Recent zbMATH articles in MSC 44Ahttps://zbmath.org/atom/cc/44A2022-09-13T20:28:31.338867ZWerkzeugBook review of: B. G. Osgood, Lectures on the Fourier transform and its applicationshttps://zbmath.org/1491.000232022-09-13T20:28:31.338867Z"Rindler, H."https://zbmath.org/authors/?q=ai:rindler.haraldReview of [Zbl 1412.42003].Characterizations of mono-components: the Blaschke and starlike typeshttps://zbmath.org/1491.300072022-09-13T20:28:31.338867Z"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.tao"Tan, Lihui"https://zbmath.org/authors/?q=ai:tan.lihuiSummary: Since the last decade, motivated by attempts of positive frequency decomposition of signals, complex periodic functions \(s(e^{it})=\rho (t)e^{i\theta (t)}\) satisfying the conditions
\[
H(\rho (t)\cos \theta (t))=\rho (t)\sin \theta (t), \quad \rho (t)\geq 0,\;\theta'(t)\geq 0,\text{ a.e.},
\]
have been sought, where \(H\) is the circular Hilbert transform and the phase derivative \(\theta '(t)\) is suitably defined and interpreted as instantaneous frequency of the signal \(\rho (t)\cos \theta (t)\). Functions satisfying the above conditions are called mono-components. Mono-components have been found to form a large pool and used to decompose and analyze signals. This note in a great extent concludes the study of seeking for mono-components through characterizing two classes of mono-components of which one is phrased as the Blaschke type and the other the starlike type. The Blaschke type mono-components are of the form \(\rho (t)\cos \theta (t)\), where \(\rho (t)\) is a real-valued (generalized) amplitude functions and \(e^{i\theta (t)}\) is the boundary limit of a finite or infinite Blaschke product. For the starlike type mono-components, we assume the condition \(\int_{0}^{2\pi }\theta'(t)dt=n\pi \), where \(n\) is a positive integer. It shows that such class of mono-components is identical with the class consisting of products between \(p\)-starlike and boundary \((n-2p) \)-starlike functions. The results of this paper explore connections between harmonic analysis, complex analysis, and signal analysis.Fractional differentiations and integrations of quadruple hypergeometric serieshttps://zbmath.org/1491.330062022-09-13T20:28:31.338867Z"Bin-Saad, Maged G."https://zbmath.org/authors/?q=ai:bin-saad.maged-gumaan"Nisar, Kottakkaran S."https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Younis, Jihad A."https://zbmath.org/authors/?q=ai:younis.jihad-aSummary: The hypergeometric series of four variables are introduced and studied by Bin-Saad and Younis recently. In this line, we derive several fractional derivative formulas, integral representations and operational formulas for new quadruple hypergeometric series.Analytical approach to a class of Bagley-Torvik equationshttps://zbmath.org/1491.340162022-09-13T20:28:31.338867Z"Mahmudov, Nazim I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu"Huseynov, Ismail T."https://zbmath.org/authors/?q=ai:huseynov.ismail-t"Aliev, Nihan A."https://zbmath.org/authors/?q=ai:aliev.nihan-a"Aliev, Fikret A."https://zbmath.org/authors/?q=ai:aliev.fikret-akhmedaliogluSummary: Multi-term fractional differential equations have been studied because of their applications in modelling, and solved using miscellaneous mathematical methods. We present explicit analytical solutions for several families of generalized multidimensional Bagley-Torvik equations with permutable matrices and two various fractional orders which are satisfying \(\alpha \in(1,2]\), \(\beta \in (0,1]\) and \(\alpha \in(1,2]\), \(\beta \in (1,2]\), both homogeneous and inhomogeneous cases. The results are obtained by means of Mittag-Leffler type matrix functions with double infinite series. In addition, we acquire general solutions of the Bagley-Torvik scalar equations with \(\frac{1}{2}\)-order and \(\frac{3}{2}\)-order derivatives. At the end, we present different examples to verify the efficiency to our main results.Operational matrix for Atangana-Baleanu derivative based on Genocchi polynomials for solving FDEshttps://zbmath.org/1491.340212022-09-13T20:28:31.338867Z"Sadeghi, S."https://zbmath.org/authors/?q=ai:sadeghi.seyed-h-hesamedin|sadeghi.seyed-hossein-h|sadeghi.sara|sadeghi.sh|sadeghi.somayeh|sadeghi.s-n|sadeghi.saeid|sadeghi.sanaz|sadeghi.samira|sadeghi.sadegh|sadeghi.s-h-hesam"Jafari, H."https://zbmath.org/authors/?q=ai:jafari.hamed-houri|jafari.hamed-mazhab|jafari.habib|jafari.hamideh|jafari.hossein|jafari.hossein.1"Nemati, S."https://zbmath.org/authors/?q=ai:nemati.somayyeh|nemati.somayeh|nemati.sepehrSummary: Recently, \textit{A. Atangana} and \textit{D. Baleanu} [``New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model'', Therm. Sci. 20, No. 2, 763--769 (2016; \url{doi:10.2298/tsci160111018a})] have defined a new fractional derivative which has a nonlocal and non-singular kernel. It is called the Atangana-Baleanu derivative. In this paper we present a numerical technique to obtain solution of fractional differential equations containing Atangana-Baleanu derivative. For this purpose, we use the operational matrices based on Genocchi polynomials together with the collocation points which help us to reduce the problem to a system of algebraic equations. An error bound for the error of the operational matrix of the fractional derivative is introduced. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.On generation of family of resolving operators for a distributed order equation analytic in sectorhttps://zbmath.org/1491.340702022-09-13T20:28:31.338867Z"Fedorov, V. E."https://zbmath.org/authors/?q=ai:fedorov.v-e.1|fedorov.v-eSummary: The questions of the existence and uniqueness of solution to the Cauchy problem for an equation in a Banach space of distributed order at most one are investigated. Necessary and sufficient conditions for the existence of a resolving family of operators of this equation analytic in the sector are obtained. An explicit form of these operators is found. Two versions of the theorem on unique solvability of the Cauchy problem for the corresponding inhomogeneous equation are obtained: with condition of increased smoothness in spatial variables (the condition of continuity in the norm of the graph of the generator of the resolving family) and with condition of increased smoothness in the time variable (the Hölder condition). Abstract results are obtained using theory of the Laplace transform and generalization some results from the theory of analytic operator semigroups and its extensions to the case of integral equations and fractional differential equations. The conditions for unique solvability of an equation in a Banach space are used to study a class of initial boundary value problems for equations with polynomials in an elliptic differential operator with respect to spatial variables.Stochastic nonlinear Schrödinger equation on half-line with boundary noisehttps://zbmath.org/1491.354042022-09-13T20:28:31.338867Z"Kaikina, Elena I."https://zbmath.org/authors/?q=ai:kaikina.elena-igorevna"Sotelo-Garcia, Norma"https://zbmath.org/authors/?q=ai:sotelo-garcia.norma"Vázquez-Esquivel, Alexis V."https://zbmath.org/authors/?q=ai:vazquez-esquivel.alexis-vSummary: We consider the stochastic nonlinear Schrödinger equations on the half-line with Neumann brown-noise boundary conditions. We establish the global existence and uniqueness of solutions to initial-boundary value problem with values in \(\mathbf{H}^1\). We are also interested in the regularity behavior of the first spatial derivative of the solutions near the origin, where the boundary data are highly irregular. To obtain optimal estimate of the stochastic boundary response we propose new method based on Laplace transform and Cauchy theory of complex analysis. Also we adopt Sthriharts estimates and the Gagliardo-Nirenberg interpolation inequalities for the case of stochastic equations on a half-line.Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functionshttps://zbmath.org/1491.420032022-09-13T20:28:31.338867Z"Isaev, Mikhail"https://zbmath.org/authors/?q=ai:isaev.mikhail-ismailovitch"Novikov, Roman G."https://zbmath.org/authors/?q=ai:novikov.roman-gSummary: We give new formulas for finding a compactly supported function \(v\) on \(\mathbb{R}^d\), \(d\geq 1\), from its Fourier transform \(\mathcal{F}v\) given within the ball \(B_r\). For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.Positive definite functions on products of metric spaces by integral transformshttps://zbmath.org/1491.420052022-09-13T20:28:31.338867Z"Franca, W."https://zbmath.org/authors/?q=ai:franca.willian"Menegatto, V. A."https://zbmath.org/authors/?q=ai:menegatto.valdir-antonioSummary: An influential theorem proved by \textit{T. Gneiting} [J. Am. Stat. Assoc. 97, No. 458, 590--600 (2002; Zbl 1073.62593)] in the beginning of the century provides a large class of continuous positive definite functions on the product \(\mathbb{R}^d \times \mathbb{R}\) commonly used in theory and applications. It turns out that the positive definite functions given by this theorem fit into what is frequently called the scale mixture approach. In other words, they are generated by certain integral transforms defined by products of parameterized positive definite functions on \(\mathbb{R}^d\) and \(\mathbb{R} \). In this paper, we consider positive definite functions on a product of metric spaces which are given by general integral transforms. We provide conditions under which the positive definite functions are either continuous or strictly positive definite. In the case in which one of the metric spaces is \(\mathbb{R}^d\), we offer constructions of continuous strictly positive definite functions defined by certain hypergeometric functions and conditionally negative definite functions. They complement and generalize the original Gneiting's contribution. Additionally, we present necessary and sufficient conditions for the strict positive definiteness of the aforementioned generalizations.Generalized fractional integral operators on Campanato spaces and their bi-predualshttps://zbmath.org/1491.420392022-09-13T20:28:31.338867Z"Yamaguchi, Satoshi"https://zbmath.org/authors/?q=ai:yamaguchi.satoshi"Nakai, Eiichi"https://zbmath.org/authors/?q=ai:nakai.eiichiSummary: In this paper we prove the boundedness of the generalized fractional integral operator \(I_\rho\) on generalized Campanato spaces with variable growth condition, which is a generalization and improvement of previous results, and then, we establish the boundedness of \(I_\rho\) on their bi-preduals. We also prove the boundedness of \(I_\rho\) on their preduals by the duality.New integral operator for solutions of differential equationshttps://zbmath.org/1491.440012022-09-13T20:28:31.338867Z"Ozyapici, Ali"https://zbmath.org/authors/?q=ai:ozyapici.ali"Karanfiller, Tolgay"https://zbmath.org/authors/?q=ai:karanfiller.tolgaySummary: This study is aimed to define general representation of integral transforms for solving differential equations. The Generalized Integral Transform consists of the well-known Laplace transform, Sumudu transform, Tarig transform and Elzaki transform, as a common coverage. Since all these transforms, respectively, promise an effective usage for solving differential equations, their corresponding theories can easily be derived by using Generalized Integral Transform. Moreover, this study shows that Generalized Integral Transform can be easily used to define a new integral operator which will provide the easiest approach to solution of the given differential equation. Some examples discussed in the paper show that the Generalized Integral Transform can be applicable for many differential equations while Laplace transform can not be applicable for the same differential equations.Lower bound of sectional curvature of Fisher-Rao manifold of beta distributions and complete monotonicity of functions involving polygamma functionshttps://zbmath.org/1491.440022022-09-13T20:28:31.338867Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher-Rao manifold of beta distributions.Boundedness and compactness of the spherical mean two-wavelet localization operatorshttps://zbmath.org/1491.440032022-09-13T20:28:31.338867Z"Mejjaoli, Hatem"https://zbmath.org/authors/?q=ai:mejjaoli.hatem"Omri, Slim"https://zbmath.org/authors/?q=ai:omri.slimSpherical means have been studied by several authors. Harmonic functions are characterized by their spherical mean values. The classic book [\textit{F. John}, Plane waves and spherical means applied to partial differential equations. New York, NY: Interscience Publishers (1955; Zbl 0067.32101)] deals with various applications of the spherical means to the theory of partial differential equations. In this paper, the authors use the properties of the Fourier transform associated with the spherical mean operator to study the boundedness and compactness of the two-wavelet localization operators associated to the spherical mean operator. The extension of one wavelet to two wavelets gives the extra degree of flexibility in signal analysis and imaging when the localization operators are used as time-varying filters. Some results of the harmonic analysis associated with the spherical mean operator and Schatten-von Neumann classes are given. The two-wavelet localization operators in the setting of the spherical mean operator are considered. The Schatten-von Neumann properties of these two localization wavelet operators are established, for trace class spherical mean two-wavelet localization operators, the traces and the trace class norm inequalities are presented.
Reviewer: S. L. Kalla (Ballwin)Discrete Lebedev-Skalskaya transformshttps://zbmath.org/1491.440042022-09-13T20:28:31.338867Z"Yakubovich, S."https://zbmath.org/authors/?q=ai:yakubovich.semyon-bSummary: Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels \(\mathrm{Re}\,K_{\alpha+in}(x)\), \(\mathrm{Im}\,K_{\alpha+in}(x)\), \(x>0\), \(n\in\mathbb{N}\), \(|\alpha|<1\), \(i\) is the imaginary unit and \(K_\nu(z)\) is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when \(\alpha=\pm 1/2\). The case \(\alpha=0\) reduces to the Kontorovich-Lebedev transform.On an extension of the Mikusiński type operational calculus for the Caputo fractional derivativehttps://zbmath.org/1491.440052022-09-13T20:28:31.338867Z"Al-Kandari, M."https://zbmath.org/authors/?q=ai:alkandari.maryam"Hanna, L. A.-M."https://zbmath.org/authors/?q=ai:hanna.latif-a-m"Luchko, Yu."https://zbmath.org/authors/?q=ai:luchko.yuri|luchko.yurii-fSummary: In this paper, a two-parameter extension of the operational calculus of Mikusiński's type for the Caputo fractional derivative is presented. The first parameter is connected with the rings of functions that are used as a basis for construction of the convolution quotients fields. The convolutions by themselves are characterized by another parameter. The obtained two-parameter operational calculi are compared each to other and some homomorphisms between the fields of convolution quotients are established.Tracial moment problems on hypercubeshttps://zbmath.org/1491.440062022-09-13T20:28:31.338867Z"Le, Cong Trinh"https://zbmath.org/authors/?q=ai:le-cong-trinh.Summary: In this paper we introduce the \textit{tracial \(K\)-moment problem} and the \textit{sequential matrix-valued \(K\)-moment problem} and show the equivalence of the solvability of these problems. Using a Haviland's theorem for matrix polynomials, we solve these \(K\)-moment problems for the case where \(K\) is the hypercube \([-1,1]^n\).Some classes of integral equations of convolutions-pair generated by the Kontorovich-Lebedev, Laplace and Fourier transformshttps://zbmath.org/1491.450052022-09-13T20:28:31.338867Z"Tuan, Trinh"https://zbmath.org/authors/?q=ai:tuan.trinhSummary: In this article, we prove the existence of a solution to some classes of integral equations of generalized convolution type generated by the Kontorovich-Lebedev (K) transform, the Laplace \((\mathcal{L})\) transform and the Fourier (F) transform in some appropriate function spaces and represent it in a closed form.Domains with algebraic \(X\)-ray transformhttps://zbmath.org/1491.520052022-09-13T20:28:31.338867Z"Agranovsky, Mark"https://zbmath.org/authors/?q=ai:agranovsky.mark-lSummary: \textit{A. Koldobsky} et al. proved in [Adv. Math. 320, 876--886 (2017; Zbl 1377.52005)] that given a convex body \(K \subset\mathbb{R}^n\), \(n\) is odd, with smooth boundary, such that the volume of the intersection \(K \cap L\) of \(K\) with a hyperplane \(L \subset\mathbb{R}^n\) (the sectional volume function) depends polynomially on the distance \(t\) of \(L\) to the origin, then the boundary of \(K\) is an ellipsoid. In even dimension, the sectional volume functions are never polynomials in \(t\), nevertheless in the case of ellipsoids their squares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity of the dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions are roots (of some power) of polynomials. In this article, we confirm this conjecture for planar domains, bounded by algebraic curves. A multidimensional version in terms of chords lengths, i.e., of \(X\)-ray transform of the characteristic function, is given. The result is motivated by Arnold's conjecture on characterization of algebraically integrable bodies.American options in the Volterra Heston modelhttps://zbmath.org/1491.911402022-09-13T20:28:31.338867Z"Chevalier, Etienne"https://zbmath.org/authors/?q=ai:chevalier.etienne"Pulido, Sergio"https://zbmath.org/authors/?q=ai:pulido.sergio"Zúñiga, Elizabeth"https://zbmath.org/authors/?q=ai:zuniga.elizabethImage reconstruction and knowledgebase tomography mechanismhttps://zbmath.org/1491.920782022-09-13T20:28:31.338867Z"Afzalipour, Laya"https://zbmath.org/authors/?q=ai:afzalipour.laya"Nikazad, Touraj"https://zbmath.org/authors/?q=ai:nikazad.touraj(no abstract)