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On the norms of polynomials in systems of periodic wavelets in the spaces \(L_p\). (English. Russian original) Zbl 0901.42023

Math. Notes 59, No. 5, 565-568 (1996); translation from Mat. Zametki 59, No. 5, 780-783 (1996).
In accordance with the multiresolution analysis on \(\mathbb{R}\) with a scaling function \(\varphi\), the author considers the following three systems of wavelets: (a) The wavelets generated by a function \(\varphi\) with compact support; (b) The Battle-Lemarié spline-wavelets; (c) The wavelets generated by a function \(\varphi\) whose Fourier transform has a compact support. For these systems of wavelets one can define polynomials \(T(y)\) which (in a sense) are similar to trigonometric polynomials.
A number of statements concerning validity of the estimate of the form \[ K_1\Biggl({1\over M} \sum^M_{k= 1}| T(y_k)|^p\Biggr)^{1/p}\leq \| T\|_p\leq K_2\Biggl({1\over M} \sum^M_{k= 1}| T(y_k)|^p\Biggr)^{1/p} \] for certain points \(y_k\), \(k= 1,\dots, M\), \(M\leq K_3N\), and every positive integer \(N\), are announced.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] B. S. Kashin and V. N. Temlyakov,Mat. Zametki [Math. Notes],56, No. 5, 57–86 (1994).
[2] B. S. Kashin, ”On trigonometric polynomials with coefficients modulo zero or one,” in:Theory of Functions and Approximations, Part 1 (Saratov, 1986) [in Russian], Saratov Gos. Univ., Saratov (1987), pp. 19–30.
[3] A. Zygmund,Trigonometric series, 2nd ed., Cambridge Univ. Press, Cambridge (1959). · Zbl 0085.05601
[4] Y. Meyer,Ondelettes, Herrman, Paris (1990).
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[6] V. V. Zhuk,Strong Approximation of Periodic Functions [in Russian], Izd. Leningrad Univ., Leningrad (1989). · Zbl 0721.42002
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