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This article is cited in 3 scientific papers (total in 3 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,
$\sum_{j\in\mathbb N}\bigl\|\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}\bigr\|_2<\infty$),
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.
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
Linking options:
https://www.mathnet.ru/eng/im2710https://doi.org/10.1070/IM2009v073n01ABEH002442 https://www.mathnet.ru/eng/im/v73/i1/p177
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Abstract page: | 491 | Russian version PDF: | 196 | English version PDF: | 14 | References: | 55 | First page: | 22 |
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