<Emphasis Type="Italic">p</Emphasis>-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL ₂(ℤ) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ₀(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.     

On Dirichlet L-functions with periodic coefficients and Eisenstein series

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of SL₂(\mathbbZ){SL_2(\mathbb{Z})} and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

Wavelet Approximation of Periodic Functions

We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and L_p norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough.     

On conditions of the average convergence (upper boundedness) of trigonometric series

Let {ie005-01} be Fourier coefficients of a function ƒ ∈ L _2π . We prove that the condition {fx005-01} is necessary for the convergence of the Fourier series of ƒ in the L-metric; moreover, this condition is sufficient under some additional hypothesis for Fourier coefficients of ƒ. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Minimal Bases in Complete 3-Lattices

In this paper, u-convergence problems for multiple Fourier series with monotone and with positive coefficients in the L _p metrics are considered.     

Coefficient Condition forL1-Convergence of Walsh–Fourier Series

In this paper we give a condition with respect to Walsh–Fourier coefficients that implies theL₁-convergence of the corresponding Walsh–Fourier series. We show that theL₁-convergence class induced by this condition contains each one of the previously known convergence classes as a proper subset. We also show that our condition implies not only theL₁-convergence but also the convergence in the dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-convergence of complex double fourier series

It is proved that the complex double Fourier series of an integrable functionf(x, y) with coefficients c_jk satisfying certain conditions, will converge in L¹-norm. The conditions used here are the combinations of Tauberian condition of Hardy-Karamata kind and its limiting case. This paper extends the result of Bray [1] to complex double Fourier series. An erratum to this article is available at .     

Fast direct solvers for Poisson equation on 2D polar and spherical geometries

A simple and efficient class of FFT‐based fast direct solvers for Poisson equation on 2D polar and spherical geometries is presented. These solvers rely on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the second‐ and fourth‐order finite difference discretizations. Using a grid by shifting half mesh away from the origin/poles, and incorporating with the symmetry constraint of Fourier coefficients, the coordinate singularities can be easily handled without pole condition. By manipulating the radial mesh width, three different boundary conditions for polar geometry including Dirichlet, Neumann, and Robin conditions can be treated equally well. The new method only needs O(MN log₂ N) arithmetic operations for M × N grid points. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 56–68, 2002     

On solution of Lamé equations in axisymmetric domains with conical points

Partial Fourier series expansion is applied to the Dirichlet problem for the Lamé equations in axisymmetric domains ??³ with conical points on the rotation axis. This leads to dimension reduction of the three‐dimensional boundary value problem resulting to an infinite sequence of two‐dimensional boundary value problems on the plane meridian domain Ω_a?? of with solutions u _n(n=0,1,2,…) being the Fourier coefficients of the solution û of the 3D BVP. The asymptotic behaviour of the Fourier coefficients u _n (n=0,1,2,…) near the angular points of the meridian domain Ω_a is fully described by singular vector‐functions which are related to the zeros α_n of some transcendental equations involving Legendre functions of the first kind. Equations which determine the values of α_n are given and a numerical algorithm for the computation of α_n is proposed with some plots of values obtained presented. The singular vector functions for the solution of the 3D BVP is obtained by Fourier synthesis. Copyright © 2004 John Wiley & Sons, Ltd     

The Dirichlet series for the exterior square <Emphasis Type="Italic">L</Emphasis>-function on GL(<Emphasis Type="Italic">n</Emphasis>)

We present an elementary derivation of the Jacquet–Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients A(m ₁,…,m _n−1).     

On the Fourier coefficients of linear fractional stable motion

Inspired by a theorem of Marcinkiewicz [J. Marcinkiewicz, On a class of functions and their Fourier series, C. R. Soc. Sci. Varsovie, 26:71–77, 1934. Reprinted in: J. Marcinkiewicz, Collected Papers (A. Zygmund (Ed.)), PaństwoweWydawnictwo Naukowe,Warsaw, 1964] stating that the maximum of the absolute values of real Fourier coefficients a _n and b _n of a function of bounded p-variation ( p \geqslant 1 ) \left( {p \geqslant 1} \right) on an interval [0, 1] is of order O(n ^−1/p ) as n → ∞, we compute the Fourier coefficients of the linear fractional stable motion (LFSM) and of the closely related Riemann–Liouville (RL) process and investigate the rate of their decay.     

Higher order Lipschitz classes of functions and absolutely convergent Fourier series

We introduce the higher order Lipschitz classes Λ _r (α) and λ _r (α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients in order that f belongs to one of the above classes. This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Approximation power of RBFs and their associated SBFs: a connection

Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long time. Regardless of the underlying manifold, for example ℝⁿ or S ⁿ, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝⁿ⁺¹, to the unit sphere S ⁿ. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent to a Sobolev space H _s(ℝⁿ⁺¹), then the restriction to S ⁿ has a native space equivalent to H _s−1/2(S ⁿ). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of these were known earlier. Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation.     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

On -convergence of Fourier series

In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of L₁-convergence for such a series, that is f−S_n=o(1), are obtained in terms of coefficients.     

Optimality of AIC in Inference About Brownian Motion

In the usual Gaussian White-Noise model, we consider the problem of estimating the unknown square-integrable drift function of the standard Brownian motion using the partial sums of its Fourier series expansion generated by an orthonormal basis. Using the squared L ₂ distance loss, this problem is known to be the same as estimating the mean of an infinite dimensional random vector with l ₂ loss, where the coordinates are independently normally distributed with the unknown Fourier coefficients as the means and the same variance. In this modified version of the problem, we show that Akaike Information Criterion for model selection, followed by least squares estimation, attains the minimax rate of convergence. An erratum to this article can be found at     

The Validity Range of Two Complex Analysis Theorems

In this article we give complete characterizations of shift-invariant uniform algebras A_S on compact abelian groups, in which two of the classical theorems for analytic functions hold, namely, Radó's theorem for analytic extension and Riemann's theorem for removable singularities. Our characterization is in terms of algebraical properties of the semigroup S of non-zero Fourier coefficients of the functions in A_S .     

Recovering fourier coefficients of some functions and factorization of integer numbers

It is shown that if a function determined on the segment [−1, 1] has a sufficiently good approximation by partial sums of its expansion over Legendre polynomial, then, given the function’s Fourier coefficients c _n for some subset of n ∈ [n ₁, n ₂], one can approximately recover them for all n ∈ [n ₁, n ₂]. A new approach to factorization of integer numbers is given as an application.