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Ufimsk. Mat. Zh., 2013, Volume 5, Issue 1, Pages 56–62 (Mi ufa186)  

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

K. A. Kirillov, M. V. Noskov

Institute of Space and Information Technologies, Siberian Federal University

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.

Keywords: cubature formulas exact for Haar polynomials, discrete Haar transform, $\Pi_0$-grids.

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English version:
Ufa Mathematical Journal, 2013, 5:1, 56–62 (PDF, 331 kB); https://doi.org/10.13108/2013-5-1-56

Bibliographic databases:

Document Type: Article
UDC: 517.518.87
Received: 20.12.2011

Citation: K. A. Kirillov, M. V. Noskov, “A version of discrete Haar transform with nodes of $\Pi_0$-grids”, Ufimsk. Mat. Zh., 5:1 (2013), 56–62; Ufa Math. J., 5:1 (2013), 56–62

Citation in format AMSBIB
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\jour Ufa Math. J.
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\pages 56--62
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