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Mat. Zametki, 2005, Volume 78, Issue 5, Pages 658–675 (Mi mz2630)  

This article is cited in 4 scientific papers (total in 4 papers)

Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion

Yu. K. Dem'yanovich

Saint-Petersburg State University

Abstract: With each infinite grid $X:…<x_{-1}<x_0<x_1<\dotsb$ we associate the system of trigonometric splines $\{\mathfrak T_j^B\}$ of class $C^1(\alpha,\beta)$, the linear space $\mathscr T^B(X)\overset{\textrm{def}}=\{\tilde u\mid\tilde u=\sum_jc_j\mathfrak T_j^B \forall c_j\in\mathbb R^1\}$, and the functionals $g^{(i)}\in(C^1(\alpha,\beta))^*$ with the biorthogonality property: $\langle g^{(i)},\mathfrak T_j^B\rangle=\delta_{i,j}$ (here $\alpha\overset{\textrm{def}}=\lim_{j\to-\infty}x_j$, $\beta\overset{\textrm{def}}=\lim_{j\to+\infty}x_j$). For nested grids $\overline X\subset X$, we show that the corresponding spaces $\mathscr T^B(\overline X)\subset\mathscr T^B(X)$ are embedded in $\mathscr T^B(X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $\mathscr T^B(X)=\mathscr T^B(\overline X)\dotplus W$ derived with the help of the system of functionals indicated above.

DOI: https://doi.org/10.4213/mzm2630

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English version:
Mathematical Notes, 2005, 78:5, 615–630

Bibliographic databases:

UDC: 518
Received: 25.08.2004

Citation: Yu. K. Dem'yanovich, “Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion”, Mat. Zametki, 78:5 (2005), 658–675; Math. Notes, 78:5 (2005), 615–630

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. K. Dem'yanovich, “Local wavelet basis for an irregular grid”, J. Math. Sci. (N. Y.), 141:6 (2007), 1618–1632  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Kosogorov O., Makarov A., “On Some Piecewise Quadratic Spline Functions”, Numerical Analysis and Its Applications (NAA 2016), Lecture Notes in Computer Science, 10187, eds. Dimov I., Farago I., Vulkov L., Springer International Publishing Ag, 2017, 448–455  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Dem'yanovich Yu.K., Makarov A.A., “Necessary and Sufficient Nonnegativity Conditions For Second-Order Coordinate Trigonometric Splines”, Vestnik St. Petersburg Univ. Math., 50:1 (2017), 5–10  crossref  mathscinet  zmath  isi  scopus  scopus
    4. E. K. Kulikov, A. A. Makarov, “Ob approksimatsii giperbolicheskimi splainami”, Chislennye metody i voprosy organizatsii vychislenii. XXXI, Zap. nauchn. sem. POMI, 472, POMI, SPb., 2018, 179–194  mathnet
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