Skopina, M. A. 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. Reviewer: B.Rubin (Jerusalem) Cited in 1 Document MSC: 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems Keywords:periodic wavelets; \(L^p\)-estimates of polynomials of wavelets; multiresolution analysis; scaling function PDFBibTeX XMLCite \textit{M. A. Skopina}, Math. Notes 59, No. 5, 565--568 (1996; Zbl 0901.42023); translation from Mat. Zametki 59, No. 5, 780--783 (1996) Full Text: DOI References: [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). [5] I. Daubechies,Ten Lectures on Wavelets, CBMS-NSR Series in Appl. Math., SIAM (1992). · Zbl 0776.42018 [6] V. V. Zhuk,Strong Approximation of Periodic Functions [in Russian], Izd. Leningrad Univ., Leningrad (1989). · Zbl 0721.42002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.