglu, Ismail"https://zbmath.org/authors/?q=ai:ekincioglu.ismail"Keskin, Cansu"https://zbmath.org/authors/?q=ai:keskin.cansu"Guner, Serap"https://zbmath.org/authors/?q=ai:guner.serapSummary: In this paper, we introduce higher order Riesz-Bessel transforms which we can express partial derivatives of order \(\alpha\) of \(I_{m,\nu}f\) for \(f\in L_{p,\nu}\). In addition, we establish a relationship between a Riesz potential with a higher order Riesz-Bessel transform related to the generalized shift operator. By using this relationship, we make some improvements of integral estimates for \(I_{m,\nu}f\) and the higher order Riesz-Bessel transform \(R_{\nu}^{m}\) in the Beppo Levi space \(BL_{p,\nu}^{m}\). We prove an estimate for the singular integral operator with convolution type generated by the generalized shift operator in the Beppo Levi spaces.Construction of bi-orthogonal two-direction vector valued multiwavelet with tectonic scale \(a\).https://zbmath.org/1449.420732021-01-08T12:24:00+00:00"Lv, Jun"https://zbmath.org/authors/?q=ai:lv.jun"Ku, Fuli"https://zbmath.org/authors/?q=ai:ku.fuli"Wang, Gang"https://zbmath.org/authors/?q=ai:wang.gang.2|wang.gang.1|wang.gang.5|wang.gang.3|wang.gang|wang.gang.4Summary: Based on the basic theory of vector-valued wavelet, through a unitary matrix, a two-direction vector-valued bi-orthogonal multiwavelet is given and its construction is obtained. Finally, an example is given.Variable fractional integral operator and its commutator on the Herz-Hardy space with variable exponent.https://zbmath.org/1449.420242021-01-08T12:24:00+00:00"Yao, Junqing"https://zbmath.org/authors/?q=ai:yao.junqing"Shi, Hui"https://zbmath.org/authors/?q=ai:shi.hui"Zhao, Kai"https://zbmath.org/authors/?q=ai:zhao.kaiSummary: Based on the definitions and basic properties of the function spaces with variable exponents and the variable fractional integral operators, by the atomic decomposition of the Herz-Hardy spaces with variable exponents, using the Hölder and Jensen inequalities, we proved the boundedness of variable fractional integral operators with homogeneous kernel and its commutators on the Herz-Hardy spaces with variable exponents.A new method of heart sound denoising based on lifting wavelet transform.https://zbmath.org/1449.920222021-01-08T12:24:00+00:00"Xiao, Miao"https://zbmath.org/authors/?q=ai:xiao.miao"Chang, Jun"https://zbmath.org/authors/?q=ai:chang.jun"Pan, Jiahua"https://zbmath.org/authors/?q=ai:pan.jiahua"Yang, Hongbo"https://zbmath.org/authors/?q=ai:yang.hongbo"Wang, Weilian"https://zbmath.org/authors/?q=ai:wang.weilianSummary: The traditional wavelet threshold function denoising can not effectively filter out the specific noise in the signal. Combining the advantages of soft and hard threshold functions, a new threshold function algorithm based on lifting wavelet is proposed to denoise the heart sound signal. Firstly, the heartbeat signal is decomposed by lifting wavelet transform. Then the wavelet coefficients are updated by the new threshold function for reconstruction. Finally, the Hilbert envelope extraction is performed to improve the denoising effect of the heart sounds after denoising the wavelet soft and hard threshold functions. The experimental results show that the proposed method for improving wavelet denoising has better filtering effect than the soft and hard threshold method, and the extracted curve envelope is more clear and smooth.On approximation of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Nörlund mean.https://zbmath.org/1449.420062021-01-08T12:24:00+00:00"Pradhan, T."https://zbmath.org/authors/?q=ai:pradhan.tejaswini"Jena, B. B."https://zbmath.org/authors/?q=ai:jena.bidu-bhusan"Paikray, S. K."https://zbmath.org/authors/?q=ai:paikray.susanta-kumar"Dutta, H."https://zbmath.org/authors/?q=ai:dutta.hemen"Misra, U. K."https://zbmath.org/authors/?q=ai:misra.uma-kantaSummary: In this paper, we have studied an estimate of the rate of convergence of Fourier series in the generalized Hölder metric \(H_{L_r}^{(\omega)}\) space by using the deferred Nörlund mean and established some new results. Our results are more advanced that the already known and unify many other results available in the literature.Frequency characteristics of Sonin-Laguerre orthogonal functions.https://zbmath.org/1449.420422021-01-08T12:24:00+00:00"Prokhorov, S. A."https://zbmath.org/authors/?q=ai:prokhorov.s-a"Kulikovskikh, I. M."https://zbmath.org/authors/?q=ai:kulikovskikh.i-mSummary: The frequency properties of the orthogonal functions in the Sonin-Laguerre basis when changing the parameters of the orthogonal basis are analyzed. Basic relations used for constructing analytical models of the correlation function and the spectral power density are given.Boundedness of commutators for the multilinear Calderón-Zygmund operators with kernels of Dini's type.https://zbmath.org/1449.420292021-01-08T12:24:00+00:00"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiang"Hu, Xi"https://zbmath.org/authors/?q=ai:hu.xiSummary: The authors give a sharp maximal estimate for the iterated commutator that is generated by multilinear Calderón-Zygmund operators with kernels of Dini's type and Lipschitz function. Furthermore, in the suitable index case, the commutator is bounded on product Lebesgue spaces.A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials.https://zbmath.org/1449.652722021-01-08T12:24:00+00:00"Fatone, Lorella"https://zbmath.org/authors/?q=ai:fatone.lorella"Funaro, Daniele"https://zbmath.org/authors/?q=ai:funaro.daniele"Manzini, Gianmarco"https://zbmath.org/authors/?q=ai:manzini.gianmarcoSummary: In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.The Wigner-Ville distribution associated with the quaternion offset linear canonical transform.https://zbmath.org/1449.420072021-01-08T12:24:00+00:00"El Kassimi, M."https://zbmath.org/authors/?q=ai:el-kassimi.m"El Haoui, Y."https://zbmath.org/authors/?q=ai:el-haoui.youssef"Fahlaoui, S."https://zbmath.org/authors/?q=ai:fahlaoui.saidThe Wigner-Ville distribution (WVD) and the quaternion offset linear canonical Fourier transform (QOLCT) are useful tools in signal analysis and image processing. The WVD is widely used for the analysis of the non-stationary signals. In the paper the authors introduce an extension of the WVD to the quaternion algebra by means of the QOLCT. The WVD-QOLC transform combines properties and flexibility of both WVD and QOLCT. The authors show how the WVD-QOLC relates to the general two-sided quaternion Fourier transform (QFT) and to the QOLCT. Using the relation between the WVD-QOLCT and the QOLCT, they obtain the inversion and Plancherel formulas for the WVD-QOLCT. Applying their earlier results related to the QFT and QOLCT, the authors establish Heisenberg's uncertainty principle and the Poisson summation formula associated with WVD-QOLCT. Finally, they formulate and prove an analogue of Lieb's theorem for the WVD-QOLCT. For this purpose they use Lieb's theorem for the quaternionic linear canonical transform.
Reviewer: Olga M. Katkova (Boston)A note on the stability of \(K\)-\(g\)-frames.https://zbmath.org/1449.420642021-01-08T12:24:00+00:00"Xiang, Zhongqi"https://zbmath.org/authors/?q=ai:xiang.zhongqiSummary: In this paper, we present a new stability theorem on the perturbation of \(K\)-\(g\)-frames by using an operator theory method. The result we obtained improves one corresponding conclusion of other authors.The duality of \(K\)-frames in Hilbert \(C^*\)-modules.https://zbmath.org/1449.460502021-01-08T12:24:00+00:00"Xiang, Zhongqi"https://zbmath.org/authors/?q=ai:xiang.zhongqi"Shi, Huangping"https://zbmath.org/authors/?q=ai:shi.huangpingSummary: The present paper studies the dual problems of \(K\)-frames in Hilbert \(C^*\)-modules. Some characterizations for \(K\)-dual Bessel sequences in Hilbert \(C^*\)-modules are obtained by using the operator theory methods, which generalize the duality theory of \(K\)-frames in Hilbert spaces.Two-dimensional Müntz-Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations.https://zbmath.org/1449.652782021-01-08T12:24:00+00:00"Sabermahani, Sedigheh"https://zbmath.org/authors/?q=ai:sabermahani.sedigheh"Ordokhani, Yadollah"https://zbmath.org/authors/?q=ai:ordokhani.yadollah"Yousefi, Sohrab-Ali"https://zbmath.org/authors/?q=ai:yousefi.sohrab-aliSummary: In this manuscript, we present a new numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Müntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works.Sums of \(K\)-Riesz frames in Hilbert spaces.https://zbmath.org/1449.420572021-01-08T12:24:00+00:00"Huang, Xinli"https://zbmath.org/authors/?q=ai:huang.xinli"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: According to the operator theory, the paper suggests a new research method to solve the problem of how the sum of a \(K\)-Riesz frame and a sequence generates a new \(K\)-Riesz frame. Then the sufficient conditions for generating a new \(K\)-Riesz frame are obtained. The results in the paper correct the remarkable results of Riesz-frames.Commutators of bilinear \(\theta\)-type Calderón-Zygmund operators on Morrey spaces over non-homogeneous spaces.https://zbmath.org/1449.420162021-01-08T12:24:00+00:00"Lu, G.-H."https://zbmath.org/authors/?q=ai:lu.gui-hua|lu.guohao|lu.genghong|lu.guanhua|lu.guanghui|lu.guang-hongThe author proves some boundedness properties for commutators which are generated by the bilinear \(\theta\)-type Calderón-Zygmund operators \(T_\theta\) and two functions \(b_1\), \(b_2\) belonging to the space that is a variant of the bounded mean oscillation class, firstly defined by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)].
Precisely, \([b_1,b_2,T_\theta]\) is bounded from the Lebesgue space \(L^p(\mu)\) into the product of Lebesgue spaces \[L^{p_1}(\mu) \times L^{p_2}(\mu), \quad \frac{1}{p}\,=\, \frac{1}{p_1}\,+\, \frac{1}{p_2}, \,\,(1 < p, p_1, p_2 < \infty)\] being \(\mu\) a Borel measure.
Moreover the boundedness of the commutator \([b_1, b_2, T_\theta]\) on the Morrey space \(M^q_p(\mu)\), \(1\,<\,q\,<\,p\,<\,\infty\) is obtained. Main tools are the definitions of geometrically doubling metric space and upper doubling metric measure space.
Reviewer: Maria Alessandra Ragusa (Catania)\(\ell^p\)-improving inequalities for discrete spherical averages.https://zbmath.org/1449.420332021-01-08T12:24:00+00:00"Kesler, R."https://zbmath.org/authors/?q=ai:kesler.robert-m"Lacey, M. T."https://zbmath.org/authors/?q=ai:lacey.michael-tFor \( \lambda ^2 \in \mathbb{N} \), let \(\mathbb{S}^d_\lambda := \{ n\in \mathbb{Z}^d \;:\; | n| = \lambda\}\). For a function \(f:\mathbb{Z} ^{d} \to \mathbb{R} \), define \[A _{\lambda } f (x) = | \mathbb{S}^d_\lambda | ^{-1} \sum_{n \in \mathbb{S}^d_\lambda } f(x-n).\] The following estimate is the main result of the paper under review: \[| A _{\lambda }| _{\ell ^{p} \to \ell ^{p'}} \leq C _{d,p, \omega (\lambda ^2 )} \lambda ^{d (1-\frac{2}p)}, \tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}, d\geq 4.\] In dimension \(d=4\) this estimate proved for odd \(\lambda ^2\). Here \(\omega (\lambda ^2)\) is the number of distinct prime factors of \(\lambda^2\).
This inequality is a discrete version of a classical inequality of \textit{W. Littman} [Partial diff. Equ., Berkeley 1971, Proc. Sympos. Pure Math. 23, 479--481 (1973; Zbl 0263.44006)] and \textit{R. S. Strichartz} [J. Funct. Anal. 5, 218--235 (1970; Zbl 0189.40701)] on the \(L^{p}\) improving property of spherical averages on \(\mathbb{R} ^{d}\).
Reviewer: Michael Perelmuter (Kyïv)Weighted estimates for two kinds of Toeplitz operators.https://zbmath.org/1449.420112021-01-08T12:24:00+00:00"Cui, Zhe"https://zbmath.org/authors/?q=ai:cui.zhe"Lin, Yan"https://zbmath.org/authors/?q=ai:lin.yanSummary: In this paper, we establish the boundedness of a class of Toeplitz operators related to strongly singular Calderón-Zygmund operators and weighted BMO functions on weighted Morrey spaces. Moreover, the boundedness of another kind of Toeplitz operators related to strongly singular Calderón-Zygmund operators and weighted Lipschitz functions on weighted Morrey spaces is also obtained.Classical orthogonal polynomials and some new properties for their centroids of zeroes.https://zbmath.org/1449.330102021-01-08T12:24:00+00:00"Aloui, B."https://zbmath.org/authors/?q=ai:aloui.baghdadi"Chammam, W."https://zbmath.org/authors/?q=ai:chammam.wathekThe aims of the paper is to highlight new properties of the centroid of the zeros of a polynomial. As a illustration, they apply these techniques to \(O\)-classical orthogonal polynomials, where \(O\) is the derivative operator \(D\) or the \(q\)-derivative \(H\).
Reviewer: Francisco Pérez Acosta (La Laguna)A sufficient condition for the finite \({\mu_{M,D}}\)-orthogonal exponentials function system.https://zbmath.org/1449.280112021-01-08T12:24:00+00:00"Li, Na"https://zbmath.org/authors/?q=ai:li.na"Li, Jianlin"https://zbmath.org/authors/?q=ai:li.jianlin|li.jianlin.1Summary: Let \({\mu_{M, D}}\) be a self-affine measure uniquely determined by the iterated function system \(\{\phi_d (x) = M^{-1} (x+d)\}_{d \in D}\). The spectrality or non-spectrality of \({\mu_{M, D}}\) is directly connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space \({L^2} (\mu_{M, D})\). In this paper, the authors provide a sufficient condition for the finite \({\mu_{M, D}}\)-orthogonal exponentials by applying the elementary matrix transformations. This sufficient condition depends only upon the determinant of the matrix \(M\), and is easy to use in the research of non-spectrality of \({\mu_{M, D}}\).Asymptotic property of wavelet estimators in fixed-design partially linear errors-in-variables models with long-range dependent errors.https://zbmath.org/1449.620552021-01-08T12:24:00+00:00"Liu, Xiang"https://zbmath.org/authors/?q=ai:liu.xiang"Hu, Hongchang"https://zbmath.org/authors/?q=ai:hu.hongchang"Yu, Xinxin"https://zbmath.org/authors/?q=ai:yu.xinxinSummary: In this paper, we consider that the random errors are the function of Gaussian random variables with stationary and long-range dependence, and we investigate a partially linear errors-in-variables (EV) model in fixed-design by the wavelet method. Under several conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distribution and weak convergence rates of the parametric and nonparametric estimators.Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear strongly singular Calderón-Zygmund operators.https://zbmath.org/1449.420152021-01-08T12:24:00+00:00"Lin, Yan"https://zbmath.org/authors/?q=ai:lin.yan"Han, Yanyan"https://zbmath.org/authors/?q=ai:han.yanyanSummary: In this paper, the authors aim to establish the sharp maximal estimates for the multilinear iterated commutators generated by BMO functions and multilinear strongly singular Calderón-Zygmund operators. As application, the boundedness of the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces can be obtained, respectively.Boundedness of the multilinear fractional integral operators on the product weighted Herz spaces with variable exponents.https://zbmath.org/1449.420252021-01-08T12:24:00+00:00"Yuan, Lingling"https://zbmath.org/authors/?q=ai:yuan.lingling"Wang, Ruimei"https://zbmath.org/authors/?q=ai:wang.ruimei"Zhao, Kai"https://zbmath.org/authors/?q=ai:zhao.kaiSummary: The definitions and some basic properties of the variable exponent Lebesgue space are mentioned. Then, by the properties of weighted Lebesgue spaces with variable exponents and the boundedness of the multilinear fractional integral operator on \({L^p}\), based on the definition of the weighted Herz spaces with variable exponent, using the real methods in harmonic analysis, the boundedness of the multilinear fractional integral operators on the product weighted Herz spaces with variable exponents is obtained.Derivative sampling expansions for the linear canonical transform: convergence and error analysis.https://zbmath.org/1449.940532021-01-08T12:24:00+00:00"Annaby, Mahmoud H."https://zbmath.org/authors/?q=ai:annaby.mahmoud-h"Asharabi, Rashad M."https://zbmath.org/authors/?q=ai:asharabi.rashad-mSummary: In recent decades, the fractional Fourier transform as well as the linear canonical transform became very efficient tools in a variety of approximation and signal processing applications. There are many literatures on sampling expansions of interpolation type for bandlimited functions in the sense of these transforms. However, rigorous studies on convergence or error analysis are rare. It is our aim in this paper to establish sampling expansions of interpolation type for bandlimited functions and to investigate their convergence and error analysis. In particular, we introduce rigorous error estimates for the truncation error and both amplitude and jitter-time errors.A wavelet frame based approach for signal reconstruction.https://zbmath.org/1449.420652021-01-08T12:24:00+00:00"Yang, Jianbin"https://zbmath.org/authors/?q=ai:yang.jianbin"Tao, Xinzhu"https://zbmath.org/authors/?q=ai:tao.xinzhuSummary: Recovering analog signals from noisy samples is a fundamental problem, which plays an important role in signal and image processing, medical engineering, control theory, etc. This paper proposes a wavelet frame based model for recovering signals from discrete samples with mixed or unknown noises, and applies the augmented Lagrangian multiplier and accelerated proximal gradient methods to solve the model. Furthermore, an error analysis of the reconstruction model is given. In the end, numerical experiments for recovering functions from noisy samples are performed to demonstrate the effectiveness of the method.On representation of Parseval frames.https://zbmath.org/1449.420612021-01-08T12:24:00+00:00"Ryabtsov, Igor' Sergeevich"https://zbmath.org/authors/?q=ai:ryabtsov.igor-sergeevichSummary: This paper investigates properties of Parseval frames in finite dimensional vector spaces, namely, the possibility of representing some frames as sums of others. A new approach in constructing arbitrary Parseval frames and the decomposition of an arbitrary frame into a sum are described. Besides there is a number of special properties of equiangular tight frames.Approximation of functions belonging to \(L[0, \infty)\) by product summability means of its Fourier-Laguerre series.https://zbmath.org/1449.420052021-01-08T12:24:00+00:00"Khatri, Kejal"https://zbmath.org/authors/?q=ai:khatri.kejal"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we have proved the degree of approximation of functions belonging to \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series at \(x=0\). The aim of this paper is to concentrate on the approximation properties of the functions in \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series associated with the function \(f\).Bochner-Riesz operators on weak Musielak-Orlicz Hardy spaces.https://zbmath.org/1449.420392021-01-08T12:24:00+00:00"Wang, Aiting"https://zbmath.org/authors/?q=ai:wang.aiting"Liu, Xiong"https://zbmath.org/authors/?q=ai:liu.xiong"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Li, Baode"https://zbmath.org/authors/?q=ai:li.baodeSummary: Let \(\varphi:\mathbb{R}^n \times [0, \infty) \to [0, \infty)\) satisfy that \(\varphi (x, \cdot)\), for any given \(x \in \mathbb{R}^n\), is an Orlicz function and \(\varphi (\cdot, t)\) is a Muckenhoupt \({A_\infty}\) weight uniformly in \(t \in (0, \infty)\). In this paper, by using the atomic decomposition of the weak Musielak-Orlicz Hardy space \({\mathrm{WH}}^\varphi (\mathbb{R}^n)\) and a subtle pointwise estimate for the non-tangential grand maximal function of the Bochner-Riesz operators \(T_R^\delta\), we obtain that \(T_R^\delta\) is bounded on \({\mathrm{WH}}^\varphi (\mathbb{R}^n)\). The result is also new even when \(\varphi (x, t):= \Phi (t)\) for all \( (x, t) \in \mathbb{R}^n\times [0, \infty)\), where \(\Phi\) is an Orlicz function.Applications of multi-resolution analysis to Besov-Q type spaces and Triebel-Lizorkin type spaces.https://zbmath.org/1449.420712021-01-08T12:24:00+00:00"Li, Pengtao"https://zbmath.org/authors/?q=ai:li.pengtao"Sun, Wenchang"https://zbmath.org/authors/?q=ai:sun.wenchangSummary: In this survey, we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces \(\dot B_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\) and Triebel-Lizorkin-Q type spaces \(\dot B_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\). We will state briefly the recent progress on the wavelet characterizations, the boundedness of Calderón-Zygmund operators and the boundary value problem of \(\dot B_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\) and \(\dot F_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\). We also present the recent developments on the well-posedness of fluid equations with small data in \(\dot B_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\) and \(\dot F_{p,q}^{\gamma_1, \gamma_2} (\mathbb{R}^n)\).Localization property for the convolution of generalized periodic functions.https://zbmath.org/1449.460292021-01-08T12:24:00+00:00"Gorodets'kyĭ, V. V."https://zbmath.org/authors/?q=ai:gorodetskyi.vasyl-v|gorodetskij.v-v"Martynyuk, O. V."https://zbmath.org/authors/?q=ai:martynyuk.olga-vSummary: The well-known Riemann localization principle for the Fourier series of summable functions is reformulated for the convolution of generalized periodic functions with families of functions, which usually coincide with kernels of certain linear methods of summation of Fourier series (for example, summation methods such as the Gauss-Weierstrass one). We call the families of functions, for which the Riemann localization holds, the families of functions of a class \(L(X)\). The necessary and sufficient conditions of belonging the family of functions to the class \(L(X)\) are found in the case where \(X\) is a sufficiently broad non-quasi-analytic class of periodic functions or \(X\) is a class of analytic periodic functions (in particular, \(X =G_{\{\beta\}}\) for \(\beta > 1\) and \(X =G_{\{\beta\}}\) if \(0 <\beta\leq1\)). The definition of ``analytic functional equal to zero on an open set'' is also substantiated; a specific example of analytic functional is given, which is 0 on \((a, b)\subset[0, 2\pi]\). The use of the obtained result in partial differential equation theory allows us to obtain a new property (localization property, the property of local convergence improvement) of many problems of mathematical physics, since such solutions are often depicted as a convolution of some family of basic functions from the space \(X\) with a function \(F\) defined at the boundary of the domain, \(F\) may be a generalized function from a space \(X'\).On the consistency of wavelet density estimators for NA samples.https://zbmath.org/1449.420752021-01-08T12:24:00+00:00"Xu, Junlian"https://zbmath.org/authors/?q=ai:xu.junlianSummary: Density estimation is considered for negative association samples \({X_1}, {X_2}, \dots, {X_n}\). A linear wavelet density estimator is defined by using the wavelet method, and the \({L^p}\) \((1 \le p \le \infty)\) mean consistency of the estimator is proved without any assumptions of smoothness for the density function. The results show that the wavelet estimator can effectively estimates the density function of NA samples. It extends the application scope of the wavelet method.On orthogonal second order finite functions associated with triangular meshes and their application in mathematical modeling.https://zbmath.org/1449.420702021-01-08T12:24:00+00:00"Leont'ev, Viktor Leonidovich"https://zbmath.org/authors/?q=ai:leontev.v-l"Kochulimov, Aleksandr Valer'evich"https://zbmath.org/authors/?q=ai:kochulimov.aleksandr-valerevichSummary: Second order orthogonal finite functions are studied, their compact supports are local sets of grid triangles. The theorem on approximating the properties of sequences of sets of such functions is formulated. The applications in algorithms of mixed numerical methods of boundary value problems solution, in algorithms of linear approximation of surfaces are considered.On the compactness of commutators of bilinear fractional maximal operator.https://zbmath.org/1449.420372021-01-08T12:24:00+00:00"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiang"Guo, Qingdong"https://zbmath.org/authors/?q=ai:guo.qingdongSummary: Let \(\mathcal{M}_\alpha\) be the bilinear fractional maximal operator and let \(\vec{b} = ({b_1}, {b_2})\) be a collection of locally integrable functions. In this paper we mainly study the compactness of commutators of bilinear fractional maximal operators on Lebesgue spaces, of which the commutators include the fractional maximal linear commutator \(\mathcal{M}_{\alpha, \Sigma \vec{b}}\) and the fractional maximal iterated commutator \(\mathcal{M}_{\alpha, \Pi \vec{b}}\). The results are new even in the linear case.Approximation properties and frames for Banach and operator spaces.https://zbmath.org/1449.460222021-01-08T12:24:00+00:00"Hu, Qianfeng"https://zbmath.org/authors/?q=ai:hu.qianfeng"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.ruiSummary: We made a survey of the main results on approximation properties and frames for Banach spaces and operator spaces. By introducing Schauder frame and completely bounded frame, we presented the equivalent characterizations of the bounded approximation properties for Banach spaces and the completely bounded approximation properties for operator spaces. We also provided some examples and the duality theory on complemented embedding, and introduced some open questions.Operator characterizations of continuous \(g\)-frames.https://zbmath.org/1449.420662021-01-08T12:24:00+00:00"Zhang, We"https://zbmath.org/authors/?q=ai:zhang.weSummary: Continuous \(g\)-preframe operator is an important operator for the application of operator theory to the study of the continuous \(g\)-frame theory. The characterization of continuous \(g\)-frames, Parseval continuous \(g\)-frames, continuous \(g\)-Riesz bases and continuous \(g\)-orthonormal bases can be realized in terms of the continuous \(g\)-preframe operator. By adopting operator methods, the new continuous \(g\)-frames, Parseval continuous \(g\)-frames, continuous \(g\)-Riesz bases and continuous \(g\)-orthonormal bases are constructed and characterized in terms of the operators. The link between continuous \(g\)-preframe operator and strong disjointness, disjointness, strongly complementary pair is built, respectively. Finally, using the established characterization results, the operator characterization of the sum preserving properties of two continuous \(g\)-frames is obtained.The sufficient conditions for the existence of tight multiple periodic frames.https://zbmath.org/1449.420632021-01-08T12:24:00+00:00"Wang, Hui"https://zbmath.org/authors/?q=ai:wang.hui.5|wang.hui.4|wang.hui|wang.hui.1|wang.hui.2"Qiu, Jinling"https://zbmath.org/authors/?q=ai:qiu.jinlingSummary: The existence problem of tight multiple cycled frames is studied. The translation operator \(S_k^\tau\) is defined for the space \(L^2[-\pi, \pi]\) of all square-integrable periodic complex-valued functions, and the periodic affine functions are constructed from the periodic refinable functions and periodic complex sequences. We put forward a sufficient condition for the existence of tight multiple cycled frames by virtue of the unitary extension principle, the time-frequency analysis method and frame multiresolution analysis. The tight frames are constructed by the unitary extension principle. Moreover, the compactly supported periodic tight frames are obtained.Wavelet adaptive pointwise density estimations with super-smooth noises.https://zbmath.org/1449.420742021-01-08T12:24:00+00:00"Wu, Cong"https://zbmath.org/authors/?q=ai:wu.cong"Zeng, Xiaochen"https://zbmath.org/authors/?q=ai:zeng.xiaochen"Wang, Jinru"https://zbmath.org/authors/?q=ai:wang.jinruSummary: This paper considers pointwise deconvolution estimation of density functions under the local Hölder condition by the wavelet method. We firstly give a lower bound of any estimator with super-smooth noises. Then the practical linear wavelet estimator is constructed to obtain the optimal convergence rate, which means that the rate coincides with the lower bound. The strong convergence rate of the defined wavelet estimator is also provided. It should be pointed out that all above estimations are adaptive.On the Riemann Zeta function, I: KS-transform.https://zbmath.org/1449.110892021-01-08T12:24:00+00:00"Ge, Liming"https://zbmath.org/authors/?q=ai:ge.limingSummary: Kadison-Singer transform (KS-transform) is introduced as a multiplicative Fourier transform associated with the multiplicative structure of natural numbers. It is a unitary operator between the Hilbert space \({L^2} ([1, \infty))\) and Hardy space \({H^2} (\Omega)\), where \(\Omega\) is a the right half complex plane with the real part great than or equal to 1/2. We also show that KS-transform maps the multiplicative convolution of two functions on \([1, \infty)\) to the usual product of functions on \(\Omega\). Riemann hypothesis is equivalent to the vanishing index of certain convolution operators.On \(G\)-Banach frames.https://zbmath.org/1449.420602021-01-08T12:24:00+00:00"Rathore, Ghanshyam Singh"https://zbmath.org/authors/?q=ai:singh.ghanshyam"Mittal, Tripti"https://zbmath.org/authors/?q=ai:mittal.triptiSummary: \textit{M. R. Abdollahpour} et al. [Methods Funct. Anal. Topol. 13, No. 3, 201--210 (2007; Zbl 1144.46010)] generalized the concepts of frames for Banach
spaces and defined \(g\)-Banach frames in Banach spaces. In the present paper, we
define various types of \(g\)-Banach frames in Banach spaces. Examples and counter
examples to distinguish various types of \(g\)-Banach frames in Banach spaces have
been given. It has been proved that if a Banach space \(\mathcal{X}\) has a Banach frame, then \(\mathcal{X}\) has a normalized tight \(g\)-Banach frame for \(\mathcal{X}\). A characterization of an exact \(g\)-Banach frame has been given. Also, we consider the finite sum of \(g\)-Banach frames and give a sufficient condition for the finite sum of \(g\)-Banach frames to be a \(g\)-Banach frame. Finally, a sufficient condition for the stability of \(g\)-Banach frames in Banach spaces which provides optimal frame bounds has been given.On alternate duals of generalized frames.https://zbmath.org/1449.420592021-01-08T12:24:00+00:00"Rajeswari, K. N."https://zbmath.org/authors/?q=ai:rajeswari.kota-nagalakshmi"George, Neelam"https://zbmath.org/authors/?q=ai:george.neelamSummary: In this paper we give a sufficient condition as to when the difference
of two \(g\)-frames is a \(g\)-frame and characterize an alternate dual \(g\)-frame of a given \(g\)-frame in a Hilbert space.Approximation of correlation function and power spectral density with Sonin-Laguerre orthogonal functions.https://zbmath.org/1449.420432021-01-08T12:24:00+00:00"Prokhorov, S. A."https://zbmath.org/authors/?q=ai:prokhorov.s-a"Kulikovskikh, I. M."https://zbmath.org/authors/?q=ai:kulikovskikh.i-mSummary: Approximate capabilities of Sonin-Laguerre orthogonal functions with predefined parameter of orthogonal basis are studied. Parameters of approximation expression are evaluated, that are used for construction of the models of correlation function and power spectral density in compliance with the minimal weighted quadratic error of approximation.Partial best approximations and magnitude of double Vilenkin-Fourier coefficients.https://zbmath.org/1449.420502021-01-08T12:24:00+00:00"Kuznetsova, Maria A."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-andreevna"Volosivets, Sergey S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: We give estimates for the magnitude of double Vilenkin-Fourier coefficients of functions from generalized Hölder spaces, some \(p\)-fluctuational spaces and bounded \(\Lambda\text{-}\Gamma\text{-}\varphi\)-fluctuation spaces. For Hölder and \(p\)-fluctuational spaces we establish the sharpness of these estimates. Also we establish relation between full and partial best approximations and Watari-Efimov type inequality concerning partial best approximation and partial modulus of continuity.Boundedness of the multilinear fractional integral operators on the Hardy spaces.https://zbmath.org/1449.420142021-01-08T12:24:00+00:00"Lin, Xiansheng"https://zbmath.org/authors/?q=ai:lin.xiansheng"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiechengSummary: In this paper we considered the boundedness of the multilinear fractional integral operators on a Hardy space. By virtue of the atomic decomposition of a Hardy space and the Hölder inequality, the boundedness of bilinear fractional integral operators and triple multilinear fractional integral operators on a Hardy space was obtained. The results extended some known conclusions.Generalized dyadic derivative and uniform convergence of its Walsh-Fourier series.https://zbmath.org/1449.420492021-01-08T12:24:00+00:00"Golubov, Boris I."https://zbmath.org/authors/?q=ai:golubov.boris-i"Volosivets, Sergey S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: In the paper the notion of dyadic \(\lambda\)-derivative is introduced for the nonnegative, nondecreasing and concave sequence \(\{\lambda_n\}_{n=0}^{\infty}\). Analogues of Bernstein inequality for Walsh polynomials and of inverse approximation theorem are established. Also the uniform convergence of Walsh-Fourier series to this \(\lambda\)-derivative is studied.On the convergence of Fejér means of some subsequences of partial sums of Walsh-Fourier series.https://zbmath.org/1449.420462021-01-08T12:24:00+00:00"Gát, György"https://zbmath.org/authors/?q=ai:gat.gyorgyFor sequences \(\alpha\) satisfying the condition \(\alpha(n+1) \geq (1 + 1/(n+1)^{\delta}) \alpha(n)\) for some \(0 < \delta < 1/2\) and every \(n \in N\), the author proves that the Fejér means of the partial sums \(S_{\alpha(n)} f\) of Walsh-Fourier series converge to \(f\) almost everywhere. This implies that the almost everywhere convergence result also holds for lacunary sequences \(\alpha\).
Reviewer: Ferenc Weisz (Budapest)Wavelet tight frames in Walsh analysis.https://zbmath.org/1449.420552021-01-08T12:24:00+00:00"Farkov, Yuri A."https://zbmath.org/authors/?q=ai:farkov.yuri-aSummary: We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1) Parseval frames for \(L^2\)-spaces on Vilenkin groups, (2) finite tight frames for the space \(\ell^2(\mathbb{Z}_N)\). In particular cases these tight frames coincide with orthogonal wavelet bases associated with the classical Walsh functions.Image segmentation based on wavelet feature descriptor and dimensionality reduction applied to remote sensing.https://zbmath.org/1449.621502021-01-08T12:24:00+00:00"da Silva, Ricardo Dutra"https://zbmath.org/authors/?q=ai:da-silva.ricardo-dutra"Schwartz, William Robson"https://zbmath.org/authors/?q=ai:schwartz.william-robson"Pedrini, Helio"https://zbmath.org/authors/?q=ai:pedrini.helioSummary: Image segmentation is a fundamental stage in several domains of knowledge, such as computer vision, medical applications, and remote sensing. Using feature descriptors based on color, pixel intensity, shape, or texture, it divides an image into regions of interest that can be further analyzed by higher level modules. This work proposes a two-stage image segmentation method that maintains an adequate discrimination of details while allowing a reduction in the computational cost. In the first stage, feature descriptors extracted using the wavelet transform are employed to describe and classify homogeneous regions in the image. Then, a classification scheme based on partial least squares is applied to those pixels not classified during the first stage. Experimental results evaluate the effectiveness of the proposed method and compares it with a segmentation approach that considers Euclidean distance instead of the partial least squares for the second stage.A functional bound for Young's cosine polynomial.https://zbmath.org/1449.260192021-01-08T12:24:00+00:00"Fong, J. Z. Y."https://zbmath.org/authors/?q=ai:fong.jolie-zhi-yi"Lee, T. Y."https://zbmath.org/authors/?q=ai:lee.tuo-yeong"Wong, P. X."https://zbmath.org/authors/?q=ai:wong.pei-xianYoung's inequality asserts that \(1+\sum_{k=1}^{n}\frac{\cos k\theta}{k}>0\) for all \(n\in\mathbb{N}\) and \(\theta\in(0,\pi).\) The paper under review is devoted to the following stronger result: \(\frac{5}{6}+\sum_{k=1}^{n}\frac{\cos k\theta}{k}\geq\frac{1}{4}(1+\cos\theta)^{2}\) for all \(n\geq2\) and \(\theta\in(0,\pi).\) Equality occurs if and only if \(n=2\) and \(\theta=\pi-\arccos\frac{1}{3}.\)
Reviewer: Constantin Niculescu (Craiova)\(\mathcal{I}_H\)-regular Borel measures on locally compact abelian groups.https://zbmath.org/1449.430032021-01-08T12:24:00+00:00"Klotz, L."https://zbmath.org/authors/?q=ai:klotz.lukasz|klotz.lutz-peter|klotz.lawrence-h"Medina, J. M."https://zbmath.org/authors/?q=ai:medina.jose-m-moral|medina.juan-miguelLet \(G\) be an LCA group, \(H\) a closed subgroup, \(\Gamma\) the dual group of \(G\) and \(\Lambda\) the annihilator group of \(H\) in \(\Gamma\). Let \(\pi_H\) be the canonical homeomorphism from \(G\) onto the factor group \(G/H\) and \(\tilde{x} := \pi_H(x)\). For a non-empty subset \(S\) in \(G\) let \(P(S)\) denote the linear space of all trigonometric \(S\)-polynomials and let \(P(x+H) =: P(\tilde{x})\), \(x \in G\). Moreover let \(\mu\) be a regular finite non-negative measure on the Borel \(\sigma\)-algebra \(B(\Gamma)\). The measure \(\mu\) is called \(J_H\)-regular if and only if \[ \bigcap_{x\in G} C_\alpha P(x + H) = \bigcap_{\tilde{x}\in G/H}C_\alpha P(\tilde{x}) = \{0\}\] and is called \(J_H\)-singular if \(C_\alpha P(\tilde{x}) = L^\alpha(\mu)\). Here \(C_\alpha P(\tilde{x})\) denotes the closure of \(P(\tilde{x})\) in \(L^\alpha(\mu)\).
A characterization of \(J_H\)-regular measures is given in terms of be Radon-Nikodym derivatives of some measures defined by elements of the annihilator. Moreover the Wold type decomposition is obtained and relations to the Whittaker-Shannon-Kotel'nikov theorem are discussed.
Reviewer: Leszek Skrzypczak (Poznań)Estimation of the error of the approximation method of measurement of integrated characteristics on the separate instant values connected with transitions of signals through the zero.https://zbmath.org/1449.420032021-01-08T12:24:00+00:00"Melent'ev, Vladimir Sergeevich"https://zbmath.org/authors/?q=ai:melentev.vladimir-sergeevich"Yaroslavkina, Ekaterina Evgen'evna"https://zbmath.org/authors/?q=ai:yaroslavkina.ekaterina-evgenevna"Bolotnova, Anna Nikolaevna"https://zbmath.org/authors/?q=ai:bolotnova.anna-nikolaevnaSummary: Approaches to the estimation of the resulting inaccuracy of the definition of integrated characteristics of signals on separate instant values, caused by a deviation of real signals from used harmonious model are considered.Distributional boundary values of generalized Hardy functions in Beurling's tempered distributions.https://zbmath.org/1449.320072021-01-08T12:24:00+00:00"Sohn, Byung Keun"https://zbmath.org/authors/?q=ai:sohn.byung-keunLet \(C\) be an open convex cone in \(\mathbb{R}^N\) and let \(T^C=\mathbb{R}^N+iC\) in \(\mathbb{C}^N\). In this paper the author defines a generalization of Hardy functions (\(1 \leq p < \infty\)) on \(T^C\) and extended tempered distribution space \(S_{w}'\) of Beurling's tempered distribution space \(S_{(w)}'\) for a weight function \(w\). The author obtains the analytical and topological properties of \(S_{w}'\) and shows that the generalized Hardy functions (\(1< p \leq 2\)), have distributional boundary values in the weak topology of \(S_{(w)}'\) using the analytical properties of \(S_{w}'\) .
Reviewer: Koichi Saka (Akita)New inequalities and erasures for continuous g-frames.https://zbmath.org/1449.420682021-01-08T12:24:00+00:00"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.3|zhang.wei.9|zhang.wei.5|zhang.wei.6|zhang.wei.18|zhang.wei.2|zhang.wei.17|zhang.wei.7"Li, Yun-Zhang"https://zbmath.org/authors/?q=ai:li.yunzhangThe concept of continuous \(g\)-frames is a generalization of both continuous frames and \(g\)-frames. The authors establish some inequalities for continuous \(g\)-frames and dual continuous \(g\)-frames. Some results on erasure of elements for continuous \(g\)-frames are also obtained.
Reviewer: Alexander Ulanovskii (Stavanger)Characterizations of certain Hankel transform involving Riemann-Liouville fractional derivatives.https://zbmath.org/1449.420092021-01-08T12:24:00+00:00"Upadhyay, S. K."https://zbmath.org/authors/?q=ai:upadhyay.santosh-kumar"Khatterwani, Komal"https://zbmath.org/authors/?q=ai:khatterwani.komalSummary: In this paper, the relation between the two dimensional fractional Fourier transform and the fractional Hankel transform is discussed in terms of radial functions. Various operational properties of the Hankel transform and the fractional Hankel transform are studied involving Riemann-Liouville fractional derivatives. The application of the fractional Hankel transform in networks with time varying parameters is given.Superconvergence analysis of local discontinuous Galerkin methods for linear convection-diffusion equations in one space dimension.https://zbmath.org/1449.652672021-01-08T12:24:00+00:00"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.2|zhang.jun.10|zhang.jun.5|zhang.jun.9|zhang.jun|zhang.jun.1|zhang.jun.3|zhang.jun.6|zhang.jun.7"Chen, Xiangling"https://zbmath.org/authors/?q=ai:chen.xianglingSummary: This paper is concerned with the superconvergence study of the local discontinuous Galerkin (LDG) method for one-dimensional time-dependent linear convection-diffusion equations, where the convection flux is taken as the upwind flux, while the diffusion fluxes chosen as the alternating fluxes. Superconvergence properties for both the solution itself and auxiliary variables are established. Precisely, we prove that, the LDG solutions are superconvergent with an order of \(k+2\) towards a particular projection of the exact solution and the auxiliary variable, and thus a \(k+1\)-th order superconvergence for the derivative approximation and a \(k+2\)-th order superconvergence for the function value approximation at a class of Radau points are obtained. Especially, we show that the convergence rate of the derivative approximation for the exact solution can reach \(k+2\) when the convection flux is the same as the diffusion flux, two order higher than the optimal convergence rate. Furthermore, a \(2k+1\)-th order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages, is also obtained under some suitable initial discretization. Numerical experiments indicate that most of our theoretical findings are optimal.Erratum to: Atomic decomposition of Hardy spaces and characterization of \(BMO\) via Banach function spaces.https://zbmath.org/1449.460252021-01-08T12:24:00+00:00"Ho, K.-P."https://zbmath.org/authors/?q=ai:ho.keang-po|ho.kwok-punErratum to \textit{K.-P. Ho} [Anal. Math. 38, No. 3, 173--185 (2012; Zbl 1289.46049)].Research and simulation of blind watermarking algorithms based on DWT-SVD of non-square image.https://zbmath.org/1449.940682021-01-08T12:24:00+00:00"Zhang, Shuai"https://zbmath.org/authors/?q=ai:zhang.shuai"Yang, Xuexia"https://zbmath.org/authors/?q=ai:yang.xuexiaSummary: In order to solve the limitation of watermarking algorithm, by which only the array image was studied, and the shortage of extracted watermark in the cropping attack, a blind watermarking algorithm was proposed, which can transform any size image in the discrete wavelet transform domain and combine with matrix singular value decomposition. By normalizing the carrier image, the image can be decomposed by any level of wavelet transform, then the problem in shearing attack was solved by embedding double watermarking in singular value. The experimental results show that the proposed algorithm can achieve blind extraction of watermark information, and it has strong robustness against many types of attacks. The watermark information extracted in the shear attack performs better than similar algorithms.Boundedness of the fractional integral operator with rough kernel and its commutator in vanishing generalized variable exponent Morrey spaces on unbounded sets.https://zbmath.org/1449.420182021-01-08T12:24:00+00:00"Mo, Huixia"https://zbmath.org/authors/?q=ai:mo.huixia"Wang, Xiaojuan"https://zbmath.org/authors/?q=ai:wang.xiaojuan"Han, Zhe"https://zbmath.org/authors/?q=ai:han.zhe|han.zhe.1Summary: In this paper, we study the boundedness of fractional integral operators and their commutators in vanishing generalized Morrey spaces with variable exponent on unbounded sets. Using the properties of variable exponent functions and the pointwise estimates of operators \(T_{\Omega, \alpha}\) and their commutators \([b, T_{\Omega, \alpha}]\) in Lebesgue spaces with variable exponent, we obtain the boundedness of fractional integral operators \(T_{\Omega, \alpha}\) and their commutators \([b, T_{\Omega, \alpha}]\) in vanishing generalized Morrey spaces with variable exponents on unbounded sets, which extend the previous results.Upper bounds for the approximation of some classes of bivariate functions by triangular Fourier-Hermite sums in the space \(L_{2,p}(\mathbb{R}^2)\).https://zbmath.org/1449.420442021-01-08T12:24:00+00:00"Shabozov, M. Sh."https://zbmath.org/authors/?q=ai:shabozov.mirgand-shabozovich"Dzhurakhonov, O. A."https://zbmath.org/authors/?q=ai:dzhurakhonov.olimdzhon-akmalovichThe authors study supprema of approximation of bivariate functions, generalizing research on the approximation of functions by algebraic polynomials on the real axis \(\mathbb{R}\) with Chebyshev weight \(\tilde{\rho}(x)=\exp{\{x^2\}}\). The main results are given in four theorems:
Theorem 1. Let \(m,N\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\), \(0<p\leq 2\), \(h\in (0,1]\) and let \(q\) be a nonnegative measurable summable function on \((0,h)\) which does not vanish identically. Then
\[\sup_{f\in L_{2,\rho}^{(r)}} \frac{N^rE_{N-1}(f)_{2,\rho}}{\left(\int_0^h \Omega_m^p(D^rf,t)_{2,\rho}q(t)dt\right)^{1/p}} = \frac{1}{\left\{\int_0^h (1-(1-t^2)^{N/2})^{mp}q(t)dt\right\}^{1/p}}. \]
Theorem 2. Let \(m\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\). Then, for an arbitrary \(N\in\mathbb{N}\)
\[\sup_{f\in L_{2,\rho}^{(r)}} \frac{N^r E_{N-1}(f)_{2,\rho}}{K(D^rf,\frac{1}{N^m})_{2,\rho}}=1.\]
Theorem 3. Let \(m,N\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\), \(k=0,1,2,\ldots,N\), \(0<p\leq 2\), \(0<H<1\) and \(q\geq 0\) be a measurable function summable on \((0,H)\) which is not equivalent to zero. Then \[\gamma_{N(N+1)/2+k}(H_{2,p}^r (\Omega_m,q);L_{2,\rho}) = E_{N-1}(HW_{2,p}^r(\Omega_m,q))_{2,\rho}=\] \[=N^{-r}\left\{ \int_0^H (1-(1-t^2)^{N/2})^{mpp}q(t)dt\right\}^{-1/p},\] where \(\gamma(\cdot)\) is any of the following widths: Bernstein, Gelfand, Kolmogorov, linear and projectional.
Theorem 4. Let \(\Phi\) be some majorant defining the class of functions \(W_{2,\rho}^r(K_m,\Phi)\), where \(r\in\mathbb{Z}_{+}\), \(m\in\mathbb{N}\). Then, for an arbitrary \(N\in\mathbb{N}\) and \(k=0,1,2,\ldots,N\)
\[\gamma_{N(N+2)/2+k}(W_{2,\rho}^r(K;\Phi);L_{2,\rho}) = E_{N-1}(W_{2,\rho}^r(K;\Phi))_{2,\rho}=N^{-r}\Phi(N^{-m}),\]
where \(\gamma_{nu}(\cdot)\) is any of the widths mentioned in Theorem 3. The specific definitions of the notations used would take to much space here.
The main concept is the space \(L_{2,\rho}=L_{2,\rho}(\mathbb{R}^2)\); real squared summable functions on \(\mathbb{R}^2\) with weight \(\rho(x)=\exp{\{-(x^2+y^2)\}}\). The system \(\{H_k(x)H_l(x)\}_{k,l\in\mathbb{Z}_{+}}\) of Hermite polynomials is orthogonal on the entire plane \(\mathbb{R}^2\) with weight \(\rho\) and the double Fourier-Hermite series is given by \[f(x,y)=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\,c_{kl}(f)H_k(x)h_l(y),\] with \[c_{kl}(f)=\int\int_{\mathbb{R}^2}\,\rho(x,y)f(x,y)H_k(x)H_l(y)dx dy\] convergence in \(L_{2,\rho}(\mathbb{R}^2)\)-sense.
Reviewer: Marcel G. de Bruin (Heemstede)Rate of convergence of wavelet series by Cesàro means.https://zbmath.org/1449.420512021-01-08T12:24:00+00:00"Ali, Mir Ahsan"https://zbmath.org/authors/?q=ai:ali.mir-ahsan"Sheikh, Neyaz A."https://zbmath.org/authors/?q=ai:sheikh.neyaz-ahmed|sheikh.neyaz-ahmadSummary: Wavelet frames have become a useful tool in time frequency analysis and image processing. Many authors worked in the field of wavelet frames and obtained various necessary and sufficient conditions. \textit{A. Ron} and \textit{Z. Shen} [J. Funct. Anal. 148, No. 2, 408--447 (1997; Zbl 0891.42018)] gave a charactarization of wavelet frames. \textit{J. J. Benedetto} and \textit{O. M. Treiber} [in: Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhäuser. 3--36 (2001; Zbl 1036.42032)], presented different works on the wavelet frames. Any function \(f\in L^2(R)\) can be expanded as an orthonormal wavelet series and pointwise convergence and uniform convergence of series have been discussed extensively by various authors [\textit{S. E. Kelly} et al., Bull. Am. Math. Soc., New Ser. 30, No. 1, 87--94 (1994; Zbl 0788.42014)]. In this paper we investigate the pointwise convergence of orthogonal wavelet series in Pringsheim's sense. Furthermore, we investigate Cesàro \(\vert C,1,1\vert\) summability and the strong Cesàro \(\vert C,1,1\vert\) summability of wavelet series.On a ``Martingale property'' of Franklin series.https://zbmath.org/1449.420482021-01-08T12:24:00+00:00"Gevorkyan, G. G."https://zbmath.org/authors/?q=ai:gevorkyan.gegham-gA nested sequence of partitions of the interval \([0,1]\) into \(n=2^\mu+\nu\) subintervals is defined by dividing \([0,1]\) into \(2^\mu\) intervals of length \(2^{-\mu}\) and then adding the midpoints of the first \(\nu\) of these intervals. Let \(S_n\) be the set of piecewise linear continuous functions on the subintervals. Then the Franklin system consists of the sequence of orthogonal functions \(\{f_n\}_{n=0}^\infty\) with \(f_n\in S_n\) orthogonal to \(S_{n-1}\) for \(n\ge2\). A function \(f\in L^2[0,1]\) can be expanded as \(f(x)=\sum_{n=0}^\infty a_nf_n(x)\) with partial sums \(\sigma_n(x)=\sum_{k=0}^n a_k f_k(x)\). This paper derives convergence results that are analogous to convergence results obtained for other systems such as the Haar system [\textit{F. G. Arutyunyan}, Dokl., Akad. Nauk Arm. SSR 42, 134--140 (1966; Zbl 0178.40802)] and martingales [\textit{R. F. Gundy}, Trans. Am. Math. Soc. 124, 228--248 (1966; Zbl 0158.35801)]. Although the Franklin system is not a martingale, it has similar properties, which are exploited here to obtain similar convergence results. The key result in this derivation is the theorem saying that \(\inf_n\sigma_n(x)>-\infty\) on \(E\) implies that \(\sup_n\sigma_n(x)<+\infty\) a.e. on \(E\).
Reviewer: Adhemar Bultheel (Leuven)O'Neil inequality for convolutions associated with Gegenbauer differential operator and some applications.https://zbmath.org/1449.420302021-01-08T12:24:00+00:00"Guliyev, Vagif S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Ibrahimov, E. J."https://zbmath.org/authors/?q=ai:ibrahimov.elman-j"Ekincioglu, S. E."https://zbmath.org/authors/?q=ai:ekincioglu.s-elifnur"Jafarova, S. Ar."https://zbmath.org/authors/?q=ai:jafarova.s-arSummary: In this paper we prove an O'Neil inequality for the convolution operator (\(G\)-convolution) associated with the Gegenbauer differential operator \({G_\lambda}\). By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the \(G\)-convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the \(G\)-fractional maximal and \(G\)-fractional integral operators from the spaces \({L_{p,\lambda}}\) to \({L_{q,\lambda}}\) and from the spaces \({L_{1,\lambda}}\) to the weak spaces \(W{L_{p,\lambda}}\).On decompositions of continuous generalized frames in Hilbert spaces.https://zbmath.org/1449.420672021-01-08T12:24:00+00:00"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.10|zhang.wei.6|zhang.wei.16|zhang.wei.4|zhang.wei.12|zhang.wei.17|zhang.wei.18|zhang.wei.3|zhang.wei.11|zhang.wei.15|zhang.wei.5|zhang.wei.2|zhang.wei.13|zhang.wei.14|zhang.wei.7|zhang.wei.9"Fu, Yanling"https://zbmath.org/authors/?q=ai:fu.yanlingSummary: This paper establishes the characterization of continuous generalized frames, Parseval continuous \(g\)-frames, continuous generalized Riesz bases and continuous generalized orthonormal bases in terms of the continuous generalized preframe operator. Using the established characterization results and decompositions of bounded operators, the representation of continuous generalized frames in terms of linear combinations of simpler ones such as continuous generalized orthonormal bases, continuous generalized Riesz bases and Parseval continuous generalized frames is studied.Boundedness characterization of maximal commutators on Orlicz spaces in the Dunkl setting.https://zbmath.org/1449.420132021-01-08T12:24:00+00:00"Guliyev, Vagif S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Mammadov, Yagub Y."https://zbmath.org/authors/?q=ai:mammadov.yagub-y"Muslumova, Fatma A."https://zbmath.org/authors/?q=ai:muslumova.fatma-aSummary: On the real line, the Dunkl operators \[{D_v} (f) (x): = \frac{{df (x)}}{{dx}}+ ({2v + 1})\frac{{f (x)-f ({-x})}}{{2x}}, \; \; \; \forall x \in \mathbb{R},\; \forall v \ge - \frac{1}{2}\] are differential-difference operators associated with the reflection group \({\mathbb{Z}_2}\) on \(\mathbb{R}\), and on \({\mathbb{R}^d}\) the Dunkl operators \(\{{{D_{k, j}}}\}_{j = 1}^d\) are the differential-difference operators associated with the reflection group \(\mathbb{Z}_2^d\) on \({\mathbb{R}^d}\). In this paper, in the setting \(\mathbb{R}\) we show that \(b \in BMO ({\mathbb{R}, d{m_v}})\) if and only if the maximal commutator \({M_{b, v}}\) is bounded on Orlicz spaces \({L_\Phi} ({\mathbb{R}, d{m_v}})\). Also in the setting \({\mathbb{R}^d}\) we show that \(b \in BMO ({{\mathbb{R}^d}, h_k^2 (x)dx})\) if and only if the maximal commutator \({M_{b, k}}\) is bounded on Orlicz spaces \({L_\Phi} ({{\mathbb{R}^d}, h_k^2 (x)dx})\).Boundedness of Marcinkiewicz integral and its commutator on weighted \(\lambda\)-central Morrey spaces.https://zbmath.org/1449.420202021-01-08T12:24:00+00:00"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Chen, Zhuanzhuan"https://zbmath.org/authors/?q=ai:chen.zhuanzhuanSummary: By applying function decompositions and properties of weights, the boundedness of Marcinkiewicz integrals and their commutators is established on the weighted \(\lambda\)-central Morrey spaces with the help of the corresponding boundedness on weighted spaces.Bounds on autocorrelation coefficients of Rudin-Shapiro polynomials.https://zbmath.org/1449.420012021-01-08T12:24:00+00:00"Allouche, J.-P."https://zbmath.org/authors/?q=ai:allouche.jean-paul"Choi, S."https://zbmath.org/authors/?q=ai:choi.seungju|choi.sangki|choi.sunhwa|choi.sulyoung|choi.seunghoon|choi.sujung|choi.sung-bin|choi.sooyoung|choi.sujin|choi.syngjoo|choi.seungwon|choi.sukwon|choi.sungwhan|choi.sangdo|choi.seungmoon|choi.seongmi|choi.sungho|choi.sungjin|choi.sumi|choi.seongsoo|choi.sunghyun|choi.seongim|choi.seungbae|choi.sekyu|choi.sangsu|choi.seungil|choi.sangmin|choi.seungtak|choi.sangbang|choi.sanghyun|choi.suhan|choi.sunghun|choi.sungchul|choi.seonmi|choi.sunwoong|choi.seongjeon|choi.seungyeob|choi.sung|choi.sunghee|choi.sungjoon|choi.sunhi|choi.soomin|choi.suhyoung|choi.sungkyu|choi.seungbok|choi.seonil|choi.soyoung|choi.soonwon|choi.sora|choi.suktae|choi.sungsub|choi.seungjin|choi.seul-hee|choi.sungyong|choi.sunghwan|choi.stephen-kwok-kwong|choi.sukjoo|choi.sangyup|choi.siwon|choi.suyoung|choi.sooyong|choi.sokhwan|choi.soodong|choi.sanga|choi.sungwoo|choi.sangbum|choi.sangjin|choi.sujeong|choi.seungho|choi.soohak|choi.sunjin|choi.sanghun|choi.sungyun"Denise, A."https://zbmath.org/authors/?q=ai:denise.alain"Erdélyi, T."https://zbmath.org/authors/?q=ai:erdelyi.tamas"Saffari, B."https://zbmath.org/authors/?q=ai:saffari.bahmanThe authors study the autocorrelation coefficients of the Rudin-Shapiro polynomials, giving certain bounds for its maximum in the interval \([0,2^n]\). The authors themselves indicate in this work that they continue working on the correlation coefficients since they have found weaker bounds for them.
Reviewer: Antonio López-Carmona (Granada)Denoising of sEMG signal based on improved wavelet algorithm.https://zbmath.org/1449.940392021-01-08T12:24:00+00:00"Ma, Dong"https://zbmath.org/authors/?q=ai:ma.dong"Yang, Zheng"https://zbmath.org/authors/?q=ai:yang.zheng"Wang, Liling"https://zbmath.org/authors/?q=ai:wang.lilingSummary: According to the poor denoising effect of the traditional wavelet method in sEMG signal, improved wavelet method and improved wavelet threshold function are proposed. The improved threshold takes account of the denoising characteristics of sEMG signals as different decomposition levels and each decomposition layer noise distribution. Experimental simulation results show that the wavelet improved threshold and wavelet improved threshold function improved the wavelet threshold method, which can obtain better denoising effect than other methods.Duals of Hardy amalgam spaces and norm inequalities.https://zbmath.org/1449.420382021-01-08T12:24:00+00:00"Ablé, Z. V. P."https://zbmath.org/authors/?q=ai:able.z-v-p"Feuto, J."https://zbmath.org/authors/?q=ai:feuto.justinThere are many generalizations of the classical Hardy spaces by taking the norm of the maximal function in certain spaces rather than in the Lebesgue ones. The author's choice for replacing is the Wiener amalgam spaces. In the paper under review, they first study characterizations of such spaces, including the atomic ones. Then they characterize the dual spaces of the generalized Hardy spaces defined in the above way. Finally, they prove that in these generalized Hardy spaces some classical singular operators, such as Calderón-Zygmund, convolution and Riesz potential operators, are bounded.
Reviewer: Elijah Liflyand (Ramat-Gan)Construction of framelet packets on \(\mathbb{Z}\) and algorithm implementation.https://zbmath.org/1449.420722021-01-08T12:24:00+00:00"Lu, Dayong"https://zbmath.org/authors/?q=ai:lu.dayong"Yi, Hua"https://zbmath.org/authors/?q=ai:yi.huaSummary: In order to facilitate the use of wavelets, the study on frame wavelets (also called framelets) and framelet packets in digital setting has been addressed in recent years. An approach to construct a class of \(J\)-stage framelet packets for \({\ell^2} (\mathbb{Z})\) was given by us previously [the authors, ``Construction of framelet packets on \(z[\text{J}]\)'', ICIC Express Letters, Part B: Applications 7, No. 1, 143--149 (2016)]. However, the detailed results on how to use them are missing. To further improve the theoretical system of \(J\)-stage framelet packets for \({\ell^2} (Z)\), and by following the previous work, the fast decomposition and reconstruction algorithms are given in this paper, with which one can establish the relationship of coefficients between different stages. For the convenience of use, the detailed data of framelet packets for \({\ell^2} (Z)\) when the number \(n\) of mother framelet ranges from 1 to 4 are constructed. Finally, a numerical experiment is given to illustrate the perfect reconstruction of framelet packets.Intrinsic Littlewood-Paley \({g_\alpha}\) operator on weighted Campanato space.https://zbmath.org/1449.420322021-01-08T12:24:00+00:00"He, Suixin"https://zbmath.org/authors/?q=ai:he.suixin"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangSummary: The boundedness of the intrinsic Littlewood-Paley \({g_\alpha}\) operator on certain weighted Campanato spaces is proved. It is also shown that the intrinsic Littlewood-Paley \({g_\alpha}\) operator is bounded on \({\mathrm{BMO}} (\mathbb{R}^n)\) and \({\mathrm{Lip}}_\alpha (\mathbb{R}^n)\).Error analysis of the method of integral characteristics measurement caused by the deviation of signals form from the harmonic model.https://zbmath.org/1449.420022021-01-08T12:24:00+00:00"Melent'ev, Vladimir Sergeevich"https://zbmath.org/authors/?q=ai:melentev.vladimir-sergeevich"Ivanov, Yuriĭ Mikhaĭlovich"https://zbmath.org/authors/?q=ai:ivanov.yurii-mikhailovich"Muratova, Vera Vladimirovna"https://zbmath.org/authors/?q=ai:muratova.vera-vladimirovnaSummary: A new method of measurement of integral characteristics on instant values of harmonic signals divided both in space and in time is considered. Results of the analysis of the methodological error caused by a deviation of the real signal from the harmonic model are obtained.Wavelet estimation for anisotropic density functions.https://zbmath.org/1449.420692021-01-08T12:24:00+00:00"Cao, Kaikai"https://zbmath.org/authors/?q=ai:cao.kaikaiSummary: This paper provides an estimation for noncompact supported density functions based on the wavelet method over an anisotropic Besov space and gives the linear wavelet estimator, then it's upper bound under \({L^p}\) \((2 \le p < +\infty)\) risk is obtained. Furthermore, we assume that the density function has independent structure so that the dimension disaster is reduced, and the corresponding proof is given.Boundedness of a class of singular integral operators and commutators on Herz-Morrey-Hardy spaces with variable exponent.https://zbmath.org/1449.420272021-01-08T12:24:00+00:00"Zhao, Huan"https://zbmath.org/authors/?q=ai:zhao.huan"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangSummary: Let \(\Omega \in {L^s} (S^{n - 1})\) for \({s \ge 1}\) be a homogeneous function of degree zero and \(b\) be BMO functions. Using the atomic decomposition theorem, we obtain the boundedness of the Calderón-Zygmund singular integral operator \({T_\Omega}\) and its commutator \([b, {T_\Omega}]\) on Herz-Morrey-Hardy spaces with variable exponent.Commutators generated by Littlewood-Paley \(g_\lambda^*\) function and local Campanato functions on generalized local Morrey spaces.https://zbmath.org/1449.420172021-01-08T12:24:00+00:00"Mo, Huixia"https://zbmath.org/authors/?q=ai:mo.huixia"Ma, Ruiqing"https://zbmath.org/authors/?q=ai:ma.ruiqingSummary: In this paper, we obtain the boundedness of Littlewood-Paley \(g_\lambda^*\) function on the generalized local Morrey space \({\mathrm{LM}}_{p, \varphi}^{\{x_0\}}\), as well as the boundedness of the commutators generated by Littlewood-Paley \(g_\lambda^*\) function and local Campanato functions.An estimate for maximal Bochner-Riesz means on Musielak-Orlicz Hardy spaces.https://zbmath.org/1449.420352021-01-08T12:24:00+00:00"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Qiu, Xiaoli"https://zbmath.org/authors/?q=ai:qiu.xiaoli"Wang, Aiting"https://zbmath.org/authors/?q=ai:wang.aiting"Li, Baode"https://zbmath.org/authors/?q=ai:li.baodeSummary: In this paper, we study the boundedness of maximal Bochner-Riesz means. By using the pointwise of maximal Bochner-Riesz means and the atomic decomposition of weak Musielak-Orlicz Hardy space, we establish the boundedness of maximal Bochner-Riesz means from weak Musielak-Orlicz Hardy space to weak Musielak-Orlicz space. This result is new even when \(\varphi (x, t): = \Phi (t)\) for all \( (x, t) \in {\mathbb{R}^n} \times [0, \infty)\), where \(\Phi \) is an Orlicz function, and it is an extension to Musielak-Orlicz spaces from the setting of the weighted spaces of a literature.Boundedness of the Calderón-Zygmund singular integral operator on Herz-type Hardy spaces with variable exponents.https://zbmath.org/1449.420102021-01-08T12:24:00+00:00"Cai, Jinling"https://zbmath.org/authors/?q=ai:cai.jinling"Tang, Canqin"https://zbmath.org/authors/?q=ai:tang.canqinSummary: Let \(\Omega \in {L^s} (S^{n - 1})\) \((s > 1)\) be a homogeneous function of degree zero, \({T_\Omega}\) be a Calderón-Zygmund singular integral operator. The boundedness of \({T_\Omega}\) and its commutator respectively from Herz-type Hardy spaces with two variable exponents to the related variable exponent Herz spaces are obtained.Using Legendre spectral element method with quasi-linearization method for solving Bratu's problem.https://zbmath.org/1449.652762021-01-08T12:24:00+00:00"Lotfi, Mahmoud"https://zbmath.org/authors/?q=ai:lotfi.mahmoud"Alipanah, Amjad"https://zbmath.org/authors/?q=ai:alipanah.amjadSummary: This work presented here is the solution of the one-dimensional Bratu's problem. The nonlinear Bratu's problem is first linearised using the quasi-linearization method and then solved by the spectral element method. We use the Legendre polynomials for interpolation. Finally, we show the results with a numerical example.Legendre neural network method for solving one dimensional convection diffusion equation.https://zbmath.org/1449.653492021-01-08T12:24:00+00:00"Yang, Yunlei"https://zbmath.org/authors/?q=ai:yang.yunlei"Hou, Muzhou"https://zbmath.org/authors/?q=ai:hou.muzhou"Luo, Jianshu"https://zbmath.org/authors/?q=ai:luo.jianshuSummary: In order to find out the numerical solution of one-dimensional convection diffusion equation, differential properties of Legendre polynomial and the matrix tensor product property were defined, a Legendre neural network method for solving one-dimensional convection diffusion equation problem was proposed. Approximate solutions of the differential equations were constructed by using Legendre neural network. The influence of the network topology in the neural network model on the numerical results was studied. Numerical experiments show that the calculation accuracy and running time of the given samples are affected by the hidden layer neurons.Pairs of dual wavelet frames on local fields.https://zbmath.org/1449.420522021-01-08T12:24:00+00:00"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younusThe author introduces the notion of orthogonal wavelet frames on local fields of positive characteristic and presents an algorithm for the construction of a pair of orthogonal wavelet frames based on polyphase matrices formed by the polyphase components of the wavelet masks. He also gives a general construction algorithm for all orthogonal wavelet tight frames on local fields of positive characteristic from a compactly supported scaling function and investigates their properties by means of the Fourier transform. The motivation for this work are the papers by \textit{F. A. Shah} [Acta Univ. Apulensis, Math. Inform. 49, 47--65 (2017; Zbl 1413.42060)] on orthogonal wavelet frames generated by Walsh polynomials, and \textit{F. A. Shah} and \textit{L. Debnath} [Analysis, München 33, No. 3, 293--307 (2013; Zbl 1277.42047)] on tight wavelet frames on local fields.
Reviewer: Richard A. Zalik (Auburn)Characterizations of product Hardy space associated to Schrödinger operators.https://zbmath.org/1449.420402021-01-08T12:24:00+00:00"Zhao, Kai"https://zbmath.org/authors/?q=ai:zhao.kai"Liu, Su-Ying"https://zbmath.org/authors/?q=ai:liu.suying"Jiang, Xiu-Tian"https://zbmath.org/authors/?q=ai:jiang.xiutianSummary: Let \(L_1\) and \(L_2\) be the Schrödinger operators on \(\mathbb{R}^n\) and \(\mathbb{R}^m\), respectively. By using different maximal functions and Littlewood-Paley \(g\) function on distinct variables, in this paper, some characterizations for functions in the product Hardy space \(H_{{L_1},{L_2}}^1( \mathbb{R}^n \times \mathbb{R}^m)\) associated to operators \(L_1\) and \(L_2\) are obtained.Estimation on the Walsh-Fejér and Walsh logarithmic kernels.https://zbmath.org/1449.420472021-01-08T12:24:00+00:00"Gát, György"https://zbmath.org/authors/?q=ai:gat.gyorgy"Lucskai, Gábor"https://zbmath.org/authors/?q=ai:lucskai.gaborSummary: The main aim of this article is to demonstrate the difference of the trigonometric and the Walsh system with respect to the behaviour of the maximal function of the Fejér kernels. Moreover, properties (positivity among others) of the Walsh logarithmic kernels are also investigated.A kind of a triangular V-system construction scheme based on linear independent function groups.https://zbmath.org/1449.330152021-01-08T12:24:00+00:00"Wang, Xianghai"https://zbmath.org/authors/?q=ai:wang.xianghai"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.10|li.wei.8|li.wei.7|li.wei.9|li.wei|li.wei.5|li.wei-wayne"Lv, Fang"https://zbmath.org/authors/?q=ai:lv.fang"Song, Chuanming"https://zbmath.org/authors/?q=ai:song.chuanmingSummary: In recent years, with the development of non-continuous orthogonal function systems, a class of orthogonal complete function systems on \({L^2}[0, 1]\), U-system and V-system, have emerged, which have strong expression ability for continuous and discontinuous signals. Triangular patches are valued for their flexibility, convenience, and adaptability in complex surface modeling, and have significant advantages in 3D geometric modeling. This paper proposes a triangular domain 1st V-system construction scheme based on linear independent function groups. First, we select a set of linearly independent function groups under the 1st-level triangulation domain, then perform Gram-Schmidt orthogonalization to obtain 12 canonical orthogonal functions, and then rotate, compress and translate the generators. Other operations generate other functions of the V-system in turn. At the same time, the process of the V-system on the triangular domain in practical application is explained. The generators of the 1st V-system based on the linear independent function group constructed in this paper avoid a large number of 0, and the spectrum information of the 3D geometric model can be extracted more effectively in the application.Boundedness of \(\theta \)-type C-Z operators on weighted variable exponent Morrey spaces.https://zbmath.org/1449.420232021-01-08T12:24:00+00:00"Yang, Yanqi"https://zbmath.org/authors/?q=ai:yang.yanqi"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangpingSummary: We obtain some boundedness results for the \(\theta \)-type Calderón-Zygmund operators \({T_\theta}\) under natural regularity assumptions on a class of generalized Lebesgue spaces with weight and variable exponent. Furthermore, the boundedness of \({T_\theta}\) is established on the weighted variable Herz and Herz-Morrey spaces based on the above conclusions. We also prove the boundedness of the corresponding commutator \([b, {T_\theta}]\) in the generalized weighted Morrey spaces with variable exponent.The approximation of Laplace-Stieltjes transforms with slow growth.https://zbmath.org/1449.300812021-01-08T12:24:00+00:00"Xu, Hongyan"https://zbmath.org/authors/?q=ai:xu.hongyan"Liu, Sanyang"https://zbmath.org/authors/?q=ai:liu.sanyangSummary: The main purpose of this paper is to study the growth and approximation on entire functions represented by Laplace-Stieltjes transforms of finite logarithmic order convergent on the whole complex plane, and obtain some results about the logarithmic order, the logarithmic type, the error, and the coefficients of Laplace-Stieltjes transforms which are generalization and improvement of the previous results.Generalized complete Bessel sequences and conditions of generalized lower semi-frames.https://zbmath.org/1449.420532021-01-08T12:24:00+00:00"Cai, Yun"https://zbmath.org/authors/?q=ai:cai.yun"Gao, Wenjun"https://zbmath.org/authors/?q=ai:gao.wenjun"Li, Dengfeng"https://zbmath.org/authors/?q=ai:li.dengfengSummary: This paper considers generalized complete Bessel sequences and generalized lower semi-frames, including discrete and continuous cases. Properties of analysis operator for generalized complete Bessel sequences are firstly discussed. Then, some sufficient and necessary conditions of generalized lower semi-frames are established. Lastly, it is concluded that dual sequences of generalized complete Bessel sequences are generalized lower semi-frames.Lipschitz type characterizations of harmonic Bergman-Orlicz spaces.https://zbmath.org/1449.420412021-01-08T12:24:00+00:00"Ma, Rumeng"https://zbmath.org/authors/?q=ai:ma.rumeng"Xu, Jingshi"https://zbmath.org/authors/?q=ai:xu.jingshiSummary: We study characterizations of harmonic Bergman-Orlicz spaces and the boundedness of difference quotients of harmonic functions on the upper half-space or the unit ball. First, we give Lipschitz type characterizations of harmonic Bergman-Orlicz spaces via the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. By these characterizations, we obtain the boundedness of difference quotients of harmonic functions on the upper half-space or the unit ball, which generalize those for harmonic Bergman spaces on the upper half-space or the unit ball.Generalization of the first kind of oscillatory integrals and its estimates.https://zbmath.org/1449.420262021-01-08T12:24:00+00:00"Yu, Yufeng"https://zbmath.org/authors/?q=ai:yu.yufengSummary: This paper focuses on the generalization of the first kind of oscillatory integrals \(I (\lambda)\), namely the one derived from replacing the exponential function of the integrand in the classical first kind of oscillatory integrals by a real function, satisfying some differential equation. Firstly, we investigate the local properties of \(I (\lambda)\), that is, how \(I (\lambda)\) develops with \(\lambda\) varying (restricting that \(\lambda > 0\)). Then we give an estimate of \(I (\lambda)\) when the phase function \(\phi (x)\) satisfies \(\left| \phi^{ (k)} (x)\right|\ge 1\).Maximal operators of Cesáro means with varying parameters of Walsh-Fourier series.https://zbmath.org/1449.420452021-01-08T12:24:00+00:00"Gát, Gy."https://zbmath.org/authors/?q=ai:gat.gyorgy"Goginava, U."https://zbmath.org/authors/?q=ai:goginava.ushangiThe Cesaro means of the Walsh-Fourier series of an integrable function \(f\) on \([0,1)\) are \(\frac1{A^{\alpha_n}_{n-1}}\sum_{j=0}^{n-1} A^{\alpha_n}_{n-1-j}\hat f(j)w_j(x)\) where \(A^{\alpha_n}_n=\frac {(1+\alpha_n)\cdots (n+\alpha_n)}{n!}\), \(w_n(x)=(-1)^{\sum_{j=0}^\infty n_jx_j}\) when \(n=\sum_{k=0}^\infty n_k2^k\), \(x=\sum_{j=0}^\infty x_j2^{-j-1}\) for \(x\in [0,1)\) and \(\hat f(j)=\int_0^1 f(x)w_j(x)\, dx\). These means can also be written as convolutions in the form \( f*K^{\alpha_n}_n \) where \(K^{\alpha_n}_n=\frac 1{A^{\alpha_n}_{n-1}}\sum_{j=0}^{n-1} A^{\alpha_n}_{n-1-j}w_j\) when the group operation used in the definition of the convolution is \(x\dot + y=\sum_{j=0}^\infty \vert x_j-y_j\vert 2^{-j-1}\). The main result of the paper is that for every \(p>0\) there is a constant \(c_p<\infty\) such that \(\Vert \sup_{N\in \mathbb{N}} \vert f*\vert K^{\alpha_N}_{2^N}\vert \Vert_{L_p}\leq c_p\Vert f^*\Vert_{L_p}\) when \(0<\alpha_N<1\) and where \(f^*(x)= \sup_{n\in \mathbb{N}} \frac 1{\vert I_n(x)\vert} \int_{I_n(x)}f(t)\, dt\vert \) where \(I_n(x)\) is the dyadic interval \([\frac {j}{2^n},\frac {j+1}{2^n})\) containing \(x\).
Reviewer: Gustaf Gripenberg (Aalto)Explicit bounds of complex exponential frames on a complex field.https://zbmath.org/1449.420622021-01-08T12:24:00+00:00"Vellucci, P."https://zbmath.org/authors/?q=ai:vellucci.pierluigiThe author considers frame sets for \(L^2(-\pi,\pi)\) of the form \(\{e_n\}=\{e^{i\lambda_n t}\}\) with \(\{\lambda_n\}_{n\in\mathbb{Z}}\) a sequence of distinct complex numbers. It is really a frame if there exist frame constants \(A,B>0\), such that \[\forall x\in L^2(-\pi,\pi):\ A\Vert x\Vert ^2\leq \sum_n\,\vert \langle x,e_n\rangle\vert ^2\leq B\Vert x\Vert ^2.\]
Let \(\alpha\) be the real number satisfying \[\sum_{k=1}^{\infty}\,\frac{\pi^{2k}\alpha^{2k}}{k!(2k+1)}+\frac{2}{3}\pi^2\alpha^2 e^{\pi^2 \alpha^2}=1\] (numerical estimates give: \(\alpha=0.249012\ldots\)).
The main result, improving upon previous results, is the following:
Assume \[\lambda_n-n\vert \leq L<\alpha,\ n=0,\pm 1,\pm 2,\ldots,\] and \[f(x)=\sum_{k=1}^{\infty}\,\frac{\pi^{2k}x^{2k}}{k!(2k+1)}+\frac{2}{3}\pi^2x^2 e^{\pi^2x^2},\ x\in\mathbb{R}.\] Then the set \({\mathcal{F}}=\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}\) is a frame for \(L^2(-\pi,\pi)\) with bounds \(A^\prime\geq (1-(f(L))^{1/2})^2\) and \(B^\prime\leq (1+(f(L))^{1/2})^2\).
Reviewer: Marcel G. de Bruin (Heemstede)Research progress of the two-dimensional linear canonical transform theory and its application.https://zbmath.org/1449.420082021-01-08T12:24:00+00:00"Song, Yu'e"https://zbmath.org/authors/?q=ai:song.yue"Bu, Hongxia"https://zbmath.org/authors/?q=ai:bu.hongxia"Li, Bingzhao"https://zbmath.org/authors/?q=ai:li.bingzhaoSummary: As a generalized form of classical Fourier transform and fractional Fourier transform, linear canonical transform (LCT) has greater flexibility and is a powerful tool for analyzing and processing non-stationary signals. In the same way, the two-dimensional LCT has good performance in processing and analyzing two dimensional signals. We first review recent developments and the research results of two-dimensional LCT systematically, mainly focusing on the latest theory of two-dimensional non-separate LCT, such as important properties, sampling and discrete theory, fast algorithm, Heisenberg uncertainty principle, and eigenfunctions theory. Based on the summary of its theory, we summarize various and latest applications of the two-dimensional LCT in many fields, including filter design, image processing, etc. As well, the application of the two-dimensional LCT for estimating two-dimensional linear frequency modulated signal is analyzed in deeply. Finally, we make a development prospect of the two-dimensional LCT. It is of great reference value for researchers to fully understand the two-dimensional LCT and can further promote its engineering applications.On the wavelet estimation of a function in a density model with non-identically distributed observations.https://zbmath.org/1449.620832021-01-08T12:24:00+00:00"Chesneau, Christophe"https://zbmath.org/authors/?q=ai:chesneau.christophe"Hosseinioun, Nargess"https://zbmath.org/authors/?q=ai:hosseinioun.nargessSummary: A density model with possible non-identically distributed random variables is considered. We aim to estimate a common function appearing in the densities. We construct a new linear wavelet estimator and study its performance for independent and dependent data (the \(\rho \)-mixing case is explored). Then, in the independent case, we develop a new adaptive hard thresholding wavelet estimator and prove that it attains a sharp rate of convergence.