Related papers: $\mathbf L^1$ completeness for Fourier series

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of…

In this paper we investigate some convergence and divergence properties of the logarithmic means of quadratical partial sums of double Fourier series of functions in the measure and in the $L$ Lebesgue norm.

Let $f\in L_{2\pi}$ be a real-valued even function with its Fourier series $ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$ and let $S_{n}(f,x), n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the…

We prove the Hardy-Littlewood theorem in two dimensions for functions whose Fourier coefficients obey general monotonicity conditions and, importantly, are not necessarily positive. The sharpness of the result is given by a counterexample,…

A classical theorem of Menshov states that every measurable function can redefined on a set of arbitrarily small Lebesgue measure, so that the resulting function has uniformly convergent Fourier series. We prove that the same is true if we…

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a…

We introduce an elementary argument to bound the $\textrm{BMO}$ seminorm of Fourier series with gaps giving in particular a sufficient condition for them to be in this space. Using finer techniques we carry out a detailed study of the…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

This paper presents a very simple explicit description of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$. The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain…

The product of any number of Legendre functions, under a restricted domain, can be expanded by the corresponding Legendre polynomials, with the coefficient being the sinc function. While an analogous expansion can be made for any number of…

We prove a version of the classical Mittag-Leffler Theorem for regular functions over quaternions. Our result relies upon an appropriate notion of principal part, that is inspired by the recent definition of spherical analyticity.

We prove Siegel-Walfisz type theorems (over long and short intervals) for the Fourier coefficients of certain automorphic $L$-functions and Rankin-Selberg $L$-functions over number fields.

We show that the Fourier transform on the Jacobian of a curve interchanges "$\delta$ functions" at the curve and the theta divisor. The Torelli theorem is an immediate consequence.

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality…

We prove an extension of the Furstenberg set theorem to families of graphs satisfying a transversality condition. We apply the result to derive bounds on $L^{p}$-norms of Fourier transforms of fractal measures supported on plane curves.

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which…

The completeness of Gaussians in a Hilbert functional space is established

We study the geometry and partial differential equations arising from the consideration of Frobenius determinants, also called-group-determinants. This leads us to address some aspects of twistor theory as well as some extensions of Bessel…

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…