Geodesic
Affine synthesis in the space $L^2(\mathbb R^d)$
Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 171-180

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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. 
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     author = {P. A. Terekhin},
     title = {Affine synthesis in the space $L^2(\mathbb R^d)$},
     journal = {Izvestiya. Mathematics },
     pages = {171--180},
     publisher = {mathdoc},
     volume = {73},
     number = {1},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a8/}
}
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P. A. Terekhin. Affine synthesis in the space $L^2(\mathbb R^d)$. Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 171-180. http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a8/