Article
Ufa Mathematical Journal
Volume 5, Number 1, pp. 56-62
A version of discrete Haar transform with nodes of $\Pi_0$-grids
Kirillov K.A., Noskov M.V.
DOI:10.13108/2013-5-1-56
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We propose a version of the two-dimensional discrete Haar transform with $2^D$ nodes forming $\Pi_0$-grids associated with the triangular partial sums of Fourier\,--\,Haar series of a given function. Due to the structure 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.