Distributional Point Values and Convergence of Fourier Series and Integrals

Abstract

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If f ∈ S (R) and x 0 ∈ R, and f is locally integrable, then f (x 0 ) = γ distributionally if and only if there exists k such that 1 2π lim x→∞ ax −xf (t)e −ix 0 t dt = γ (C, k) , for each a > 0, and similarly in the case when f is a general distribution. Here (C, k) means in the Cesàro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by lim x→∞ −x≤n≤ax a n e inx 0 = γ (C, k) .We also show that under some extra conditions, as if the sequence {a n } ∞ n=−∞ belongs to the space l p for some p ∈ [1, ∞) and the tails satisfy the estimate ∞ |n|≥N |a n | p = O N 1−p , as N → ∞, the asymmetric partial sums converge to γ . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.

The structure of quasiasymptotics of Schwartz distributions

“…Therefore, in the case at infinity the only unknown structural theorem was for quasiasymptotics whose degrees are negative integers. In a recent paper [22] a structural theorem for the quasiasymptotic behavior of degree -1 with respect to the trivial slowly varying function, L ≡ 1, was obtained, the technique employed was based on the concept of asymptotically homogeneous functions of degree 0 with respect to the trivial slowly varying function previously used in [6] to characterized Lojasiewicz point values of periodic distributions. In the case at the origin, only partial results were known under restrictions on the degree of the quasiasymptotic and boundedness of L [14].…”

“…Some results of G. Walter [26,27] are obtained by this method. We also discuss the case of jump behavior of distributions at a point [8,21,22].…”

“…The quasiasymptotics of distributions have shown to be of importance in several areas such as mathematical physics [24,29], abelian and tauberian theory of integral transforms [16,24,25], asymptotic behavior of solutions of partial differential equations [5,8,21,24,28] and summability of trigonometric series and integrals [6,8,21,22,26,27].…”

“…Since its introduction, the study of the structure of the quasiasymptotics has deserved a special place [5,8,10,11,12,13,14,15,16,19,22,24]. S. Lojasiewicz introduced the value of a distribution at a point, and he provided the corresponding structural theorem for it.…”

“…In Section 2, the structural theorems will be stated. I then discuss in Section 3 a particular case of the structural theorems within the context of summability of Fourier series and integrals; actually it should be mentioned that many of the ideas to study the structure of quasiasymptotics, specially in the case of degrees in Z − , were inspired by techniques previously applied by the author and R. Estrada in [6,21,22] at studying the value of a distribution at a point and distributional jump behaviors in connection with trigonometric series and integrals. In Section 4, the structure of quasiasymptotics at infinity is applied to show that the quasiasymptotic behavior holds in smaller spaces than S , namely on some spaces of the type K β .…”