This article is cited in 10 papers

Absolute convergence of double series of Fourier–Haar coefficients for functions of bounded $p$-variation

B. I. Golubov

Chair of Higher Mathematics, Moscow Institute of Physical Technologies (State University), Dolgoprudnyi, Moscow Region, Russia

Abstract: We consider functions of two variables of bounded $p$-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of the Fourier–Haar series for functions of one variable, provided that they have a bounded Wiener $p$-variation or belong to the class $\operatorname{Lip}\alpha$. We show that the obtained results are unimprovable. We also formulate $N$-dimensional analogs of the main result and its corollaries.

Keywords: double Haar system, Fourier–Haar coefficients, functions of two variables of bounded $p$-variation.

UDC: 517.521

Received: 16.06.2011

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2012, 56:6, 1–10

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