The Multichannel Deconvolution Problem: A Discrete Analysis

We study the multichannel deconvolution problem (MDP) in a discrete setting by developing the theory for converting the method used in the continuous setting in [36]. We give a method for solving the MDP when the convolvers are characteristic functions, derive the explicit form of the linear system, and obtain an upper bound on the condition number of the system in a particular case. We compare the Schiske reconstruction [28] to our solution in the discrete setting, and give an explicit formula for the corresponding error. We then give the algorithm for solving the general MDP and discuss in detail the local reconstruction aspects of the problem. Finally, we describe a method for improving the reconstruction by regularization and give some explicit estimates on error bounds in the presence of noise.     

N-dimensional characteristic problem for the Mangeron equation of non-integer order

In the paper we consider a n-dimensional characteristic problem for a certain partial differential equation of non-integer order.We prove the existence and uniqueness of a solution of the problem in the spaces of integrable and continious functions, respectively. Morever, we give sufficient conditions under which the set of solutions is not empty and relatively compact in the space of integrable functions     

Exact Solutions of Some Discrete Deconvolution Problems

In this work, we study the existence of solutions of the deconvolution problems in the discrete setting. More precisely, we prove the existence of solutions of the discrete multichannel deconvolution problems DMDP with convolvers being the characteristic functions of finite sets of positive integers. Also, we provide the reader with a simple method and a fast algorithm for finding the closed forms of the discrete deconvolvers with minimal supports that constitute exact solutions of the DMDP. Moreover, we show that unlike the singular value decomposition scheme, the multichannel deconvolution scheme based on the use of these discrete deconvolvers is not very sensitive to small 2-norm perturbation of the data. Finally, we show how to generalize our method for solving the 2-D version of the DMDP.     

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.     

Opérateurs d'extension linéaires explicites dans des intersections de classes ultradifférentiables

B. S. Mityagin proved that the Chebyshev polynomials form a Schauder basis of the space of C ^∞ functions on the interval [–1,1]. Whereof he deduced an explicit continuous linear extension operator. These results were extended, by A. Goncharov, to compact sets without Markov's property. On the reverse, M. Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this paper, we generalize these works to the intersections of ultradifferentiable classes of functions built on the model of the non quasianalytic intersection of Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type.     

Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

We obtain Gevrey regular mild solutions to the incompressible Navier–Stokes equations in R ⁿ with periodic boundary condition in a subset of the variables. The method is based on an extension of Young's convolution inequality in weighted Lebesgue spaces of measurable functions defined on locally compact abelian groups. This generalizes and provides a unified treatment of the Gevrey regularity result of Foias and Temam in the space periodic case and those of Le Jan and Sznitman and Lemarié–Rieusset in the whole space with no boundary.     

Opérateurs linéaires continus d'extension dans des intersections de classes ultradifférentiables

Mityagin proved that the Tchebyshev polynomials form a Schauder basis of the space of C^∞ functions on the interval [?1,1]. Thus, he deduced an explicit continuous linear extension operator. These results were extended, by Goncharov, to compact sets which do not satisfy the Markov's inequalities. On the other hand, Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this Note, we generalize these works to ultradifferentiable classes of functions built on the model of the intersection of non quasi-analytic Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type. To cite this article: P. Beaugendre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Universality limits for random matrices and de Branges spaces of entire functions

We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [−1,1].     

On the Semi-norm of Radial Basis Function Interpolants

Radial basis function interpolation has attracted a lot of interest in recent years. For popular choices, for example thin plate splines, this problem has a variational formulation, i.e. the interpolant minimizes a semi-norm on a certain space of radial functions. This gives rise to a function space, called the native space. Every function in this space has the property that the semi-norm of an arbitrary interpolant to this function is uniformly bounded. In applications it is of interest whether a sufficiently smooth function belongs to the native space. In this paper we give sufficient conditions on the differentiability of a function with compact support, in the case of cubic, linear and thin plate splines. In the case of multiquadrics and Gaussian functions, it is shown that the only compactly supported function that satisfies these conditions is identically zero.     

Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Gevrey class regularity for the viscous Camassa–Holm equations

We prove that the solutions of the three-dimensional viscous Camassa–Holm equations with periodic boundary conditions are analytic in time with values in a Gevrey class of functions (for the space variable).     

Gevrey空间中抽象柯西-柯瓦列夫斯卡娅定理:能量方法

本文研究了非线性柯西问题的适定性问题.利用经典的能量法和抽象柯西-柯瓦列夫斯卡娅定理,得到非线性柯西问题在Gevrey空间中是适定的.推广了已有文献在非线性柯西问题适定性方面的研究.     

On the well-posedness of weakly hyperbolic equations with time-dependent coefficients

In this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots. We also derive the corresponding well-posedness results in the space of Gevrey Beurling ultradistributions.     

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle(see Qiu, 2014), we prove a new version of Ekeland variational principle(briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results(no matter scalar-valued case,or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.     

Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the “complete” attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when there are no exact ordinary solutions, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactifier (an example is given showing that the similar property may be absent for other variants of the extension of the original problem).     

Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well-posed. So we find an additional assumption on the state-dependent delay function to guarantee the well-posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor.     

Equivalent forms of Levi condition

In this paper we consider hyperbolic differential operators with characteristic roots of constant multiplicity and we prove the equivalence of some conditions, called Levi conditions, for the correctness of the Cauchy problem inC ^∞ and in Gevrey classes.

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Sommario In questo articolo prendiamo in considerazione operatori differenziali iperbolici con caratteristiche di molteplicità costante e dimostriamo l’equivalenza di alcune condizioni, note come condizioni di Levi, necessarie e sufficienti per la buona positura del problema di Cauchy nelle classiC ^∞ e Gevrey.

     

Bounds for a domain containing all compact invariant sets of the system describing the laser–plasma interaction

In this paper we consider the localization problem of compact invariant sets of the system describing the laser–plasma interaction. We establish that this system has an ellipsoidal localization for simple restrictions imposed on its parameters. Then we improve this localization by applying other localizing functions. In addition, we give sufficient conditions under which the origin is the unique compact invariant set.