Authors:Xiangcheng Zheng, Hong Wang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time [math], in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at [math] or have certain singularity at [math] like its constant-order fractional analogue if the variable order has a non-integer value at time [math]. Citation: Analysis and Applications PubDate: 2020-03-02T08:00:00Z DOI: 10.1142/S0219530520500013

Authors:Joost A. A. Opschoor, Philipp C. Petersen, Christoph Schwab Pages: 1 - 56 Abstract: Analysis and Applications, Ahead of Print. Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “[math]-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN size for univariate, piecewise Gevrey functions with point singularities are established. Combined with recent results on ReLU DNN approximation of rational, oscillatory, and high-dimensional functions, this corroborates that continuous, piecewise affine ReLU DNNs afford algebraic and exponential convergence rate bounds which are comparable to “best in class” schemes for several important function classes of high and infinite smoothness. Using composition of DNNs, we also prove that radial-like functions obtained as compositions of the above with the Euclidean norm and, possibly, anisotropic affine changes of co-ordinates can be emulated at exponential rate in terms of the DNN size and depth without the curse of dimensionality. Citation: Analysis and Applications PubDate: 2020-02-21T08:00:00Z DOI: 10.1142/S0219530519410136

Authors:Yuan-Hang Su, Wan-Tong Li, Fei-Ying Yang Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [math] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [math]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [math] is in a different range. In particular, for the case [math], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition. Citation: Analysis and Applications PubDate: 2020-02-14T08:00:00Z DOI: 10.1142/S0219530519500222

Authors:Gilles Blanchard, Nicole Mücke Pages: 1 - 14 Abstract: Analysis and Applications, Ahead of Print. We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales. Citation: Analysis and Applications PubDate: 2020-02-11T08:00:00Z DOI: 10.1142/S0219530519500258

Authors:Fan Jia, Jun Liu, Xue-Cheng Tai Pages: 1 - 19 Abstract: Analysis and Applications, Ahead of Print. Convolutional neural networks (CNNs) have achieved prominent performance in a series of image processing problems. CNNs become the first choice for dense classification problems such as semantic segmentation. However, CNNs predict the class of each pixel independently in semantic segmentation tasks, spatial regularity of the segmented objects is still a problem for these methods. Especially when given few training data, CNN could not perform well in the details, isolated and scattered small regions often appear in all kinds of CNN segmentation results. In this paper, we propose a method to add spatial regularization to the segmented objects. In our method, the spatial regularization such as total variation (TV) can be easily integrated into CNN network and it produces smooth edges and eliminate isolated points. We apply our proposed method to Unet and Segnet, which are well-established CNNs for image segmentation, and test them on WBC and CamVid datasets, respectively. The results show that the details of predictions are well improved by regularized networks. Citation: Analysis and Applications PubDate: 2020-02-10T08:00:00Z DOI: 10.1142/S0219530519410148

Authors:Annegret Glitzky, Matthias Liero, Grigor Nika Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. This work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A “generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss–Fermi statistics and mobility functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder’s fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis. Citation: Analysis and Applications PubDate: 2020-02-07T08:00:00Z DOI: 10.1142/S0219530519500246

Authors:Yichen Dai, Weiwei Hu, Jiahong Wu, Bei Xiao Pages: 1 - 44 Abstract: Analysis and Applications, Ahead of Print. The Littlewood–Paley decomposition for functions defined on the whole space [math] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [math] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [math]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [math] with [math] in the literature do not behave well on the Lebesgue space [math] except for [math]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [math]. As an application of the tools developed here, we study the periodic weak solutions of the [math]-dimensional Boussinesq equations with the fractional dissipation [math] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [math]-weak solutions for any [math] and the existence and uniqueness of the [math]-weak solutions when [math] for [math]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. Citation: Analysis and Applications PubDate: 2020-02-06T08:00:00Z DOI: 10.1142/S0219530519500234

Authors:Nan Liu, Boling Guo Pages: 1 - 46 Abstract: Analysis and Applications, Ahead of Print. The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou. Citation: Analysis and Applications PubDate: 2020-01-23T08:00:00Z DOI: 10.1142/S0219530519500210

Authors:Junping Li, Lan Cheng, Anthony G. Pakes, Anyue Chen, Liuyan Li Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. Large deviation rates are determined for quantities associated with a Markov branching process [math] having offspring mean [math] and split rate [math]. The principal quantities examined are [math] and [math], where [math] is the almost sure limit of an appropriately normed version of [math]. Modifications and conditional versions are examined. Some of this requires determination of the asymptotic behavior of harmonic moments [math]. Citation: Analysis and Applications PubDate: 2020-01-21T08:00:00Z DOI: 10.1142/S0219530519500209

Authors:Fusheng Lv, Jun Fan Pages: 1 - 18 Abstract: Analysis and Applications, Ahead of Print. Correntropy-based learning has achieved great success in practice during the last decades. It is originated from information-theoretic learning and provides an alternative to classical least squares method in the presence of non-Gaussian noise. In this paper, we investigate the theoretical properties of learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and correntropy loss. By choosing an appropriate scale parameter of Gaussian kernel, we show the polynomial decay of approximation error under a Sobolev smoothness condition. In addition, we employ a tight upper bound for the uniform covering number of Gaussian RKHS in order to improve the estimate of sample error. Based on these two results, we show that the proposed algorithm using varying Gaussian kernel achieves the minimax rate of convergence (up to a logarithmic factor) without knowing the smoothness level of the regression function. Citation: Analysis and Applications PubDate: 2019-11-11T06:55:29Z DOI: 10.1142/S0219530519410124

Authors:Chenxi Chen, Yunmei Chen, Xiaojing Ye Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We consider a class of convex decentralized consensus optimization problems over connected multi-agent networks. Each agent in the network holds its local objective function privately, and can only communicate with its directly connected agents during the computation to find the minimizer of the sum of all objective functions. We propose a randomized incremental primal-dual method to solve this problem, where the dual variable over the network in each iteration is only updated at a randomly selected node, whereas the dual variables elsewhere remain the same as in the previous iteration. Thus, the communication only occurs in the neighborhood of the selected node in each iteration and hence can greatly reduce the chance of communication delay and failure in the standard fully synchronized consensus algorithms. We provide comprehensive convergence analysis including convergence rates of the primal residual and consensus error of the proposed algorithm, and conduct numerical experiments to show its performance using both uniform sampling and important sampling as node selection strategy. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:24Z DOI: 10.1142/S0219530519410082

Authors:Liren Huang, Chunguang Liu, Lulin Tan, Qi Ye Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. In this paper, we generalize the representer theorems in Banach spaces by the theory of nonsmooth analysis. The generalized representer theorems assure that the regularized learning models can be constructed by the nonconvex loss functions, the generalized training data, and the general Banach spaces which are nonreflexive, nonstrictly convex, and nonsmooth. Specially, the sparse representations of the regularized learning in 1-norm reproducing kernel Banach spaces are shown by the generalized representer theorems. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:23Z DOI: 10.1142/S0219530519410100

Authors:L. Agud, J. M. Calabuig, E. A. Sánchez Pérez Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. Let [math] be a finite measure space and consider a Banach function space [math]. Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels [math] that define integration maps [math] between Köthe–Bochner spaces of Hilbert-valued functions [math] We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces [math] and [math] — where [math] — under the assumption of [math]-concavity of [math] Equivalently, a new kernel obtained by multiplying [math] by scalar functions can be given in such a way that the kernel operator is defined from [math] to [math] in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:18Z DOI: 10.1142/S0219530519500179

Authors:Wei Yan, Yongsheng Li, Jianhua Huang, Jinqiao Duan Pages: 1 - 54 Abstract: Analysis and Applications, Ahead of Print. The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation ut + Dx α∂ xu + ∂x−1∂ y2u + 1 2∂x(u2) = 0,α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces [math] with [math] and [math]. Second, we prove that the Cauchy problem is globally well-posed in [math] with [math] if [math]. Finally, we show that the Cauchy problem is globally well-posed in [math] with [math] if [math] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352]. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:18Z DOI: 10.1142/S0219530519500180

Authors:Bao-Huai Sheng, Jian-Li Wang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. [math]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [math]-functionals in [math] spaces with [math]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [math]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:17Z DOI: 10.1142/S0219530519500192

Authors:Xin Zhong Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. We study the Cauchy problem of nonhomogeneous magneto-micropolar fluid system with zero density at infinity in the entire space [math]. We prove that the system admits a unique local strong solution provided the initial density and the initial magnetic field decay not too slowly at infinity. In particular, there is no need to require any Choe–Kim type compatibility condition for the initial data. Citation: Analysis and Applications PubDate: 2019-09-09T08:02:45Z DOI: 10.1142/S0219530519500167

Authors:Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [math] convergence, and quantitative estimates with respect to the [math] norm, [math]-functionals and [math]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested. Citation: Analysis and Applications PubDate: 2019-08-19T06:31:09Z DOI: 10.1142/S0219530519500155

Authors:Ingo Gühring, Gitta Kutyniok, Philipp Petersen Pages: 1 - 57 Abstract: Analysis and Applications, Ahead of Print. We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently approximate Sobolev regular functions if the approximation error is measured with respect to weaker Sobolev norms. In this context, we first establish upper approximation bounds by ReLU neural networks for Sobolev regular functions by explicitly constructing the approximate ReLU neural networks. Then, we establish lower approximation bounds for the same type of function classes. A trade-off between the regularity used in the approximation norm and the complexity of the neural network can be observed in upper and lower bounds. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations. Citation: Analysis and Applications PubDate: 2019-08-19T06:31:08Z DOI: 10.1142/S0219530519410021

Authors:Palle E. T. Jorgensen, James F. Tian Pages: 1 - 31 Abstract: Analysis and Applications, Ahead of Print. We study reduction schemes for functions of “many” variables into system of functions in one variable. Our setting includes infinite dimensions. Following Cybenko–Kolmogorov, the outline for our results is as follows: We present explicit reduction schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in “many” variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation. Citation: Analysis and Applications PubDate: 2019-08-05T06:44:19Z DOI: 10.1142/S021953051941001X

Authors:Ahmed Abdeljawad, Sandro Coriasco, Joachim Toft Pages: 1 - 61 Abstract: Analysis and Applications, Ahead of Print. We deduce one-parameter group properties for pseudo-differential operators [math], where [math] belongs to the class [math] of certain Gevrey symbols. We use this to show that there are pseudo-differential operators [math] and [math] which are inverses to each other, where [math] and [math]. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorphisms between them. For each weight functions [math] moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) [math] is an isomorphism from [math] to [math] for every [math]. Citation: Analysis and Applications PubDate: 2019-07-30T02:40:53Z DOI: 10.1142/S0219530519500143

Authors:Wei Shen, Zhenhuan Yang, Yiming Ying, Xiaoming Yuan Pages: 1 - 41 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function depending on pairs of instances among which notable examples are bipartite ranking, metric learning, area under ROC curve (AUC) maximization and minimum error entropy (MEE) principle. Our contribution is twofolded. Firstly, we establish the stability results for SGD for pairwise learning in the convex, strongly convex and non-convex settings, from which generalization errors can be naturally derived. Secondly, we establish the trade-off between stability and optimization error of SGD algorithms for pairwise learning. This is achieved by lower-bounding the sum of stability and optimization error by the minimax statistical error over a prescribed class of pairwise loss functions. From this fundamental trade-off, we obtain lower bounds for the optimization error of SGD algorithms and the excess expected risk over a class of pairwise losses. In addition, we illustrate our stability results by giving some specific examples of AUC maximization, metric learning and MEE. Citation: Analysis and Applications PubDate: 2019-07-24T03:00:36Z DOI: 10.1142/S0219530519400062

Authors:Huiping Li, Song Li, Yu Xia Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the noisy phase retrieval problem which occurs in many different areas of science and physics. The PhaseMax algorithm is an efficient convex method to tackle with phase retrieval problem. On the basis of this algorithm, we propose two kinds of extended formulations of the PhaseMax algorithm, namely, PhaseMax with bounded and non-negative noise and PhaseMax with outliers to deal with the phase retrieval problem under different noise corruptions. Then we prove that these extended algorithms can stably recover real signals from independent sub-Gaussian measurements under optimal sample complexity. Specially, such results remain valid in noiseless case. As we can see, these results guarantee that a broad range of random measurements such as Bernoulli measurements with erasures can be applied to reconstruct the original signals by these extended PhaseMax algorithms. Finally, we demonstrate the effectiveness of our extended PhaseMax algorithm through numerical simulations. We find that with the same initialization, extended PhaseMax algorithm outperforms Truncated Wirtinger Flow method, and recovers the signal with corrupted measurements robustly. Citation: Analysis and Applications PubDate: 2019-07-24T03:00:33Z DOI: 10.1142/S0219530519400049

Authors:Michel Chipot, Jérôme Droniou, Gabriela Planas, James C. Robinson, Wei Xue Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We treat three problems on a two-dimensional “punctured periodic domain”: we take [math], where [math] and [math] is the closure of an open connected set that is star-shaped with respect to [math] and has a [math] boundary. We impose periodic boundary conditions on the boundary of [math], and Dirichlet boundary conditions on [math]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [math], and examine the behavior of solutions as [math]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [math] with periodic boundary conditions. Citation: Analysis and Applications PubDate: 2019-07-05T08:54:22Z DOI: 10.1142/S0219530519500118

Authors:Ben Duan, Zhen Luo, Yan Zhou Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established. Citation: Analysis and Applications PubDate: 2019-07-05T08:54:22Z DOI: 10.1142/S021953051950012X

Authors:Mourad E. H. Ismail Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [math]-Hermite, [math]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials. Citation: Analysis and Applications PubDate: 2019-06-14T06:00:54Z DOI: 10.1142/S0219530519500088

Authors:Ana F. Loureiro, Walter Van Assche Pages: 1 - 62 Abstract: Analysis and Applications, Ahead of Print. We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [math]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [math]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [math]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni. Citation: Analysis and Applications PubDate: 2019-06-14T06:00:53Z DOI: 10.1142/S0219530519500106

Authors:Yu-Tian Li, Xiang-Sheng Wang, Roderick Wong Pages: 1 - 34 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the asymptotic behavior of the Wilson polynomials [math] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [math] is fixed and (ii) when the variable is rescaled as [math] with [math]. Case (ii) has two subcases, namely, (a) zero-free zone ([math]) and (b) oscillatory region [math]. Corresponding results are also obtained in these cases (iii) when [math] lies in a neighborhood of the transition point [math], and (iv) when [math] is in the neighborhood of the transition point [math]. The expansions in the last two cases hold uniformly in [math]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations. Citation: Analysis and Applications PubDate: 2019-05-16T02:23:12Z DOI: 10.1142/S0219530519500076

Authors:Elena Cordero, S. Ivan Trapasso Pages: 1 - 38 Abstract: Analysis and Applications, Ahead of Print. The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role. Citation: Analysis and Applications PubDate: 2019-03-25T02:23:53Z DOI: 10.1142/S0219530519500052