Recent zbMATH articles in MSC 44Ahttps://zbmath.org/atom/cc/44A2023-11-13T18:48:18.785376ZWerkzeugSuperoscillations and Fock spaceshttps://zbmath.org/1521.300632023-11-13T18:48:18.785376Z"Alpay, Daniel"https://zbmath.org/authors/?q=ai:alpay.daniel"Colombo, Fabrizio"https://zbmath.org/authors/?q=ai:colombo.fabrizio"Diki, Kamal"https://zbmath.org/authors/?q=ai:diki.kamal"Sabadini, Irene"https://zbmath.org/authors/?q=ai:sabadini.irene"Struppa, Daniele C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloSummary: In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these special wave functions can be constructed also by computing the approximating sequence of the normalized Hermite functions. First, we start by treating the case when a superoscillating sequence is multiplied by the Gaussian function. Then, we extend these calculations to the case of normalized Hermite functions leading to interesting relations with Weyl operators. In particular, we show that the Segal-Bargmann transform maps superoscillating sequences onto a superposition of coherent states. Following this approach, the computations lead to a specific linear combination of the normalized reproducing kernels (coherent states) of the Fock space. As a consequence, we obtain two new integral Bargmann-type representations of superoscillating sequences. We also investigate some results relating superoscillation functions with Weyl operators and Fourier transform.
{\copyright 2023 American Institute of Physics}On the theory of orthogonal polynomials for the weight \(x^\nu \exp (-x-t/x)\). IIhttps://zbmath.org/1521.330042023-11-13T18:48:18.785376Z"Yakubovich, S."https://zbmath.org/authors/?q=ai:yakubovich.semyon-bSummary: Orthogonal polynomials for the weight \(x^\nu \exp (-x-t/x)\), \(x,t>0\), \(\nu \in \mathbb{R}\) are investigated without the use of the Chen-Ismail ladder operators approach. In this part we derive explicit representations, recurrence relations for coefficients, the generating function and Rodrigues-type formula.
For Part I, see [ibid. 33, No. 9, 735--746 (2022; Zbl 07582201)].Arbitrary order differential equations with fuzzy parametershttps://zbmath.org/1521.340042023-11-13T18:48:18.785376Z"Allahviranloo, Tofigh"https://zbmath.org/authors/?q=ai:allahviranloo.tofigh"Salahshour, Soheil"https://zbmath.org/authors/?q=ai:salahshour.soheilSummary: In the last decades, some generalization of theory of ordinary differential equations has been considered to the arbitrary order differential equations by many researchers, the so-called theory of arbitrary order differential equations (often called as fractional order differential equations [FDEs]). Because of the ability for modeling real phenomena, arbitrary order differential equations have been applied in various fields such as control systems, biosciences, bioengineering, and references therein. In this chapter, the authors propose arbitrary order differential equations with respect to another function using fuzzy parameters (initial values and the unknown solutions). The generalized fuzzy Laplace transform is applied to obtain the Laplace transform of arbitrary order integral and derivative of fuzzy-valued functions to solve linear FDEs. To obtain the large class of solutions for FDEs, the concept of generalized Hukuhara differentiability is applied.
For the entire collection see [Zbl 1439.74003].Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operatorshttps://zbmath.org/1521.340072023-11-13T18:48:18.785376Z"Alharthi, Nadiyah Hussain"https://zbmath.org/authors/?q=ai:alharthi.nadiyah-hussain"Atangana, Abdon"https://zbmath.org/authors/?q=ai:atangana.abdon"Alkahtani, Badr S."https://zbmath.org/authors/?q=ai:alkahtani.badr-saad-tSummary: Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.Asymptotic behavior of fractional-order nonlinear systems with two different derivativeshttps://zbmath.org/1521.340082023-11-13T18:48:18.785376Z"Chen, Liping"https://zbmath.org/authors/?q=ai:chen.liping"Xue, Min"https://zbmath.org/authors/?q=ai:xue.min"Lopes, António"https://zbmath.org/authors/?q=ai:lopes.antonio-m"Wu, Ranchao"https://zbmath.org/authors/?q=ai:wu.ranchao"Chen, YangQuan"https://zbmath.org/authors/?q=ai:chen.yangquanSummary: This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag-Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.A computational approach to exponential-type variable-order fractional differential equationshttps://zbmath.org/1521.340092023-11-13T18:48:18.785376Z"Garrappa, Roberto"https://zbmath.org/authors/?q=ai:garrappa.roberto"Giusti, Andrea"https://zbmath.org/authors/?q=ai:giusti.andreaSummary: We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique.Modification of optimal homotopy asymptotic method for multi-dimensional time-fractional model of Navier-Stokes equationhttps://zbmath.org/1521.351352023-11-13T18:48:18.785376Z"Jan, Himayat Ullah"https://zbmath.org/authors/?q=ai:jan.himayat-ullah"Ullah, Hakeem"https://zbmath.org/authors/?q=ai:ullah.hakeem"Fiza, Mehreen"https://zbmath.org/authors/?q=ai:fiza.mehreen"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Mohamed, Abdullah"https://zbmath.org/authors/?q=ai:mohamed.abdullah"Mousa, Abd Allah A."https://zbmath.org/authors/?q=ai:mousa.abd-allah-a(no abstract)Generalization of statistical limit-cluster points and the concepts of statistical limit inferior-superior on time scales by using regular integral transformationshttps://zbmath.org/1521.400052023-11-13T18:48:18.785376Z"Yalçin, Ceylan"https://zbmath.org/authors/?q=ai:yalcin.ceylan-turanSummary: With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.Zalcman's problem and related two-radii theoremshttps://zbmath.org/1521.420082023-11-13T18:48:18.785376Z"Volchkov, Valery"https://zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vitaly"https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: Let \(G\) be the group of conformal automorphisms of the unit disc \(\mathbb{D} = \{z\in\mathbb{C}:|z|<1\}\). For \(r > 0\), we put \(B_r = \{z\in\mathbb{D}:|z|<\tanh r\}\). Denote by \(\overline{B}_r\) the closure of the disc \(B_r\), and by \(\partial B_r\) its boundary. Let \(\chi_r\) be the characteristic function (indicator) of \(B_r\). Assume that \(r_1, r_2\in (0, +\infty)\) are fixed and \(R > \max\{r_1, r_2\}\). We study the holomorphicity problem for functions \(f\in C(B_R)\) satisfying the condition
\[
\int\limits_{\partial B_{r_j}} f(g(z)) dz=0
\]
for all \(g\in G\) such that \(g \overline{B}_{r_j}\subset B_R\), \(j=1, 2\). We find the exact conditions for holomorphicity in terms of size \(B_R\) and properties of zeros of generalized spherical transforms of functions \(\chi_{r_1}\) and \(\chi_{r_2}\). In particular, a strengthening of the Berenstein-Pascuas theorem [\textit{C. Berenstein} and \textit{D. Pascuas}, Isr. J. Math. 86, No. 1--3, 61--106 (1994; Zbl 0827.30001)] on two radii is obtained.Special affine multiresolution analysis and the construction of orthonormal wavelets in \(L^2(\mathbb{R})\)https://zbmath.org/1521.420272023-11-13T18:48:18.785376Z"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmad"Lone, Waseem Z."https://zbmath.org/authors/?q=ai:lone.waseem-zThis paper merges special affine Fourier transforms (SAFT) and wavelet transforms. This involves several areas in harmonic analysis which we summarize. Recall that for \(p \geq 1\) the Lebesgue space \(L^p(\mathbb R)\) consists of measurable \(f : \mathbb R \rightarrow \mathbb C\) that satisfy \(\int_{\mathbb R}|f(x)|^p dx < \infty\). The Cauchy-Schwarz inequality implies that \(L^2(\mathbb R)\) admits the scalar product
\[
\tag{1} \left<f,g\right> := \int_{\mathbb R} f(x) \, \overline{g(x)} \, dx
\]
since \(|\left<f,g\right>|^2 \leq \left<f,f\right> \left<g,g\right>\). The Riesz-Fisher theorem shows that \(L^2(\mathbb R)\) is complete (every Cauchy sequence converges) under the associated norm \(\|f\|_2 := \sqrt {\left<f.f\right>}\) thus \(L^2(\mathbb R)\) is a Hilbert space.
The Fourier transform \(\mathcal F : L^1(\mathbb R) \cap L^2(\mathbb R) \rightarrow L^2(\mathbb R)\) is defined by the integral
\[
\tag{2} (\mathcal F f)(y) := \frac{1}{\sqrt 2\pi} \int_{\mathbb R} f(x)e^{-ixy} dx.
\]
\(\mathcal F\) is norm preserving and since \(L^1(\mathbb R) \cap L^2(\mathbb R)\) is a dense subset of \(L^2(\mathbb R)\), \(\mathcal F\) extends to a unitary operator on \(L^2(\mathbb R)\). We define \(\widehat f := \mathcal Ff, \, f \in L^2(\mathbb R)\).
SAFT comprise the set of operators \(\mathcal O_M\) on \(L^2(\mathbb R)\) parameterized by \(2\) by \(3\) real matrices \( M = \left(\begin{array}{cccc} A & B & : & p \\
C & D & : & q \end{array} \right)\) satisfying the unimodular condition \(AD-BA = 1\) and defined by the integral transform
\[
\tag{3} \mathcal O_M\left[ f \right](\omega) := \int_{\mathbb R} f(t)\, \mathcal K_M(t,\omega) \, dt,
\]
where
\[
\tag{4} \mathcal K_M(t,\omega) := \frac{1}{\sqrt{2\pi i B}} \exp \left[\frac{i(At^2+2t(p-\omega) -2\omega(Dp-Bq) + D(\omega^2+p^2)}{2B} \right].
\]
Clearly \(\mathcal O_M = \mathcal U_M(\omega)\mathcal F \mathcal V_M(t)\) where \(\mathcal F\) is the Fourier transform operator and \(\mathcal U_M(\omega)\) and \(\mathcal V_M(t)\) are multiplication by functions having modulus \(1\). Thus \(\mathcal O_M\) is unitary and thus satisfies Parseval's identity:
\[
\tag{5} \left<\mathcal O_M f, \mathcal O_Mg\right> = \left< f, g\right>, \ \ f, g \in L^2(\mathbb R).
\]
For \(p = q = 0\) the SAFT are the set of linear canonical transforms (LCT) of \(L^2(\mathbb R^n)\) for \(n = 1\). The LCT were discovered by \textit{A. Weil} [Acta Math. 111, 143--211 (1964; Zbl 0203.03305)] who observed that (for \(n = 1\)) they give a projective representation of the group \(SL(2,\mathbb R)\) or, equivalently, a representation of its two-to-one covering group called the metaplectic group. Subsequently, they were re-discovered by physicists and used in optics and quantum mechanics. \textit{Y. Han} and \textit{W. Sun} [Appl. Anal. 101, No. 14, 5156--5170 (2022; Zbl 1504.42093)] studied windowed versions of LCT for \(n \geq 1\). The parameters \(p\) and \(q\) allow a time-shifting and frequency-modulation. SAFT are also called offset LCT in references of the paper.
The continuous wavelet transform (CWT) is explained in detail by \textit{I. Daubechies} [Ten lectures on wavelets. Philadelphia, PA: SIAM (1992; Zbl 0776.42018), p. 24--30]. A function \(\psi \in L^2(\mathbb R)\) will be called a wavelet if it satisfies the following admissibility condition:
\[
\tag{6} C_{\psi} := 2\pi \int_{\mathbb R} \frac{|\widehat \psi(x)|^2}{x}dx < \infty.
\]
This implies that \(\widehat \psi(0) = \int_{\mathbb R} \psi(x)\, dx = 0\) so \(\psi\) has some oscillatory property so is called a wavelet. Define the affine group
\[
\tag{7} G := \left \{ \left[ \begin{array}{cc} a & b \\
0 & 1 \end{array} \right] : a , b \in \mathbb R, a \neq 0 \} \right \}
\]
and let \(L^2(G)\) denote the Hilbert space of measurable functions \(F : G \mapsto \mathbb C\) with respect to the left-invariant Haar measure \(a^{-2}dadb\) on \(G\). Define the unitary representation of \(G\) on \(L^2(\mathbb R)\) by
\[
\tag{8} (U_{a,b}f)(x) := |a|^{-1/2} f((x-b)/a), \ \ x \in \mathbb R.
\]
The wavelet transform \(T_\psi : L^3(\mathbb R) \mapsto L^2(G)\) is defined by
\[
\tag{9} (T_\psi f)(a,b) := \frac{1}{\sqrt C_\psi} \left<f, U_{a,b}\psi\right>.
\]
Parseval's identity for \(\mathcal F\) and the identity \(\widehat {U_{a,b}\psi}(y) = |a|^{1/2}e^{-iby} \widehat \psi(ay)\) gives \((T_\psi f)(a,b) = \widehat F_a(-b)\) where \(F_a(y) = |a|^{1/2} \widehat f(y)\overline {\widehat \psi(ay)}\) and \((T_\psi g)(a,b) \widehat G_a(-b)\) where \(G_a(y) = |a|^{1/2} \widehat g(y)\overline {\widehat \psi(ay)}\). Therefore
\begin{align*}
\left<T_\psi f, T_\psi g\right>_{L^2(G)} &: = C_{\psi}^{-1} \, \int_{G} \left<f, U_{a,b}\psi\right> \overline {\left<g, U_{a,b}\psi\right> } \, \frac{dadb}{a^2}\\
&= C_{\psi}^{-1} \, \int_{G} \widehat F_a(-b) \overline {\widehat G_a(-b)} \, \frac{dadb}{a^2} = C_{\psi}^{-1} \, \int_{G} F_a(b) \overline {G_a(b)} \, \frac{dadb}{a^2} \\
&= C_{\psi}^{-1} \, \int_{G} \widehat f(b) \overline {\widehat g(b)} |\psi(ay)|^2 \, \frac{dadb}{a} = \int_{G} \widehat f(b) \overline {\widehat g(b)} db = \left<f,g\right>.
\end{align*}
Therefore \(T_\psi : L^2(\mathbb R) \mapsto L^2(G)\) is an isometry so its inverse \(T_\psi : L^2(G) \mapsto L^2(\mathbb R)\) is its adjoint:
\[
\tag{10} T_\psi^{-1}H = \int_{G} H(a,b)\, U_{a,b}\psi \, \frac{dadb}{a^2}, \ \ H \in L^2(G).
\]
Define the subset \(\Lambda := \{(2^n,k2^m) : n, k \in \mathbb Z\}\) of \(G\). Note that \(\Lambda\) is not a subgroup. Orthonormal wavelet bases for \(L^2(\mathbb R)\) are sets of the form \(\{U_{a,b}\psi : (a,b) \in \Lambda\}\) and the associated discrete wavelet transform (DWT) \(D_\psi : L^2(\mathbb R) \mapsto \ell^2(\Lambda)\) is:
\[
\tag{11} (D_\psi f)(a,b) := \left<f, U_{a,b}\psi\right>, \ \ f \in L^22(\mathbb R), (a,b) \in \Lambda.
\]
The first such basis, invented by \textit{A. Haar} [Math. Ann. 69, 331--371 (1910; JFM 41.0469.03)], is formed by choosing the mother wavelet
\[
\tag{12} \psi(x) =
\begin{cases}
1, &0 \leq x < \frac{1}{2} \\
-1, &\frac{1}{2} \leq x < 1 \\
0, &\text{otherwise}. \end{cases}
\]
Daubechies [loc. cit.] first describes the construction of orthonormal bases for which \(\psi\) has compact support and has a regularity of any specified degree. For this to hold \(\psi\) must satisfy conditions much more restrictive than the admissibility condition described by equation (6). The \(\psi\) is constructed from a function \(\varphi\) called a scaling function and \(\varphi\) is constructed from finite sequences called conjugate quadrature filters which play an important role in digital signal processing [Daubechies, loc. cit., p. 163]. These discrete and continuous wavelet transforms have bear little relationship to each other.
Review by Section
Section 1 says that the SAFT is not well localized in time-frequency due to the infinite support of the kernel function \(\mathcal K_M\) and suggests that combining it with wavelet transforms will remedy this weakness. It also describes how wavelets are related to multiresolution analysis.
Section 3 defines the SAFT and graphically illustrates its special cases: fractional Fourier transforms, Fresnel transforms, and linear canonical transforms. It also defines the special affine convolution and derives a convolution theorem analogous to that of classical convolution.
Section 3 describes for certain \(\psi \in L^2(\mathbb R)\) the function
\[
\tag{13} \psi_{a,b}^M(x) := U_{a,b}\psi(x) \, \exp \left[ \frac{-i(At^2 + Dp^2 - A(b/a)^2)}{2B} \right]
\]
and the associated continuous special affine wavelet transform \(\mathcal W_\psi^M : L^2(\mathbb R) \mapsto L^2(G)\) defined by
\[
\tag{14} (\mathcal W_\psi^Mf)(a,b) := \left<f,\psi_{a,b}^M\right>, \ \ f \in L^2(\mathbb R), (a,b) \in G.
\]
In Example 3.2 these transforms are applied to the function \(f(t) = e^{-(\alpha t + \beta t^2)}\), \(\alpha \in \mathbb R\), \(\beta > 0\) with \(\psi(t) = \exp \left[ -\left( i \gamma t - \frac{t^2}{2} \right) \right]\), \(\gamma > 0\). for various matrices \(M\). The proposition shows that a certain admissibility condition on \(\psi\) implies that \(\mathcal W_{\psi}^M\psi \in L^2(G)\) and defines an associated \(C_\psi\) analogous to the ordinary CWT. Theorem 3.5 shows that \(\frac{1}{\sqrt C_\psi} \mathcal W_\psi^M\) is unitary and Theorem 3.6 gives its inverse which is completely analogous to equation (10). Section 4 extends the classical concept of multiresolution analysis (MRA) associated with orthonormal wavelet bases to special affine MRA by applying the unitary transformations in Theorem 3.6 to classical scaling functions and wavelets. Numerous special cases are computed in detail. Example 4.5 illustrates using the Haar wavelet.
Section 5 describes the fast wavelet algorithm associated with the special affine MRA. It is completely analogous to the classical fast wavelet algorithm. Incidentally, this is the only algorithm for computing discrete wavelet transforms that I have seen discussed in the literature. Section 6 discusses the role of scaling functions in special affine MRA and derives numerous identities and inequalities that may be useful for developing numerical algorithms.
The paper uses extensive detailed algebraic computations. It is well-referenced. It should be noted that the SAFT combines with wavelets by multiplying them by functions having modulus \(1\). It plays no role in the construction of orthonormal wavelets that it requires deep properties of conjugate filters as previously discussed. Rather existing wavelets are used to window the SAFT which is useful to examine localized properties in signals. An example is localized chirping in radar signals used to obtain the high bandwidth required for range resolution using low peak power.
Reviewer: Wayne M. Lawton (Krasnoyarsk)A convergent version of Watson's lemma for double integralshttps://zbmath.org/1521.440012023-11-13T18:48:18.785376Z"Ferreira, Chelo"https://zbmath.org/authors/?q=ai:ferreira.chelo"López, José L."https://zbmath.org/authors/?q=ai:lopez.jose-luis"Pérez Sinusía, Ester"https://zbmath.org/authors/?q=ai:perez-sinusia.esterSummary: A modification of Watson's lemma for Laplace transforms \(\int_0^{\infty} f(t) e^{-zt} \,\mathrm{d}t\) was introduced in [\textit{N. Nielsen}, Handbuch der Theorie der Gammafunktionen. Leipzig: B. G. Teubner (1906; JFM 37.0450.01)], deriving a new asymptotic expansion for large \(|z|\) with the extra property of being convergent as well. Inspired in that idea, in this paper we derive asymptotic expansions of two-dimensional Laplace transforms \(F(x,y) := \int_0^{\infty}\int_0^{\infty} f(t,s) e^{-xt-ys}\, \mathrm{d}t \, \mathrm{d}s\) for large \(|x|\) and \(|y|\) that are also convergent. The expansions of \(F(x,y)\) are accompanied by error bounds. Asymptotic and convergent expansions of some special functions are given as illustration.Uncertainty principles for the continuous Kontorovich Lebedev wavelet transformhttps://zbmath.org/1521.440022023-11-13T18:48:18.785376Z"Dades, Abdelaali"https://zbmath.org/authors/?q=ai:dades.abdelaali"Daher, Radouan"https://zbmath.org/authors/?q=ai:daher.radouan"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othmanSummary: The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Kontorovich Lebedev wavelet transform. We also analyse the concentration of this transform on sets of finite measure.The two-sided quaternionic Dunkl transform and Hardy's theoremhttps://zbmath.org/1521.440032023-11-13T18:48:18.785376Z"Essenhajy, Mohamed"https://zbmath.org/authors/?q=ai:essenhajy.mohamed"Fahlaoui, Said"https://zbmath.org/authors/?q=ai:fahlaoui.saidSummary: In this paper we study the two-sided quaternionic Dunkl transform. We establish its fundamental properties, such as Plancherel and inversion formula. Finally, we apply the Dunkl transform properties to establish an analogue of Hardy's theorem.Interpolation based formulation of the oscillatory finite Hilbert transformshttps://zbmath.org/1521.440042023-11-13T18:48:18.785376Z"Zaman, Sakhi"https://zbmath.org/authors/?q=ai:zaman.sakhi"Nawaz, Faiza"https://zbmath.org/authors/?q=ai:nawaz.faiza"Khan, Suliman"https://zbmath.org/authors/?q=ai:khan.suliman"Zaheer-ud-Din"https://zbmath.org/authors/?q=ai:zaheeruddin.m(no abstract)A pair of Barut-Girardello type transforms and allied pseudo-differential operatorshttps://zbmath.org/1521.440052023-11-13T18:48:18.785376Z"Mandal, U. K."https://zbmath.org/authors/?q=ai:mandal.upain-kumar"Prasad, Akhilesh"https://zbmath.org/authors/?q=ai:prasad.akhileshThe Barut-Girardello transform was introduced in 1971 by \textit{A. O. Barut} and \textit{L. Girardello} [Comm. Math. Phys. 21, 41--55 (1971; Zbl 0214.38203)] with the modified Bessel function of first kind \(I_\nu\) as its kernel. Later \textit{A. Torre} [Int. Trans. Spec. Funct. 19, No. 4, 277--292 (2008; Zbl 1145.44002)] gave two new versions of the same and studied some of their properties. The authors in the present paper advance these works and introduce the modified versions of the conventional BG-transform and the first and second BG-transforms and investigate a number of their interesting properties. They define the modified BG-transform of an integrable function \(f\) as
\[
({\mathcal{G}{_{\nu ,n}}f})(y)= {2^{\frac{1}{n}}}\int_0^\infty {\sqrt {xy} {I_\nu }({{2^{\frac{1}{n}}}xy}){e^{-\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{1}
\]
where \(\nu \ge \frac{-1}{2}\) and \(n \in \mathbb{N}\). The modified first and second BG-transforms of an integrable function \(f\) are respectively defined as
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}f})(y)= {2^{\frac{1}{n}}}{y^{2\mu + 1}}\int_0^\infty {{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{ - \frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{2}
\]
\[
({\mathcal{G}{_{2,\nu ,\mu ,n}}f})(y)= {2^{\frac{1}{n}}}\int_0^\infty {{x^{2\mu + 1}}{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{ - \frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{3}
\]
where \(\nu\) and \(n\) are as in (1) and \(\mu\) is any real number. The inversion formulas for (2) and (3) are respectively given by
\[
({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}f})(y)= {({ - 1})^{\nu + 1}}{2^{\frac{1}{n}}}{y^{2\mu + 1}}\int_0^\infty {{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx \tag{4}
\]
and
\[
({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}f})(y)= {({ - 1})^{\nu + 1}}{2^{\frac{1}{n}}}\int_0^\infty {{x^{2\mu + 1}}{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx. \tag{5}
\]
The relations between (2) and (3) are given by
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}f})(y)= {y^{2\mu + 1}}({\mathcal{G}{_{2,\nu ,\mu ,n}}{x^{ - ({2\mu + 1})}}f})(y)
\]
and
\[
({\mathcal{G}{_{2,\nu ,\mu ,n}}f})(y)= {y^{ - ({2\mu + 1})}}({\mathcal{G}{_{1,\nu ,\mu ,n}}{x^{2\mu + 1}}f})(y),
\]
while those between (4) and (5) are given by
\[
({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}f})(y)= ({{{\left\{ {\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}} \right\}}^\prime }f})(y)= {y^{2\mu + 1}}({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}{x^{ - ({2\mu + 1})}}f})(y)
\]
and
\[
({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}f})(y)= ({{{\left\{ {\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}} \right\}}^\prime }f})(y)= {y^{ - ({2\mu + 1})}}({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}{x^{2\mu + 1}}f})(y).
\]
The authors also give Parseval's and mixed type of Parseval's relations for these two types of BG-transforms. By writing the differential operator \(M_{1,\nu,\mu,n}\) associated to the BG-transforms as
\[
{M_{1,\nu ,\mu ,n}} = \frac{{{d^2}}}{{d{x^2}}} + \left({\frac{{1 + 2\mu }}{x} + 2{x^{n - 1}}}\right)\frac{d}{{dx}} + \left({\frac{{{\mu ^2} - {\nu ^2}}}{{{x^2}}} + {x^{2(n - 1)}} + ({2\mu + n}){x^{n - 2}}}\right)
\]
and its adjoint \(M_{2,\nu,\mu,n}\) as
\[
{M_{2,\nu ,\mu ,n}} = \frac{{{d^2}}}{{d{x^2}}} - \left({\frac{{1 + 2\mu }}{x} + 2{x^{n - 1}}}\right)\frac{d}{{dx}} + \left({\frac{{{{({\mu + 1})}^2} - {\nu ^2}}}{{{x^2}}} + {x^{2n - 2}} + ({2\mu + 2 - n}){x^{n - 2}}}\right)
\]
the authors deduce the relations
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}{M_{2,\nu ,\mu ,n}}f})(y)= {2^{\frac{2}{n}}}{y^2}({\mathcal{G}{_{1,\nu ,\mu }}f})(y)\quad \text{and}\quad (({\mathcal{G}{_{2,\nu ,\mu ,n}}{M_{1,\nu ,\mu ,n}}f})(y)= {2^{\frac{2}{n}}}{y^2}({\mathcal{G}{_{2,\nu ,\mu }}f})(y).
\]
In Section 2 of the paper the authors introduce some further differential operators \(A_{1,\nu,\mu,n}\), $B_{1,\nu,\mu,n}$, $A_{2,\nu,\mu,n}$, \(B_{2,\nu,\mu,n}\) which are closely related to the operators \(M_{1,\nu,\mu,n}\) and \(M_{2,\nu,\mu,n}\) in the sense \(A_{1,\nu,\mu,n}B_{1,\nu,\mu,n}=M_{1,\nu,\mu,n}\) and \(B_{2,\nu,\mu,n}A_{2,\nu,\mu,n}=M_{2,\nu,\mu,n}\) and generalize these two latter operators up to the \(r^{\mathrm{th}}\) order in Lemma~2.1. In Section~3 the authors introduce Zemanian type function spaces like those defined in the works of \textit{M.~Linares Linares} and \textit{J.~M.~R. Mendez Pérez} [Bull. Calcutta Math. Soc. 83, 447--546 (1991; Zbl 0759.46039)] and \textit{A.~Prasad} and \textit{K.~Mahato} [Rend. Circ. Mat. Palermo 65, No.~2, 209--241 (2016; Zbl 1366.46025)] and discuss the continuity of differential operators and the BG-type transforms on these spaces. The translation and convolution operators associated with the modified BG-type transforms are studied in Section 4 and a number of stimulating results are proven about them. Two pseudo-differential operators involving the BG-type transforms are defined in Section~5 and their continuity theorems are proved. Lastly the authors discuss the applicability of the modified BG-transform to solve a differential equation and a pseudo-differential equation.
Reviewer: Lalit Mohan Upadhyaya (Dehradun)On the set of convergence for Mellin-Barnes integral representing solutions to the tetranomial algebraic equationhttps://zbmath.org/1521.440062023-11-13T18:48:18.785376Z"Antipova, Irina A."https://zbmath.org/authors/?q=ai:antipova.i-a"Zykova, Tat'yana V."https://zbmath.org/authors/?q=ai:zykova.tatyana-vSummary: In the present paper we give the detailed description of the set of convergence for Mellin-Barnes integral representing solutions to the tetranomial algebraic equation.An LT-BEM formulation for problems of anisotropic functionally graded materials governed by transient diffusion-convection-reaction equationhttps://zbmath.org/1521.742892023-11-13T18:48:18.785376Z"Azis, M. I."https://zbmath.org/authors/?q=ai:azis.mohammad-ivan(no abstract)Thermocapillary motion of two viscous liquids in a cylindrical pipehttps://zbmath.org/1521.761192023-11-13T18:48:18.785376Z"Andreev, Viktor K."https://zbmath.org/authors/?q=ai:andreev.viktor-konstantinovich"Kuznetsov, Vladimir V."https://zbmath.org/authors/?q=ai:kuznetsov.vladimir-vSummary: A study is made of an invariant solution of the equations motion of a viscous heat-conducting fluids, which is treated as unidirectional motion in a circular pipe with a common interface under the action of the thermocapillary force. A priori estimates of the velocity and temperature are obtained. The steady state is determined and it is shown that, at larger times, this state is the limiting one. It was established that the thermocapillary effect with surface curvature can induce the return flow. Using Laplace transformation properties the exact analytical solution was constructed. Some examples of numerical reconstruction of the velocities fields depending on geometric and physical parameters were considered.\(k\)-ambiguity function in the framework of offset linear canonical transformhttps://zbmath.org/1521.940112023-11-13T18:48:18.785376Z"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younus"Dar, Aamir H."https://zbmath.org/authors/?q=ai:dar.aamir-hSummary: A new version of ambiguity function (AF) associated with the offset linear canonical transform (OLCT) is considered in this paper. This new version of AF coined as the \(k\)-AF associated with the OLCT \((k\)-AFOL) is defined based on the OLCT and the fractional instantaneous auto-correlation. A natural magnification effect characterized by the extra degrees of freedom of the OLCT and by a factor \(k\) on the frequency axis enables the \(k\)-AFOL to have flexibility to be used in cross-term reduction. Firstly, we defined the \(k\)-AF associated with the OLCT \((k\)-AFOL), and establish its relationship with the \(k\)-Wigner distribution in OLCT domain. Later on, we define the basic properties including the scaling, conjugate-symmetry, shifting, marginal and Moyal's formulae of \(k\)-AFOL in depth. The results show that \(k\)-AFOL can be viewed as one of the generalizations of the classical AF which has elegance, simplicity and flexibility in the frequency marginal property. The novelty of our paper lies in applications part, where we have shown how the proposed transform is used for the detection of single-component and bi-component linear frequency-modulated (LFM) signals.