Summability and uniqueness of double trigonometric integrals

Convergence, Uniqueness, and Summability of Multiple Trigonometric Series

In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let E be a subset of positive (Lebesgue) measure of the k dimensional torus. The principal result is that the convergence of a trigonometric series on E forces the boundedness of the partial sums alm...

A Class of Double Integrals Involving Gaussian and Trigonometric Factors

Summability of Multi-Dimensional Trigonometric Fourier Series

We consider the summability of oneand multi-dimensional trigonometric Fourier series. The Fejér and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space Hp to Lp, for all p > p0, where p0 depends on the summability method and the dimensio...

Integrals and Summable Trigonometric Series

is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function ƒ(x), then f(x) is necessarily integrable and the coefficients, an and bn, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (completely totalisable, [3])...

Some Divergent Trigonometric Integrals

1. Introduction. Browsing through an integral table on a dull Sunday afternoon some time ago, I came across four divergent trigonometric inte-grals. (See (1) and (2) below.) I was intrigued as to how these divergent integrals ended up in a respectable table. Tracing their history, it turned out they were originally " evaluated " when some convergent integrals, (5) and (6), were differentiated u...