Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

CONTRIBUTIONS TO ABSTRACT ANALYTIC NUMBER THEORY

Abstract

This paper reports on some recent contributions to the theory of multiplicative arithmetic semigroups, which have been initiated by John Knopfmacher's work on abstract analytic number theory. They concern weighted inversion theorems of the Wiener type, mean-value theorems for multiplicative functions, and Ramanu-jan expansions.     

关于各类插值样条函数

去年在第二届全国逼近论会议上,浙江大学论文集中提出各种形式的插值样条,给出了收敛阶的估计以及一些渐近展开。但结果不大精确,且每一种都需冗繁的计算。 本文应用我们多次用过的方法,统一处理这些问题。由此看出,我们的方法不仅简洁,而且可普遍适用于许多类似问题。     

The existence theorems of the random solutions for random Hammerstein equation with random kernel (II)

In this paper, we use the random fixed theorems of cone expansion and compression of random operator to obtain the existence theorems of random solutions for random Hammerstein integral equation of polynomial type with random kernel.     

Polynomials on Riesz spaces

In this paper we investigate polynomial mappings on Riesz spaces. We give a characterization of positivity of homogeneous polynomials in terms of forward differences. Finally we prove Hahn-Banach type extension theorems for positive and regular polynomial mappings.     

Series expansions of analytic functions

Many of the classical polynomial expansions of analytic functions share a common property: the space of “expandable” functions is a Banach space isometrically isomorphic to the space of complex sequences with limit 0. Under the isometries, these polynomial expansions all correspond to essentially the same biorthogonal expansion in this sequence space. Sufficient conditions for such an isometry to exist are obtained, and convergence properties of the expansions are studied. The results obtained also apply to expansions other than polynomial expansions.     

Infinite dimensional Cauchy-Kowalevski and Holmgren type theorems

The aim of this paper is to show Cauchy-Kowalevski and Holmgren type theorems with an infinite number of variables. We adopt von Koch and Hilbert's definition of analyticity of functions as monomial expansions. Our Cauchy-Kowalevski type theorem is derived by modifying the classical method of majorants.Based on this result, by employing some tools from abstract Wiener spaces, we establish our Holmgren type theorem.     

Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

特殊Hermite展开的乘子定理

本文通过建立与特殊Hermite展开相对应的Littlewood-Paley分解和相关的扭曲卷积核的L2估计,得到特殊Hermite展开的乘子定理,作为该结果的应用,给出了Hermite函数及Laguerre函数展开的乘子定理。     

Jackson-type Theorems on a Finite Interval with Weights Having Non-symmetric Inner Singularities

We consider Jackson-type theorems for weighted polynomial approximation when the weight has certain type of zeros inside and/or at the end-points of the interval. The approximating polynomials are constructed from truncated convolution integrals.     

Fuzzy度量空间型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Fuzzy度量空间扩张型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Bilateral generating functions for Jacobi polynomials

In this paper, we have obtained three theorems on generating functions. We derive from these theorems a large number of bilateral generating functions for Jacobi polynomials. Certain interesting expansions of triple hypergeometric series are also obtained from one of the theorems.     

A numerical method for solving singular boundary value problems

Summary A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.     

Describing functions: Atomic decompositions versus frames

The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces. Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabortype expansions for modulation spaces, and sampling theorems for wavelet and Gabor transforms.     

Two expansion theorems by diagonal elements for the permanent and the determinant

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

EIGENFUNCTION EXPANSION FOR STURM-LIOUVILLE PROBLEMS WITH TRANSMISSION CONDITIONS AT ONE INTERIOR POINT

The purpose of this article is to extend some spectral properties of regular SturmLiouville problems to the special type discontinuous boundaxy-value problem,which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions.We construct the resolvent operator and Green's function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L_2[a,b].     

Transplantation Theorems��A Survey

In this survey article we overview transplantation theorems for several types of continuous and discrete orthogonal expansions. These include: Hankel and Dunkl transforms, and Fourier-Bessel, Jacobi and Laguerre expansions. We also discuss the idea of transference of transplantation and point out how a notion of conjugacy for orthogonal expansions may be interpreted as a generalized transplantation.