Constructing radial kernels with higher-order generalized Strang-Fix conditions

The paper provides an approach for constructing multivariate radial kernels satisfying higher-order generalized Strang-Fix conditions from a given univariate generator. There are three key features of the approach. First, the kernels are explicitly expressed only by the derivatives of the f-form of the generator without computing any Fourier transforms. Second, it includes the radial kernels with the highest-order generalized Strang-Fix conditions. Finally, it requires only computing univariate derivatives of the f-form. Therefore, the approach is simple, efficient and easy to implement. As examples, the paper constructs radial kernels from some commonly used generators, including the Gaussian functions, the inverse multiquadric functions and compactly supported positive definite functions.     

The study of dual integral equations with generalized Legendre functions

Closed form solutions for dual integral equations involving generalized Legendre functions as kernels are obtained. Connected to these dual integral equations an exact solution for dual integral equations involving sine functions as kernels is also obtained. Properties of generalized Legendre functions and the inversion theorem for the generalized Mehler-Fock transforms are used to obtain the solution of dual integral equations     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

Norm Inequalities for Fractional Integral Operators on Generalized Weighted Morrey Spaces

Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds,the fractional integral operators with rough kernels and fractional maximal operators with rough kernels as special cases,we prove that if these operators are bounded on weighted Lebesgue spaces and satisfy some local pointwise control,then these operators and the commutators of these operators with a BMO functions are also bounded on generalized weighted Morrey spaces.     

Generalized integral operators and Schwartz kernel type theorems

In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz' result is contained in our main theorem.     

Nonlinear Commutation Relations: Representations by Point-Supported Operators

We present a class of non-Lie commutation relations admitting representations by point-supported operators (i.e., by operators whose integral kernels are generalized point-supported functions). For such relations we construct all operator-irreducible representations (up to equivalence). Each representation is realized by point-supported operators in the Hilbert space of antiholomorphic functions. We show that the reproducing kernels of these spaces can be represented via hypergeometric series and the theta function, as well as via their modifications. We construct coherent states that intertwine abstract representations with irreducible representations.     

Estimation of the uniform modulus of continuity of the generalized Bessel potential

We study generalized Bessel potentials constructed by means of convolutions of functions with kernels that generalize the classical Bessel-Macdonald kernels. In contrast to the classical case, nonpower singularities of kernels in a neighborhood of the origin are admitted. The integral properties of functions are characterized in terms of decreasing rearrangements. The differential properties of potentials are described by the kth-order moduli of continuity in the uniform norm. An order-sharp upper estimate is established for the modulus of continuity of a potential. Such estimates play an important role in the theory of function spaces. They allow one to establish sharp embedding theorems for potentials, find majorants of the moduli of continuity, and estimate the approximation numbers of embedding operators.     

Approximation by generalized Shannon sampling operators generated by band-limited kernels

In this paper we consider some generalized Shannon sampling operators, which are defined by band–limited kernels. In particular, we use dilated an averaged versions of some previously known kernels. We give also some examples of using sampling operators in imaging applications. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

We present a new technique for explicit construction of Cauchy kernels and Cauchy integral representations for a class of generalized analytic functions and p-analytic functions.     

Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions

We establish sharp estimates for the best approximations of convolution classes by entire functions of exponential type. To obtain these estimates, we propose a new method for testing Nikol’ski?-type conditions which is based on kernel periodization with an arbitrarily large period and ensuing passage to the limit. As particular cases, we obtain sharp estimates for approximation of convolution classes with variation diminishing kernels and generalized Bernoulli and Poisson kernels.     

Some remarks on Newton–Sabatier methods

It is shown how Newton–Sabatier methods (arising in inverse scattering at fixed energy) can be related to spectral measures and typically when they correspond to regular potentials. A number of spectral formulae for various transmutation kernels are also given in terms of general Kontorovi?–Lebedev theory and connections to generating functions, generalized orthogonal polynomials, etc. are indicated.     

具有核的态射广义Moore-Penrose逆

本文研究了具有核的态射广义Moore-Penrose逆.利用加边态射的可逆性,获得了态射广义Moore-Penrose逆存在的一些新的充要条件及相关表达式,推广了态射Moore-Penrose逆的相应结论.     

The principal kernels of semifields of continuous positive functions

This work is devoted to the study of kernel lattices of semifields of continuous positive functions defined on some topological space. It is established that they are lattices with a pseudo-complement. New characterizations of the following properties of topological spaces are obtained in terms of kernels, predominantly principal kernels, and semifields of continuous functions: to be an F-space, to be a P-space, basic and extremal disconnectedness, pseudo-compactness, and finiteness.     

Certain integral transforms with the Macdonald function Kν(z) in the kernels

Four integral transforms are considered with the Macdonald function in their kernels. Existence and inversion theorems are proved and Parseval equalities are established in a Hilbert space, in a Wiener normed ring, and in the space of generalized Gelfand-Shilov functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 990–993, July, 1990.     

Asymptotic formulae for multivariate Kantorovich type generalized sampling series

In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines.     

Extended Kac-Akhiezer formulae and the Fredholm determinant of finite section Hilbert-Schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators onL ₂-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels, which are not necessarily of convolution nature and for domains in ? ⁿ .     

Functions orthogonal to polynomials and their application in axially symmetric problems in physics

We investigate properties of sets of functions comprising countably many elements A_n such that every function A_n is orthogonal to all polynomials of degrees less than n. We propose an effective method for solving Fredholm integral equations of the first kind whose kernels are generating functions for these sets of functions. We study integral equations used to solve some axially symmetric problems in physics. We prove that their kernels are generating functions that produce functions in the studied families and find these functions explicitly. This allows determining the elements of the matrices of systems of linear equations related to the integral equations for considering the physical problems.     

Use of the results of auxiliary experiments in solving problems in the linear theory of viscoelasticity

The approximate method of solving problems of the theory of linear viscoelasticity with arbitrary creep and relaxation kernels, proposed in [2], is substantiated and generalized. The essence of this method consists in the approximation of the functions depending on the Laplace — Carson transforms of the mechanical characteristics of a viscoelastic body by means of certain combinations of the transforms of the creep and relaxation kernels. The expressions obtained as a result of the approximation enable the inverse transforms of the unknown functions to be found without difficulty.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 963–969, November–December, 1968.