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Authors:Andreas Debrouwere, Bojan Prangoski Pages: 1 - 50 Abstract: Analysis and Applications, Ahead of Print. We obtain Gabor frame characterizations of modulation spaces defined via a broad class of translation-modulation invariant Banach spaces of distributions. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterized by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalizes and unifies several fundamental results for the classical modulation spaces [math] and the amalgam spaces [math]. Due to the absence of solidity assumptions on the Banach spaces defining these generalized modulation spaces, the methods used for the spaces [math] (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution. Citation: Analysis and Applications PubDate: 2022-12-28T08:00:00Z DOI: 10.1142/S0219530522500178

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Authors:Zhan Yu Pages: 1 - 41 Abstract: Analysis and Applications, Ahead of Print. We study nonconvex (composite) optimization and learning problems where the decision variables can be split into blocks of variables. Random block projection is a popular technique to handle this kind of problem for its remarkable reduction of the computational cost from the projection. This powerful method has not been well proposed for the situation that first-order information is prohibited and only zeroth-order information is available. In this paper, we propose to develop different classes of zeroth-order stochastic block coordinate type methods. Zeroth-order block coordinate descent (ZS-BCD) is proposed for solving unconstrained nonconvex optimization problem. For composite optimization, we establish the zeroth-order stochastic block mirror descent (ZS-BMD) and its associated two-phase method to achieve the complexity bound for the composite optimization problem. Furthermore, we also establish zeroth-order stochastic block coordinate conditional gradient (ZS-BCCG) method for nonconvex (composite) optimization. By implementing ZS-BCCG method, in each iteration, only (approximate) linear programming subproblem needs to be solved on a random block instead of a rather costly projection subproblem on the whole decision space, in contrast to the existing traditional stochastic approximation methods. In what follows, an approximate ZS-BCCG method and corresponding two-phase ZS-BCCG method are proposed. This is also the first time that a two-phase BCCG method has been developed to carry out the complexity analysis of nonconvex composite optimization problem. Citation: Analysis and Applications PubDate: 2022-12-28T08:00:00Z DOI: 10.1142/S021953052250018X

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Authors:ntonino De Martino, Kamal Diki Pages: 1 - 42 Abstract: Analysis and Applications, Ahead of Print. This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms. Citation: Analysis and Applications PubDate: 2022-12-21T08:00:00Z DOI: 10.1142/S0219530522500191

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Authors:Jianwen Huang, Feng Zhang, Jianjun Wang, Hailin Wang, Xinling Liu, Jinping Jia Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with double noise model is larger by a factor to [math], where [math] are the dimensions of the matrix and [math] is the number of measurements. Additionally, this paper discusses the reconstruction of low-rank matrices in the setting, presents sufficient conditions based on the associating null space property to guarantee the robust recovery and obtains the number of measurements. Furthermore, for the non-Gaussian noise scenario, we further explore it and give the corresponding result. The simulation experiments conducted, on the one hand show effect of noise variance on recovery performance, on the other hand demonstrate the verifiability of the proposed model. Citation: Analysis and Applications PubDate: 2022-11-30T08:00:00Z DOI: 10.1142/S0219530522500154

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Authors:Yirui Zhao, Dachun Yang, Yangyang Zhang Pages: 1 - 92 Abstract: Analysis and Applications, Ahead of Print. In this paper, the authors introduce a class of mixed-norm Herz spaces, [math], which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on [math]. Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on [math]. As applications, the authors develop various real-variable theory of Hardy spaces associated with [math] by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of [math] and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem. Citation: Analysis and Applications PubDate: 2022-11-30T08:00:00Z DOI: 10.1142/S0219530522500166

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Authors:Hà Quang Minh Pages: 1 - 57 Abstract: Analysis and Applications, Ahead of Print. This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is strictly weaker than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert–Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the kernel Gaussian–Sinkhorn divergence, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert–Schmidt norm convergence, we obtain dimension-independent convergence rates for finite sample approximations of the kernel Gaussian–Sinkhorn divergence, of the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent, is exponentially faster than the worst case scenario in the literature. Citation: Analysis and Applications PubDate: 2022-11-26T08:00:00Z DOI: 10.1142/S0219530522500142

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Authors:Mingyang Gu, Song Li, Junhong Lin Pages: 1 - 15 Abstract: Analysis and Applications, Ahead of Print. We investigate the problems of reconstruction signals’ distinct subcomponents, that are approximately sparse in terms of morphologically different dictionaries, from a few noisy linear measurements. We show that under common assumptions on a restricted isometric property adapted to the composed dictionary and a mutual coherence condition between the two different dictionaries, the two distinct subcomponents can be stably and approximately recovered via the unconstrained [math]-split analysis, with an error bound that is in general unimprovable for the Gaussian noises. Citation: Analysis and Applications PubDate: 2022-09-16T07:00:00Z DOI: 10.1142/S0219530522500130

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Authors:Yuesheng Xu Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter [math] multiple of the [math] norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction. Citation: Analysis and Applications PubDate: 2022-07-23T07:00:00Z DOI: 10.1142/S0219530522500105

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Authors:Aksel Bergfeldt, Wolfgang Staubach Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. We prove the global regularity of multilinear Schrödinger integral operators with non-degenerate phase function that are associated to nonlinear Schrödinger equations, with Banach domain and target spaces. Citation: Analysis and Applications PubDate: 2022-07-21T07:00:00Z DOI: 10.1142/S0219530522500099

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Authors:Brahim Alouini, Hichem Hajaiej Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. The purpose of this paper is to study the dynamics of solutions to an extended Gross–Pitaevskii equation that models the formation of droplets in a dipolar Bose–Einstein condensate (BEC). The formation of these droplets has been recently discovered by driving the BEC into the strongly dipolar regime. Surprisingly, instead of collapsing, the system formed stable droplets. So far, no rigorous mathematical explanation has been proved. To the best of our knowledge, only experimental results have been obtained. The goal of this paper is to validate this breakthrough discovery. Many predictions/conjectures properties of these droplets have been stated by some research groups in physics and engineering. In particular, it has been claimed that the stability of these droplets is a consequence of the presence of the damping term in the extended Gross–Pitaevskii equation under study. This term describes the three-body loss process. To accurately model the dynamics of formation of these droplets, it is necessary to consider a time-dependent harmonic trapping potential as well as other terms with different types of nonlinearity among them that describe the Lee–Huang–Yang (LHY). This presents some challenges that will be solved in this paper. Citation: Analysis and Applications PubDate: 2022-07-21T07:00:00Z DOI: 10.1142/S0219530522500117

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Authors:Joachim Toft, Divyang G. Bhimani, Ramesh Manna Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. We deduce trace properties for modulation spaces (including certain Wiener-amalgam spaces) of Gelfand–Shilov distributions.We use these results to show that [math]dos with amplitudes in suitable modulation spaces, agree with normal type [math]dos whose symbols belong to (other) modulation spaces. In particular we extend earlier trace results for modulation spaces, to include quasi-Banach modulation spaces. We also apply our results to extend earlier results on Schatten-von Neumann and nuclear properties for [math]dos with amplitudes in modulation spaces. Citation: Analysis and Applications PubDate: 2022-07-05T07:00:00Z DOI: 10.1142/S0219530522500063

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Authors:Hans Volkmer Pages: 1 - 11 Abstract: Analysis and Applications, Ahead of Print. The asymptotic expansion of the generalized hypergeometric function [math] as [math] involves a coefficient sequence [math]. Explicit formulas are given for this sequence when [math]. The result is based on an integral representation of the generalized hypergeometric function that allows application of Watson’s lemma. Citation: Analysis and Applications PubDate: 2022-06-28T07:00:00Z DOI: 10.1142/S0219530522500087

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Authors:Jiaohui Xu, Tomás Caraballo Pages: 1 - 37 Abstract: Analysis and Applications, Ahead of Print. This paper deals with fractional stochastic nonlocal partial differential equations driven by multiplicative noise. We first prove the existence and uniqueness of solution to this kind of equations with white noise by applying the Galerkin method. Then, the existence and uniqueness of tempered pullback random attractor for the equation are ensured in an appropriate Hilbert space. When the fractional nonlocal partial differential equations are driven by colored noise, which indeed are approximations of the previous ones, we show the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as [math]. Citation: Analysis and Applications PubDate: 2022-06-23T07:00:00Z DOI: 10.1142/S0219530522500075

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Authors:Víctor Almeida, Jorge J. Betancor, Lourdes Rodríguez-Mesa Pages: 1 - 31 Abstract: Analysis and Applications, Ahead of Print. By [math] we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential [math] defined in [math]. In this paper, we establish weighted [math]-inequalities for the maximal, variation, oscillation and jump operators associated with [math], where [math] and [math] denotes the Weyl fractional derivative. The range of values [math] that works is different when [math] and when [math]. Citation: Analysis and Applications PubDate: 2022-05-13T07:00:00Z DOI: 10.1142/S0219530522500038

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Authors:Xianguo Geng, Wenhao Liu, Kedong Wang, Mingming Chen Pages: 1 - 37 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation. Based on the corresponding Lax pair, the original Riemann–Hilbert problem is constructed by introducing some spectral function transformations and variable transformations, and the solution of the complex nonlinear transverse oscillation equation is transformed into the solution of the resulted Riemann–Hilbert problem. Various Deift–Zhou contour deformations and the motivation behind them are given, from which the original Riemann–Hilbert problem is further transformed into a solvable model problem. The long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation is obtained by using the nonlinear steepest decent method. Citation: Analysis and Applications PubDate: 2022-05-10T07:00:00Z DOI: 10.1142/S021953052250004X

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Authors:Yuhui Chen, Wei Luo, Zheng-an Yao Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider an initial-value problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) model. This model is regard to the dynamic mechanical properties of polymeric fluids. We shall mainly investigate two different cases, one is without damping on the stress tensor (i.e. [math]) and the other is [math]. Under some suitable assumptions on the initial data, we show that the solutions of the PTT model will blow up in finite time. This result also reveals a special solution. Furthermore, we prove the global existence and the optimal time decay rates of strong solutions to the PTT model when the initial data close enough to this special solution. In particular, it also works for the simpler case of the Oldroyd-B model. Citation: Analysis and Applications PubDate: 2022-05-10T07:00:00Z DOI: 10.1142/S0219530522500051

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Authors:Liang Chen, Yang Wang, Haizhang Zhang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. The reconstruction of a bandlimited function from its finite sample data is fundamental in signal analysis. It is well known that oversampling of a bandlimited function leads to exponential convergence in its reconstruction. A simple and efficient Gaussian regularized Shannon sampling formula has been proposed in G. W. Wei [Quasi wavelets and quasi interpolating wavelets, Chem. Phys. Lett. 296 (1998) 215–222] with such an exponential convergence ability. We show that all hyper-Gaussian regularized formulas share this desired property. The analysis is built on estimates on the Fourier transform of the hyper-Gaussian functions. We also establish error bounds for the reconstruction of derivatives of a univariate bandlimited function, and for multivariate bandlimited functions. Citation: Analysis and Applications PubDate: 2022-01-22T08:00:00Z DOI: 10.1142/S0219530521500342

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Authors:Long Huang, Ferenc Weisz, Dachun Yang, Wen Yuan Pages: 279 - 328 Abstract: Analysis and Applications, Volume 21, Issue 02, Page 279-328, March 2023. Let [math], [math] be the mixed-norm Lebesgue space, and [math] an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator [math] from [math] to itself or to the weak mixed-norm Lebesgue space [math] under some sharp assumptions on [math] and [math], the authors show that the [math]-mean of [math] converges to [math] almost everywhere over the diagonal if the Fourier transform [math] of [math] belongs to some mixed-norm homogeneous Herz space [math] with [math] being the conjugate index of [math]. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of [math]-means to the unrestricted case. Finally, the authors show that the [math]-mean of [math] converges over the diagonal to [math] at all its [math]-Lebesgue points if and only if [math] belongs to [math], and a similar conclusion also holds true for the unrestricted convergence at strong [math]-Lebesgue points. Observe that, in all these results, those Herz spaces to which [math] belongs prove to be the best choice in some sense. Citation: Analysis and Applications PubDate: 2021-06-29T07:00:00Z DOI: 10.1142/S0219530521500135 Issue No:Vol. 21, No. 02 (2021)