Multiple Haar Basis and its Properties

Abstract

In the Lebesgue spaces $L_p ([0, 1]^d ), 1 ≤ p ≤ ∞$, for $d ≥ 2$, we define a multiple basis system of functions $H^d  = (h_n )_{n = 1}^{∞}$. This system has the main properties of the well-known one-dimensional Haar basis $H$. In particular, it is shown that the system $H^d$ is a Schauder basis in the spaces $L_p ([0, 1]^d ),\; 1 ≤ p < ∞$.