Majorants and uniqueness of series in the Franklin system

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 condition mes{t∈[0,1]∶s(t)>λ}=o(1/λ) as λ→∞, 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 anA-integral.