Recovery of the Coefficients of Multiple Haar and Walsh Series

Abstract

A family of multidimensional generalized Perron type integrals is constructed. It is shown that these integrals solve the problem of recovering, by generalized Fourier formulae, the coefficients of multiple Haar and Walsh series of some class. This class includes in particular series convergent $\rho$-regularly everywhere except some countable set $E \subset G^{d}$. It is shown that some properties of rectangularly convergent multiple Haar and Walsh series do not hold for the $\rho$-regular convergence.