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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 1, Pages 177–186 (Mi izv2710)  

This article is cited in 2 scientific papers (total in 2 papers)

Affine synthesis in the space $L^2(\mathbb R^d)$

P. A. Terekhin

Saratov State University named after N. G. Chernyshevsky

Abstract: We establish some theorems on the representation of functions $f\in L^2(\mathbb R^d)$ by series of the form $f=\sum_{j\in\mathbb N}\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}$ that are absolutely convergent with respect to the index $j$ (that is, $\sum_{j\in\mathbb N}\|\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}\|_2<\infty$), where $\psi_{j,k}(x)=|{\det a_j}|^{1/2}\psi(a_jx-bk)$, $j\in\mathbb N$, $k\in\mathbb Z^d$, is an affine system of functions. We prove the validity of the Bui–Laugesen conjecture on the sufficiency of the Daubechies conditions for a positive solution of the affine synthesis problem in the space $L^2(\mathbb R^d)$. A constructive solution is given for this problem under a localization of the Daubechies conditions.

Keywords: representation of functions by series, affine system, affine synthesis.

DOI: https://doi.org/10.4213/im2710

Full text: PDF file (488 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:1, 171–180

Bibliographic databases:

UDC: 517.51
MSC: 41A15, 41A65, 94A20, 42C15, 42C30, 42C40, 46B15, 46C05, 46E35
Received: 25.07.2007

Citation: P. A. Terekhin, “Affine synthesis in the space $L^2(\mathbb R^d)$”, Izv. RAN. Ser. Mat., 73:1 (2009), 177–186; Izv. Math., 73:1 (2009), 171–180

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. A. Terekhin, “Banach frames in the affine synthesis problem”, Sb. Math., 200:9 (2009), 1383–1402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Bendory T., Dekel Sh., Feuer A., “Robust recovery of stream of pulses using convex optimization”, J. Math. Anal. Appl., 442:2 (2016), 511–536  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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