Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Convolution operators on expanding polyhedra: The limits of norms of inverse operators and pseudospectra

We consider the limits of norms of inverse operators and pseudospectra convolution operators on expanding polyhedra. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Hypoelliptic Jacobi Convolution Operators on Schwartz Distributions

In this paper we characterize the hypoellipticity of Jacobi convolution operators on Schwartz distributions. In the proof of the main result of this paper the positivity ofthe convolution structure for the inverse of the Jacobi transform plays an essential role. We also study hypoelliptic convolution equations on Chébli-Triméche hypergroups.Mathematics Subject Classification 2000: 46F12     

球面卷积算子逼近

研究d维欧氏空间R~d中单位球面上卷积算子的逼近问题.利用球面乘子理论以及K-泛函与光滑模等价关系,建立一类球面卷积算子逼近的正、逆定理.特别地,给出了逼近的强型逆向不等式,从而揭示了该类球面卷积算子的本质逼近阶.此外,作为应用,给出了球面Jackson-Matsuoka卷积算子与Abel-Poisson卷积算子逼近上、下界的相同阶估计.     

Perturbation Stability of Coherent Riesz Systems under Convolution Operators

We study the orthogonal perturbation of various coherent function systems (Gabor systems, Wilson bases, and wavelets) under convolution operators. This problem is of key relevance in the design of modulation signal sets for digital communication over time-invariant channels. Upper and lower bounds on the orthogonal perturbation are formulated in terms of spectral spread and temporal support of the prototype, and by the approximate design of worst-case convolution kernels. Among the considered bases, the Weyl–Heisenberg structure which generates Gabor systems turns out to be optimal whenever the class of convolution operators satisfies typical practical constraints.     

Real Interpolation of Compact Bilinear Operators

We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kühn and Schonbek. We work with the assumption that \(T:(A_0+A_1) \times (B_0+B_1) \longrightarrow E_0+E_1\) is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderón–Zygmund operators.     

卷积算子的交换子

利用Fourier变换估计和对算子的局部性质做精细的分析,建立与一类卷积算子交换子相关的一些算子的LP有界性结果．作为这些结果的应用,得到了与球平均算子交换子相关的离散极大算子以及一类低维集上奇异积分算子交换子等的LP有界性．     

Convolution Operators on Discrete Hardy Spaces

We establish the connection between the boundedness of convolution operators on H^p(ℝ^N) and some related operators on H^p(ℤ^N). The results we obtain here extend the already known for L^p spaces with p > 1. We also study similar results for maximal operators given by convolution with the dilation of a fixed kernel. Our main tools are some known results about functions of exponential type already presented in [BC1] that, in particular, allow us to prove a sampling theorem for functions of exponential type belonging to Hardy spaces     

Bessel Potential Operators for Canonical Lipschitz Domains

We construct convolution operators which define isomorphisms between SOBOLEV spaces of distributions supported on a canonical LIPSCHITZ domain. These operators are used for reduction of order of WIENER -HOPF equations or pseudodifferential equations on a canonical LIPSCHITZ domain.     

Reduction of Weighted Bilinear Inequalities with Integration Operators on the Cone of Nondecreasing Functions

We present the reduction methods for characterizing bilinear weighted inequalities on the Lebesgue cones of nondecreasing functions on a half-axis with integration operators.     

On Integral Operators of Generalized Wiener-Hopf-Type

A class of singular integral operators on the positive halfaxis, constituted by the one-sided HILBERT transformation, the WIENER -HOPF operators, the multiplicative convolution operators, and some multiplication operators, generated by continuous functions, are studied with BANACH algebra methods. With the help of symbol functions the FREDHOLM operators wil be characterized. Moreover one gets information about an index formula and FREDHOLM inverses.     

Integrability Properties of Maximal Convolution Operators

In this article we study integrability properties of maximal convolution operators satisfying restricted weak type (p, p) estimates.     

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators

We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L₂(0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Fixed-Point Methods for a Certain Class of Operators

We introduce in this paper a new class of nonlinear operators, which contains, among others, the class of operators with semimonotone additive inverse and also the class of nonexpansive mappings. We study this class and discuss some of its properties. Then we present iterative procedures for computing fixed points of operators in this class, which allow for inexact solutions of the subproblems and relative error criteria. We prove weak convergence of the generated sequences in the context of Hilbert spaces. Strong convergence is also discussed.     

Singular Measures and Convolution Operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.     

Convolution Integral Operators

We establish some algebraic, metric, and spectral properties of convolution integral operators in L₂(?^N).     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Convergence and Rate of Approximation in BVΦ for a Class of Integral Operators

We obtain estimates and convergence results with respect to ?-variation in spaces BV_Φ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.     

Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.