Some Lambert Series Expansions of Products of Theta Functions

Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) = . Ramanujan has given a number of Lambert series expansions such as

Expansions in Series of Products of Eigenfunctions

A method is presented to establish expansions of analytic functions in series of m-fold products of special functions of Mathematical Physics. The idea is to “multiply” vector-valued solutions of first order differential systems in a suitable way and to construct the first order differential system which the “product” satisfies. Then an expansion theorem for the corresponding Floquet eigenvalue problem can be proved.     

m-continued Fraction Expansions of Multi-Laurent Series

The simple continued fraction expansion[7] of a single real number gives the best solution to its rational approximation problem.     

Fourier Expansions of Piecewise Smooth Functions

Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions.     

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ? ^N such that $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is connected. It is known that, for each w?∈?Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.     

Two-Point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points.     

Brownian Continuity Modulus Via Series Expansions

Beginning with the series representation in terms of Haar functions, we give a simplified proof of the Lévy modulus of continuity for standard Brownian motion.     

Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions

We consider a series with respect to a multiplicative Price system or a generalized Haar system and assume that the martingale subsequence of its partial sums converges almost everywhere. In this paper we prove that, under certain conditions imposed on the majorant of this sequence, the series is a Fourier series in the sense of the A-integral (or its generalizations) of the limit function if the series is considered as a series with respect to a system with supp _n < . In similar terms, we also present sufficient conditions for a series to be a Fourier series in the sense of the usual Lebesgue integral. We give an example showing that the corresponding assertions do not hold if supp _n = .     

Uniform Asymptotic Expansions of Confluent Hypergeometric Functions

Asymptotic expansions are derived for the confluent hypergeometricfunctions M(a, b, x) and U(a, b, x) for large b. The resultsare uniformly valid with respect to = x/b in a neighbourhoodcontaining = 1; a is a fixed parameter. The expansions arederived from integral representations and contain paraboliccylinder functions and asymptotic series.     

Integrality of Power Expansions Related to Hypergeometric Series

In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z),g(z) in powers of z so that f(z) and satisfy a hypergeometric equation under a special choice of parameters, we prove that the series in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's p-adic technique.     

Universal Overconvergence of Polynomial Expansions of Harmonic Functions

For each compact subset K of ^N let (K) denote the space of functions that are harmonic on some neighbourhood of K. The space (K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of ^N such that 0Ω and ^N\Ω is connected. It is shown that there exists a series ∑H_n, where H_n is a homogeneous harmonic polynomial of degree n on ^N, such that (i) ∑H_n converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑H_n are dense in (K) for every compact subset K of ^N\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.     

Some Polynomial Expansions for Functions of Several Variables

In the present paper three new expansion formulae are givenfor a function of several variables, which is defined by a multiplepower series with arbitrary terms. It is also indicated howthese general results can be applied to derive various knownor new multiplication theorems for certain hypergeometric functionsof one and more variables.     

Some Expansions in Basic Fourier Series and Related Topics

We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product.     

Expansion of Analytic Functions in Series of Floquet Solutions of First Order Differential Systems

In this paper we study boundary eigenvalue problems for first order systems of ordinary differential equations of the form \[zy'\left( z \right) = \left( {\lambda A_1 \left( z \right) + A_0 \left( z \right)} \right)y\left( z \right),\,\,y\left( {ze^{2\pi i} } \right) = e^{2\pi iv} y\left( z \right)\] for z ? S^log, where S is a ring region around zero, S^log denotes the Riemann surface of the logarithm over S, the coefficient matrix functions A₁(z) and A₀(z) are holomorphic on S, and v is a complex number. The eigenfunctions of this eigenvalue problem are the Floquet solutions of the differential system with v as characteristic exponent. For an open subset S₀ of S, the notion of A₁-convexity of the pair (S₀, S) is introduced. For A₁-convex pairs (S₀, S) it is shown that the expansion into eigenfunctions and associated functions of holomorphic functions on S^log, satisfying the monodromy condition y(ze^2πi) = e^2πivy(z), converges regularly on S^log₀ and is unique. If S is a pointed neighbourhood of 0 and A₁(z) is holomorphic in SU{0}, it is shown that there is a pointed neighbourhood S₀ of 0 such that (S₀, S) is A₁-convex. It follows from the results of this paper that many expansions of analytic functions in terms of special functions can be considered as eigenfunction expansions of this kind.     

Hermite展开与弱函数乘法的性质

根据丁夏畦院士利用Hermite展开定义的弱函数和广义弱函数以及函数的乘法等概念,来进一步研究弱函数乘法的相关性质,并证明了弱函数的乘法满足交换律、分配律和Leibniz法则,但不满足结合律。     

Uniform Asymptotic Expansions for Ellipsoidal Wave Functions II. Ellipsoidal Wave Functions

This paper is concerned with the determination of uniform asymptoticexpansion for ellipsoidal wave functions. An alternative normalizationconvention for ellipsoidal wave functions is suggested. An illustrativetable of the leading terms of one type of ellipsoidal wave functionis supplied.     

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let _t be the heat kernel on G; that is, _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on . Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on in the limit of small t, where the Hermite functions are viewed as functions on via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of . For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.     

多进小波展开的Gibbs函数的渐近性质

本文研究了一类多进小波展开的Gibbs函数的渐近性质，特别地，包括了多进Daubechies型小波．