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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2009_73_1_a8,
     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/}
}
TY  - JOUR
AU  - P. A. Terekhin
TI  - Affine synthesis in the space $L^2(\mathbb R^d)$
JO  - Izvestiya. Mathematics 
PY  - 2009
SP  - 171
EP  - 180
VL  - 73
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a8/
LA  - en
ID  - IM2_2009_73_1_a8
ER  - 
%0 Journal Article
%A P. A. Terekhin
%T Affine synthesis in the space $L^2(\mathbb R^d)$
%J Izvestiya. Mathematics 
%D 2009
%P 171-180
%V 73
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a8/
%G en
%F IM2_2009_73_1_a8
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/

[1] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., 61, SIAM, Philadelphia, PA, 1992 | MR | Zbl | Zbl

[2] E. Hernández, G. Weiss, A fisrt course on wavelets, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1996 | MR | Zbl

[3] V. I. Filippov, P. Oswald, “Representation in $L_p$ by series of translates and dilates of one function”, J. Approx. Theory, 82:1 (1995), 15–29 | DOI | MR | Zbl

[4] Ch. K. Chui, Qiyu Sun, “Affine frame decompositions and shift-invariant spaces”, Appl. Comput. Harmon. Anal., 20:1 (2006), 74–107 | DOI | MR | Zbl

[5] H.-Q. Bui, R. S. Laugesen, “Affine systems that span Lebesgue spaces”, J. Fourier Anal. Appl., 11:5 (2005), 533–556 | DOI | MR | Zbl

[6] H.-Q. Bui, R. S. Laugesen, Affine synthesis and coefficient norms for Lebesgue, Hardy and Sobolev spaces, , 2007 http:// www.math.uiuc.edu/\allowbreakl̃augesen/ publications.html

[7] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Teoriya vspleskov, Fizmatlit, M., 2005

[8] O. Christensen, An introduction to frames and Riesz bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2003 | MR | Zbl

[9] G. Weiss, E. N. Wilson, “The mathematical theory of wavelets”, Twentieth century harmonic analysis – a celebration, Proceedings of the NATO Advanced Study Institute (Il Ciocco, Italy, 2000), NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer Acad. Publ., Dordrecht, 2001, 329–366 | MR | Zbl