Abstract
Abstract In the paper, we investigate the relation between the properties of functions and their Fourier–Haar coefficients. We show that for some classes of functions Fourier–Haar coefficients have constant signs and order of magnitude. In 1964, Golubov proved in [B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 1964, 1271–1296] that if f ( x ) ∈ C ( 0 , 1 ) {f(x)\in C(0,1)} , then its Fourier–Haar coefficients have constant signs when f ( x ) {f(x)} is a nonincreasing function on [ 0 , 1 ] {[0,1]} , and in some cases those coefficients have a certain order of magnitude. In the present paper, we continue to investigate the properties of functions which follow from the behavior of their Fourier–Haar coefficients.
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