Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
Papers published on a yearly basis
Papers
More filters
56 citations
Posted Content•
TL;DR: For each prime $p, the distribution of the Fourier coefficients of the Hecke eigenforms of large weight for the full modular group has been determined in this paper.
Abstract: For each prime $p$, we determine the distribution of the $p^{th}$ Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution.
56 citations
01 Feb 1958
TL;DR: In this paper, the Fourier series of an integrable function f on (0, 2r) was shown to diverge almost everywhere in order to give an affirmative answer to the almost everywhere theorem.
Abstract: 1. By a well known theorem of Kolmogoroff there is a function whose Fourier series diverges almost everywhere. Actually, Kolmogoroff's proof was later generalized so that the Fourier series diverged everywhere [2, p. 175]; but we shall be concerned only with the almost everywhere theorem here. The proof involves rather severe restrictions on the orders of the partial sums which are shown to diverge. The following problem connected with this theorem was suggested to the author by Professor A. Zygmund. Given a sequence I p, } of positive integers increasing to oo, can an integrable function f on (0, 2r) be constructed so that the partial sums of its Fourier series of order p, diverge almost everywhere ? The object of our paper is to give an affirmative answer to this question. Let sp(x; f) denote the pth partial sum of the Fourier series of the function f at the point x.
56 citations
TL;DR: In this article, the Christoffel functions Λ n (d μ) associated with a general nonnegative measure μ on R d were studied and the asymptotics of Θ n ( d μ) were derived for μ supported on [− 1,1] d.
56 citations
Journal Article•
TL;DR: In this paper, the Fourier series is used to describe the outline of a miospore and the changes in shape brought about by different preparation techniques are not constant from species to species.
Abstract: Fourier series is a mathematical expression of sine and cosine waves which can be used to describe completely the outline of a miospore. The method of obtaining the Fourier series for a single miospore is presented and an illustration is included to show the application of Fourier series to the solution of palynologic problems. Investigation suggests that definable differences in the outlines of various species of pine pollen can be used to differentiate among them, but the technique used to prepare pine pollen for microscopic examination affects pollen shape. Furthermore, the changes in shape brought about by different preparation techniques are not constant from species to species. These differences have been quantitatively evaluated by means of a Fourier series and appropriate statistical tests. Further investigation suggests that under certain conditions, composite pollen shapes can be generated from the Fourier series of several grains and can be used to locate the areas on the periphery of a grain which are most variable from species to species.
56 citations