Geodesic
Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 261-275.

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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 $\psi_{kl}(j)=\psi_k(j-l2^k)$, $l\in\mathbb Z$, $k=1,2\dots$ 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). 
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A. B. Pevnyi. Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 261-275. http://geodesic.mathdoc.fr/item/SJVM_2004_7_3_a7/

[1] Petukhov A. P., Vvedenie v teoriyu bazisov vspleskov, Izd-vo SPbGTU, SPb., 1999

[2] Chui K., Vvedenie v veivlety, Mir, M., 2001

[3] Novikov I. Ya., Stechkin S. B., “Osnovy teorii vspleskov”, Uspekhi matem. nauk, 53:6(324) (1998), 53–128 | MR | Zbl

[4] Berkolaiko M. Z., Novikov I. Ya., “O beskonechno gladkikh pochti-vspleskakh s kompaktnym nositelem”, Matem. zametki, 56:3 (1994), 3–12 | MR | Zbl

[5] Zheludev V. A., Pevnyi A. B., “Kardinalnaya interpolyatsiya diskretnymi splainami”, Vestnik Syktyvkarskogo un-ta. Ser. 1, 1999, no. 3, 159–172 | MR | Zbl

[6] Pevnyi A. B., Zheludev V. A., “On wavelet analysis in the discrete splines space”, Proceedings Second Int. Conf. “Tools for Math. Modeling'99”, 4, SPTU, St. Petersburg, 1999, 181–195 | MR

[7] Pevnyi A. B., Zheludev V. A., “On interpolation by discrete splines with equidistant nodes”, J. Approx. Theory, 102 (2000), 286–301 | DOI | MR | Zbl

[8] Aldroubi A., Eden M., Unser M., “Discrete spline niters for multiresolutions and wavelets of $l_2$”, SIAM J. Math. Anal., 25:5 (1994), 1412–1432 | DOI | MR | Zbl

[9] Rioul O., “A discrete-time multiresolution theory”, IEEE Trans. Signal Processing, 41 (1993), 2591–2606 | DOI | Zbl

[10] Zheludev V. A., “Integral representation of slowly growing equidistant splines”, Approximation Theory and Applications, 14:4 (1998), 66–88 | MR | Zbl

[11] Subbotin Yu. N., “O svyazi mezhdu konechnymi raznostyami i sootvetstvuyuschimi proizvodnymi”, Trudy MIAN, 78, 1965, 24–42 | MR | Zbl

[12] Schoenberg I. J., “Cardinal interpolation and spline functions. II”, J. Approx. Theory, 2:2 (1969), 167–206 | DOI | MR | Zbl

[13] Pevnyi A. B., Zheludev V. A., “Construction of wavelet analysis in the space of discrete splines using Zak transform”, J. Fourier Analysis and Application, 8:1 (2002), 55–77 | MR

[14] Zyuzin M. B., “O primenenii filtrov s postoyannymi koeffitsientami”, Chislennye metody i matematicheskoe modelirovanie, Sb. nauchnykh trudov, ed. V. P. Ilin, VTs SO AN SSSR, Novosibirsk, 1990, 73–84

[15] Ichige K., Kamada M., “An approximation for discrete $B$-splines in time domain”, IEEE Signal Processing Letters, 4:3 (1997), 82–84 | DOI

[16] Schoenberg I. J., Cardinal spline interpolation, SIAM, Philadelphia, 1973 | MR | Zbl