Poisson kernels and sparse wavelet expansions

Brandolese, Lorenzo

doi:10.1090/S0002-9939-05-07893-7

Poisson kernels and sparse wavelet expansions
HTML articles powered by AMS MathViewer

by Lorenzo Brandolese PDF

Proc. Amer. Math. Soc. 133 (2005), 3345-3353 Request permission

Abstract:

We give a new characterization of a family of homogeneous Besov spaces by means of atomic decompositions involving poorly localized building blocks. Our main tool is an algorithm for expanding a wavelet into a series of dilated and translated Poisson kernels.

References

Lorenzo Brandolese, Atomic decomposition for the vorticity of a viscous flow in the whole space, Math. Nachr. 273 (2004), 28–42. MR 2084955, DOI 10.1002/mana.200310194
Lorenzo Brandolese and Yves Meyer, On the instantaneous spreading for the Navier-Stokes system in the whole space, ESAIM Control Optim. Calc. Var. 8 (2002), 273–285. A tribute to J. L. Lions. MR 1932953, DOI 10.1051/cocv:2002021
Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), no. 4, 737–785. MR 1175690, DOI 10.2307/2374796
David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard, Wavelet shrinkage: asymptopia?, J. Roy. Statist. Soc. Ser. B 57 (1995), no. 2, 301–369. With discussion and a reply by the authors. MR 1323344, DOI 10.1111/j.2517-6161.1995.tb02032.x
Sergeĭ Yu. Dobrokhotov and Andreĭ I. Shafarevich, Some integral identities and remarks on the decay at infinity of the solutions to the Navier-Stokes equations in the entire space, Russian J. Math. Phys. 2 (1994), no. 1, 133–135. MR 1297948
Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300, DOI 10.1090/cbms/079
Stéphane Jaffard and Yves Meyer, On the pointwise regularity of functions in critical Besov spaces, J. Funct. Anal. 175 (2000), no. 2, 415–434. MR 1780484, DOI 10.1006/jfan.2000.3605
Stéphane Jaffard, Yves Meyer, and Robert D. Ryan, Wavelets, Revised edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Tools for science & technology. MR 1827998, DOI 10.1137/1.9780898718119
G. Kyriazis and P. Petrushev, New bases for Triebel-Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002), no. 2, 749–776. MR 1862566, DOI 10.1090/S0002-9947-01-02916-6
Y. Meyer, Un problème mathématique lié à la compression (manuscript).
Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. MR 1085487
V. V. Peller, Description of Hankel operators of the class ${\mathfrak {S}}_{p}$ for $p>0$, investigation of the rate of rational approximation and other applications, Mat. Sb. (N.S.) 122(164) (1983), no. 4, 481–510 (Russian). MR 725454
Pencho Petrushev, Multivariate $n$-term rational and piecewise polynomial approximation, J. Approx. Theory 121 (2003), no. 1, 158–197. MR 1963001, DOI 10.1016/S0021-9045(02)00060-6
K. Schneider, M. Farge, Numerical simulation of a mixing layer in an adaptive wavelet basis, C. R. Acad. Sci. Paris t. 328, Série II, b, Mécanique des fluides, 263–269 (2000).