Related papers: Sur les s\'eries de Fourier des fonctions continue…

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s\textgreater{}\tfrac{1}{2}$.…

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights…

Let $\rho: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $\rho(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(\tau)$ of integral weight associated to…

We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of…

For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.

This is an expository talk on a topic of classical analysis, arising from the VMO theory of the topological degree due to Br\'ezis and Nirenberg (1995). We sketch the history of the subject and some of its recent developments. The paper is…

The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of…

We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to…

It is shown that that for every Darboux function $F$ there is a non-constant continuous function $f$ such that $F+f$ is still Darboux. It is shown to be consistent --- the model used is iterated Sacks forcing --- that for every Darboux…

This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that…

This paper is concerned with the spectral characteristics of quaternionic positive definite functions on the real line. We generalize the Stone's theorem to the case of a right quaternionic linear one-parameter unitary group via two…

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…

We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any…

It is well known that the Fourier--Bohr coefficients of regular model sets exist and are uniformly converging, volume-averaged exponential sums. Several proofs for this statement are known, all of which use fairly abstract machinery. For…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated…

In this paper we present results on convergence and Ces\`{a}ro summability of Multiple Fourier series of functions of bounded generalized variation.

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property…

We consider an aspect of the open problem: Does every square-integrable function on SU(2) have an almost everywhere convergent Fourier series? Let 0 < alpha < 1. We show that to each countable set E in SU(2) there corresponds an…