Fourier expansions at cusps

The Ramanujan Journal - In this article, we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain...     

The differential Fourier transform method

We suggest the two new discrete differential sine and cosine Fourier transforms of a complex vector which are based on solving by a finite difference scheme the inhomogeneous harmonic differential equations of the first order with complex coefficients and of the second order with real coefficients, respectively. In the basic version, the differential Fourier transforms require by several times less arithmetic operations as compared to the basic classicalmethod of discrete Fourier transform. In the differential sine Fourier transform, the matrix of the transformation is complex,with the real and imaginary entries being alternated, whereas in the cosine transform, the matrix is purely real. As in the classical case, both matrices can be converted into the matrices of cyclic convolution; thus all fast convolution algorithms including the Winograd and Rader algorithms can be applied to them. The differential Fourier transform method is compatible with the Good–Thomas algorithm of the fast Fourier transform and can potentially outperform all available methods of acceleration of the fast Fourier transform when combined with the fast convolution algorithms.     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

On the equisummability of Hermite and Fourier expansions

We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.     

Series expansions for functional differential equations

In this paper we prove convergence results for the series expansion of the solution to a linear functional differential equation. The results are consequences of an analysis of eigenfunction expansions for the generator of the solution map. This abstract approach unifies the treatment of retarded and neutral functional differential equations.The research of S.M. Verduyn Lunel has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences     

Uniform convergence of expansions into a multiple trigonometric Fourier series or a Fourier integral

Questions of convergence almost everywhere of expansions into a multiple trigonometric Fourier series or a Fourier integral are studied for functions from L_p, p≥1, with summation over rectangles. Moreover, a “generalized localization principle,” understood in the sense of convergence almost everywhere, is considered in the paper.     

Rapidly computing sparse Legendre expansions via sparse Fourier transforms

In this paper, we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function f : [?1, 1] → ? with a near-optimal linear combination of s Legendre polynomials of degree ≤ N in just \((s \log N)^{\mathcal {O}(1)}\)-time. When s ? N, these algorithms exhibit sublinear runtime complexities in N, as opposed to traditional Ω(NlogN)-time methods for computing all of the first N Legendre coefficients of f. Theoretical as well as numerical results demonstrate the effectiveness of the proposed methods.     

Fourier expansions of functions with bounded variation of several variables

In the first part of the paper we establish the pointwise convergence as for convolution operators under the assumptions that has integrable derivatives up to an order and that with . We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.

     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.     

Asymptotic expansions of Fourier transforms of functions with logarithmic singularities

Asymptotic expansion as x → +∞ is obtained for the infinite Fourier integral $\text{F(x) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) e}{}^{\mathrm{ixt}}\text{dt}$, in which $\text{?(t)}$ has a logarithmic singularity of the type t^α?1(?ln t)^β at the origin. Here, Re α > 0 and β is an arbitrary complex number.     

The rate of convergence of Fourier expansions in the plane: a geometric viewpoint

Let A be an appropriate planar domain and let f be a piecewise smooth function on . We discuss the rate of convergence of in terms of the interaction between the geometry of A and the geometry of the singularities of f. The most subtle case is when x belongs to the singular set of f. Received: 21 December 2000; in final form: 4 September 2001 / Published online: 4 April 2002     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Biorthogonal expansions in root functions of differential operators

We consider the problem on the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators. We introduce the notion of almost basis property of systems of root functions of a linear operator. We demonstrate the necessity to use a new method, earlier introduced by the authors, for defining associated functions.