GENERALIZED ABSOLUTE CONVERGENCE OF SINGLE AND DOUBLE VILENKIN-FOURIER SERIES AND RELATED RESULTS

We consider the Vilenkin orthonormal system on a Vilenkin group G and the Vilenkin-Fourier coefficients (f) over cap (n), n is an element of N, of functions f is an element of L-p(G) for some 1 < p <= 2. We obtain certain sufficient conditions for the finiteness of the series Sigma(infinity)(n=1) an| (f) over cap (n)|(r), where {a(n)} is a given sequence of positive real numbers satisfying a mild assumption and 0 < r < 2. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of f and give multiplicative analogue of some results due to Moricz (2010), Moricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020).