Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter is the eigenvalue of ) for a cubic phase.

As applications we find sets of asymptotic relations for divisor functions.

     

On the relations among best approximation and Fourier coefficients

We discuss the relations among the best approximation E _n (f) and the Fourier coefficients of a function, under the conditions that and , where MVBVS^* is the class of the so-called Strong Mean Value Bounded Variation Sequences. This work was supported by NSERC RCD grant and AARMS of Canada (Yu D S), by NSERC of Canada (Zhou P), and by the National Natural Science Foundation of China (Grant No. 10471130) and the Open Fund (Grant No. PLN0613) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) (Zhou S P)     

Mean-square estimate of automorphic L-functions

Let f be a holomorphic Hecke cusp form with even integral weight k≥2 for the full modular group,and letχbe a primitive Dirichlet character modulo q.Let Lf(s,χ)be the automorphic L-function attached to f andχ-We study the mean-square estimate of Lf(s,χ)and establish an asymptotic formula.     

On hiding messages in the oversampled Fourier coefficients

A system using an oversampled Fourier transform for hiding data is given in [J.R. Miotke, L. Rebollo-Neira, Oversampling of Fourier coefficients for hiding messages, Appl. Comput. Harmon. Anal. 16 (2004) 203-207]. When viewed as a cryptographic algorithm, we demonstrate here that the system is susceptible to a known plaintext attack.     

On summability of Fourier coefficients of functions from Lebesgue space

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses.     

On the number of Fourier coefficients that determine a Hilbert modular form

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

     

The equiconvergence of expansions in eigenfunctions and associated functions of an integral operator with involution

In this paper we study an integral operator with involution. We solve the problem on the exact inversion of this operator, we obtain and study the integro-differential system for the Fredholm resolvent and, finally, we prove the theorem on the equiconvergence of expansions in eigenfunctions and associated functions, in the usual trigonometric system.     

Fourier coefficients of restrictions of eigenfunctions

Let {e_j} be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g). Let H ? M be a submanifold and {ψk} be an orthonormal basis of Laplace eigenfunctions of H with the induced metric. We obtain joint asymptotics for the Fourier coefficients ■ of restrictions γHe_j of e_j to H. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum {(μ_k, λ_j)}_j,...     

Average behavior of Fourier coefficients of cusp forms over sum of two squares

Let ${\lambda }_f(n)$ denote the nth normalized Fourier coefficient of the classical holomorphic cusp form of even integral weight $k\ge 2$ for the full modular group $\mathit{SL}(2,\mathbb{Z})$. In this paper, we investigate the average behavior of the power sum$\sum _{a^2+b^2\le x}{\lambda }_f{(a^2+b^2)}^{?}$ for $x\ge 1$, $2\le ?\le 8$ and $a,b,?\in \mathbb{Z}$.     

Theta correspondence of automorphic characters

This paper describes the lifting of automorphic characters of O(3)(A) to . It does so by matching the image of this lift with the lift of automorphic characters from O(1)(A) to . Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let V be a 3-dimensional quadratic vector space and U a certain 1-dimensional quadratic space. To an automorphic form IV(χ,φ) determined by the Schwartz function φ∈S(V(A)) in the lift of the character χ we match an automorphic form IU(μ,φ₀) determined by the Schwartz function φ₀∈S(U(A)) in the lift of the character μ. Our work shows that, the space U is explicitly determined by the character χ. The character μ is explicitly determined by the space V and the function φ₀ is given by an orbital integral involving φ.     

The cancellation of Fourier coefficients of cusp forms over different sparse sequences

Let λ f(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ).In this paper,we established nontrivial estimates for ∑n≤xλf(ni)λf(nj),where 1 ≤ i j ≤ 4.     

Fourier coefficients of cusp forms and automorphic f-functions

In the beginning of the paper, we review the summation formulas for the Fourier coefficients of holomorphic cusp forms for , =SL(2,), associated with L-functions of three and four Hecke eigenforms. Continuing the known works on the L-functions L_f,,(s) of three Hecke eigenforms, we prove their new properties in the special case of L_f,,(s). These results are applied to proving an analogue of the Siegel theorem for the L-function L_f(s) of the Hecke eigenform f(z) for (with respect to weight) and to deriving a new summation formula. Let f(z) be a Hecke eigenform for of even weight 2k with Fourier expansion . We study a weight-uniform analogue of the Hardy problem on the behavior of the sum _p≤xa(p) log p and prove new estimates from above for the sum _nxa(F(n))², where F(x) is a polynomial with integral coefficients of special form (in particular, F(x) is an Abelian polynomial). Finally, we obtain the lower estimate

A note on certain divisibility properties of the Fourier coefficients of normalized Eisenstein series

In this note, we shall prove certain divisibility properties of the Fourier coefficients of a class of normalized Eisenstein series modulo certain prime powers.     

Asymptotic behaviour of Verblunsky coefficients

Let , ζh≠ζk, h≠k and |ζj|=1, j=1,…,m, and consider the polynomials orthogonal with respect to , , where μ is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szeg? function has an analytic extension beyond |z|<1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained.     

On Fourier coefficients of automorphic forms of symplectic groups

 In this paper we study certain properties of Fourier coefficients of cuspidal representations on symplectic groups. We prove that every cuspidal representation has a nontrivial Fourier coefficient with respect to a certain type of unipotent class. Received: 17 October 2002 / Revised version: 8 December 2002 Published online: 24 April 2003     

Local approximation on surfaces with discontinuities,given limited order Fourier coefficients

We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries.     

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑m12+m22+m32≤xλ(m12+m22+m32)Λ(m12+m22+m32) and ∑m12+m22+m32≤xa(m12+m22+m32)Λ(m12+m22+m32).     

On CAP Representations for Even Orthogonal Groups Ⅰ:A Correspondence of Unramified Representations

The authors prove the local unramified correspondence for a new type of construction of CAP representations of even orthogonal groups by a generalized automorphic descent method. This method is expected to work for all classical groups.     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

In this article we establish the analogue of a theorem of Kuznetsov (theorem 6 of [3]) in the case of 3-dimensional hyperbolic space. We also consider a generalization of this result for higher dimensional hyperbolic spaces and discuss the relevant ingredients of a proof. Dedicated to the memory of Professor K G Ramanathan