A version of discrete Haar transform with nodes of $\Pi_0$-grids

Abstract

A version of the two-dimensional discrete Haar transform with $2^D$ nodes forming $\Pi_0$-grid associated with the triangular partial sums of Fourier–Haar series of a given function is proposed. Due to the structure the of $\Pi_0$-grids, the computation of coefficients of this discrete transform is based on a cubature formula with $ 2 ^ D $ nodes being exact for Haar polynomials of degree at most $ D $, owing to that all the coefficients $A_{m_1,m_2}^{(j_1, j_2)}$ of the constructed transform coincide with the Fourier–Haar coefficients $c_{m_1, m_2}^{(j_1, j_2)}$ for Haar polynomials of degree at most $D-\max \{m_1, m_2 \}$ ($ {0 \leqslant m_1 + m_2 \leqslant d }$, where ${ d \leqslant D }$). The standard two-dimensional discrete Haar transform with $ 2 ^ D $ nodes does not possess this property.