Geodesic
On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 106-125.

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

In the paper [9] we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single layer potential equation in a rectangle. This phenomenon is closely related on the one hand to the properties of the hyperbolic crosses approximation method and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper we establish several results on the approximation for the hyperbolic crosses and on the best $N$-term approximations by linear combinations of Haar functions in the $H^s$-norms, $-1$; this provides a theoretical base for our numerical research. To the author best knowledge, the negative smoothness case $s0$ was not studied earlier. 
@article{CMFD_2007_25_a9,
     author = {P. Oswald},
     title = {On $N${-Termed} {Approximations} in $H^s${-Norms} with {Respect} to the {Haar} {System}},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {106--125},
     publisher = {mathdoc},
     volume = {25},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/}
}
TY  - JOUR
AU  - P. Oswald
TI  - On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2007
SP  - 106
EP  - 125
VL  - 25
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/
LA  - ru
ID  - CMFD_2007_25_a9
ER  - 
%0 Journal Article
%A P. Oswald
%T On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System
%J Contemporary Mathematics. Fundamental Directions
%D 2007
%P 106-125
%V 25
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/
%G ru
%F CMFD_2007_25_a9
P. Oswald. On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 106-125. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/

[1] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[2] Golubov B. I., “Ryady po sisteme Khaara”, Itogi nauki i tekhniki. Matem. analiz, VINITI, M., 1971, 109–146 | MR

[3] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[4] Tribel G., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[5] Ulyanov P. L., “O ryadakh po sisteme Khaara”, Matem. sb., 63(105) (1964), 356–391

[6] Dahmen W., “Wavelet and multiscale methods for operator equations”, Acta Numerica, 6 (1997), 55–228 | DOI | MR | Zbl

[7] DeVore R. A., “Nonlinear approximation”, Acta Numerica, 7 (1998), 51–150 | DOI | MR | Zbl

[8] DeVore R. A., Lorentz G. G., Constructive Approximation, Springer-Verlag, New York, 1993 | MR

[9] DeVore R. A., Konyagin S. V., Temlyakov V. N., “Hyperbolic wavelet approximation”, Constr. Approx., 14 (1998), 1–26 | DOI | MR

[10] DeVore R. A., Temlyakov V. N., “Some remarks on greedy algorithms”, Adv. Comput. Math., 5:2–3 (1996), 173–187 | DOI | MR | Zbl

[11] Griebel M., Knapek S., Optimized approximation spaces for operator equations, Preprint SFB 256, Universität Bonn, 1998

[12] Griebel M., Oswald P., Schiekofer T., “Sparse grids for boundary integral equations”, Numer. Math., 83:2 (1999), 279–312 | DOI | MR | Zbl

[13] Lions J.-L., Magenes E., Non-homogeneous boundary value problems and applications, v. 1, Springer-Verlag, Berlin, 1972

[14] Oswald P., Multilevel finite element approximation: theory and applications, Teubner, Stuttgart, 1994 | MR | Zbl

[15] Oswald P., “Multilevel norms for $H^{-1/2}$”, Computing, 61 (1998), 235–255 | DOI | MR | Zbl

[16] Von Petersdorff T., Stephan E. P., “Decomposition in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron”, Math. Nachr., 149 (1990), 71–104 | DOI | MR

[17] Von Petersdorff T., Stephan E. P., “Regularity of mixed boundary value problems in ${\mathbb R}^3$ and boundary element methods on graded meshes”, Math. Meth. Appl. Sci., 12 (1990), 229–249 | DOI | MR | Zbl

[18] Stephan E. P., The h-p boundary element method for solving $2$- and $3$-dimensional problems, Preprint, Univ. Hannover, 1995 | MR

[19] Temlyakov V. N., Approximation of periodic functions, Nova Sci. Publ., New York, 1993 | MR | Zbl

[20] Temlyakov V. N., “Nonlinear $m$-term approximation with regard to the multivariate Haar system”, East J. Approx., 4 (1998), 87–106 | MR | Zbl

[21] Temlyakov V. N., “The best $m$-term approximation and greedy algorithms”, Adv. Comput. Math., 8:3 (1998), 249–265 | DOI | MR | Zbl

[22] Weidmann J., Linear operators in Hilbert spaces, Springer-Verlag, New York, 1980 | MR

[23] Zenger C., “Sparse grids in parallel algorithms for partial differential equations”, Proc. 6th GAMM Seminar, Kiel, ed. W. Hackbusch, Vieweg, Braunschweig, 1991, 241–251 | MR