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 Sib. Zh. Vychisl. Mat., 2004, Volume 7, Number 3, Pages 261–275 (Mi sjvm162)

Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines

A. B. Pevnyi

Syktyvkar State University, Faculty of Mathematics

Abstract: A non-stationary multiresolution analysis $\{V_k\}_{k\geq 0}$ $\ell^2(\mathbb Z)$ in the space $\ell^2(\mathbb Z)$ is performed, the subspaces $V_k$ consisting of discrete splines. In each $V_k$, there is a function $\varphi_k$ such that the system $\{\varphi_k(\cdot-l2^k):l\in\mathbb Z\}$ forms the Riesz base of $V_k$. A system of wavelets $\psi_{kl}(j)=\psi_k(j-l2^k)$, $l\in\mathbb Z$, $k=1,2…$ is not generated by shifts and dilations of the unique function. The subspaces $W_k=\operatorname{span}\{\psi_{kl}:l\in\mathbb Z\}$ form an orthogonal expansion of the space: $\ell^2(\mathbb Z)=\oplus^{\infty}_{k=1}W_k$.
The space $V_k$ is the same as the space of discrete splines $S_{p,2^k}$ of order $p$ with a distance between the knots $2^k$. For every $p$, a multiresolution analysis is obtained (for $p=1$ – the Haar multiresolution analysis).

Key words: discrete splines, discrete wavelets, multiresolution analysis.

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Bibliographic databases:
UDC: 519.65

Citation: A. B. Pevnyi, “Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines”, Sib. Zh. Vychisl. Mat., 7:3 (2004), 261–275

Citation in format AMSBIB
\Bibitem{Pev04} \by A.~B.~Pevnyi \paper Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines \jour Sib. Zh. Vychisl. Mat. \yr 2004 \vol 7 \issue 3 \pages 261--275 \mathnet{http://mi.mathnet.ru/sjvm162} \zmath{https://zbmath.org/?q=an:1068.65152}