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Mat. Zametki, 2009, Volume 86, Issue 3, Pages 429–444 (Mi mz8502)  

This article is cited in 11 scientific papers (total in 11 papers)

Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type

E. A. Rodionov, Yu. A. Farkov

Russian State Geological Prospecting University

Abstract: Suppose that $\omega(\varphi, \cdot )$ is the dyadic modulus of continuity of a compactly supported function $\varphi$ in $L^2(\mathbb R_+)$ satisfying a scaling equation with $2^n$ coefficients. Denote by $\alpha_\varphi$ the supremum for values of $\alpha>0$ such that the inequality $\omega(\varphi,2^{-j})\le C2^{-\alpha j}$ holds for all $j\in\mathbb N$. For the cases $n=3$ and $n=4$, we study the scaling functions $\varphi$ generating multiresolution analyses in $L^2(\mathbb R_+)$ and the exact values of $\alpha_\varphi$ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in $L^2(\mathbb R_+)$ corresponding to the scaling function $\varphi$ coincides with $\alpha_\varphi$.

Keywords: Daubechies wavelet, multiresolution analysis, the space $L^2(\mathbb R_+)$, Walsh series, binary entire function, Haar function, modulus of continuity, dyadic scaling function

DOI: https://doi.org/10.4213/mzm8502

Full text: PDF file (519 kB)
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English version:
Mathematical Notes, 2009, 86:3, 407–421

Bibliographic databases:

UDC: 517.518.3+517.965
Received: 23.07.2008
Revised: 20.01.2009

Citation: E. A. Rodionov, Yu. A. Farkov, “Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type”, Mat. Zametki, 86:3 (2009), 429–444; Math. Notes, 86:3 (2009), 407–421

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. A. Farkov, S. A. Stroganov, “The use of discrete dyadic wavelets in image processing”, Russian Math. (Iz. VUZ), 55:7 (2011), 47–55  mathnet  crossref  mathscinet  elib
    2. Farkov Yu.A., Maksimov A.Yu., Stroganov S.A., “On Biorthogonal Wavelets Related to the Walsh Functions”, Int. J. Wavelets Multiresolut. Inf. Process., 9:3 (2011), 485–499  crossref  mathscinet  zmath  isi  elib  scopus
    3. Stroganov S.A., “Otsenka gladkosti nizkochastotnykh mikroseismicheskikh kolebanii s pomoschyu diadicheskikh veivletov”, Geofizicheskie issledovaniya, 13:1 (2012), 17–22  elib
    4. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
    5. S. F. Lukomskii, “Step refinable functions and orthogonal MRA on Vilenkin groups”, J. Fourier Anal. Appl., 20:1 (2014), 42–65  crossref  mathscinet  zmath  isi  elib  scopus
    6. S. F. Lukomskii, “Riesz multiresolution analysis on zero-dimensional groups”, Izv. Math., 79:1 (2015), 145–176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Farkov Yu. Lebedeva E. Skopina M., “Wavelet Frames on Vilenkin Groups and Their Approximation Properties”, Int. J. Wavelets Multiresolut. Inf. Process., 13:5 (2015), 1550036  crossref  mathscinet  zmath  isi  elib  scopus
    8. Lukomskii S.F. Berdnikov G.S., “N-Valid Trees in Wavelet Theory on Vilenkin Groups”, Int. J. Wavelets Multiresolut. Inf. Process., 13:5 (2015), 1550037  crossref  mathscinet  zmath  isi  elib  scopus
    9. Krivoshein A.V., Lebedeva E.A., “Uncertainty Principle For the Cantor Dyadic Group”, J. Math. Anal. Appl., 423:2 (2015), 1231–1242  crossref  mathscinet  zmath  isi  elib  scopus
    10. M. A. Karapetyants, “Subdivision schemes on the dyadic half-line”, Izv. Math., 84:5 (2020), 910–929  mathnet  crossref  crossref  mathscinet  isi  elib
    11. M. A. Karapetyants, V. Yu. Protasov, “Spaces of Dyadic Distributions”, Funct. Anal. Appl., 54:4 (2020), 272–277  mathnet  crossref  crossref  isi
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