Majorants and uniqueness of series in the Franklin system

Abstract

ABSTRACT. It is proved that if a series in the Franklin system converges almost everywhere to a function f (t) and the distribution function of the majorant of partial sums satisfies the conditionas A -* co, then this series is a Fourier series for Lebesgue integrable functions f(t). In the general case the coefficients of the series are reconstructed by means of an A-integral. (1) A--*-i-ooThe following question arises in connection with Theorem I. Let the Frank!in series aJ.Ct)satisfy condition (1) and converge almost everywhere (a.e.) to some function f(t). Can the coefBcients of the series be expressed in terms of the function f(t)? This problem is treated in w of this article. It can be shown that if the sequence {At}k~=1 that provides the lower bound in condition (1) is known, then the coefficients an can be expressed in terms of f(t). Namely, the following theorem holds.

Uniqueness Theorems for Multiple Franklin Series

“…Therefore, according to (34)-(37), we obtain (11). Now let's prove that for any m ∈ N d 0 the coefficient a m can be found by (5). Assume m = (m 1 , m 2 , .…”

“…If the sums σ q n (x) converge in measure to a function f ∈ L[0, 1] d and for some sequence λ k → +∞ the condition (4) holds, then ( 2) is the Fourier-Franklin series of f . Not that similar questions for series by Franklin system and generlized Franklin system were considered in [4][5][6][7].…”