Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.     

Estimates for the equiconvergence rate of spectral expansions on an interval

We study the convergence rate of biorthogonal expansions of functions in series in systems of root functions of a broad class of second-order ordinary differential operators on a finite interval. The above-mentioned expansions are compared with the expansions of the same functions in trigonometric Fourier series in an integral or uniform metric on any interior compact set of the basic interval and on the entire interval. We prove the dependence of the equiconvergence rate of the expansions in question on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

Dependence of estimates of the local convergence rate of spectral expansions on the distance from an interior compact set to the boundary

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

Componentwise uniform equiconvergence of expansions in root vector functions of the Dirac operator with the trigonometric expansion

We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in root vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved.     

Series expansions for computing Bessel functions of variable order on bounded intervals

Based on a continuity property of the Hadamard product of power series we derive results concerning the rate of convergence of the partial sums of certain polynomial series expansions for Bessel functions. Since these partial sums are easily computable by recursion and since cancellation problems are considerably reduced compared to the corresponding Taylor sections, the expansions may be attractive for numerical purposes. A similar method yields results on series expansions for confluent hypergeometric functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary condition

We study issues related to the uniform convergence of the Fourier series expansions of Hölder class functions in the system of eigenfunctions corresponding to a spectral problem obtained from a mixed problem for the heat equation. We prove a theorem on the equiconvergence of these expansions with expansions in a well-known orthonormal basis.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplace's method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.     

Rational series for multiple zeta and log gamma functions

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We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers.

Video

For a video summary of this paper, please click here or visit http://youtu.be/2i5PQiueW_8.     

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.      

On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We study the uniform convergence, on a closed interval, of spectral expansions of Hölder functions in a given complete and minimal system of eigenfunctions corresponding to a spectral problem with spectral parameter in a boundary condition. We consider boundary conditions of the third kind and subject the function to be expanded to a condition of nonlocal type ensuring the uniform convergence. We prove a theorem stating that expansions in the entire system of eigenfunctions of the problem are possible without any additional conditions.     

On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We analyze the uniform convergence of the Fourier series expansions of Hölder functions in the system of eigenfunctions of a spectral problem with squared spectral parameter in a boundary condition. To this end, we first prove a theorem on the equiconvergence of such expansions with those in a well-known orthonormal basis.     

Orthogonal properties for a typical class of hypergeometric functions

We present three orthogonal properties for a typical class of hypergeometric functions. We employ orthogonal properties to generate a theory concerning infinite series expansions involving our hypergeometric functions.     

Prolate spheroidal wave functions, an introduction to the Slepian series and its properties

For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.     

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.     

Equisummability in the sense of M. Riesz of expansions in certain systems of exponential functions

We establish the equisummability in the M. Riesz sense of Fourier series expansions in two systems of exponential functions, these latter being the characteristic functions of boundary-value problems for a first-order differential equation.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Greedy Algorithms with Prescribed Coefficients

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. We concentrate on studying algorithms that provide expansions into a series. We call such expansions greedy expansions. It was pointed out in our previous article that there is a great flexibility in choosing coefficients of greedy expansions. In that article this flexibility was used for constructing a greedy expansion that converges in any uniformly smooth Banach space. In this article we push the flexibility in choosing the coefficients of greedy expansions to the extreme. We make these coefficients independent of an element f ∈ X. Surprisingly, for a properly chosen sequence of coefficients we obtain results similar to the previous results on greedy expansions when the coefficients were determined by an element f.     

Some estimates for the error in mixed Fourier-Bessel expansions of functions of two variables

Some issues concerning expansions of functions of two variables in mixed Fourier-Bessel series are considered. In particular, the rate of their convergence in the classes of functions characterized by generalized moduli of continuity are estimated, and estimates of the remainder terms are obtained.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.