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Base d’ondelettes sur les groupes de Lie stratifiés. (Basis of ondelettes on stratified Lie groups). (French) Zbl 0711.43004

The author generalizes the theory of ondelettes, started by Meyer and Grossmann for \({\mathbb{R}}^ n\) to general stratified nilpotent Lie groups G, constructing a hilbertian basis of \(L^ 2(G)\) formed with regular and oscillating functions, “uniformly” localized in space and frequency (as in the ondelettes theory). On certain groups this basis is composed of finitely many functions and the functions obtained from them by “dyadic” dilations-translations. The way of constructing this basis lies in a multi-scale analysis composed from generalized spline-surfaces spaces.
Reviewer: H.Rindler

MSC:

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
22E25 Nilpotent and solvable Lie groups
42C15 General harmonic expansions, frames
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References:

[1] DAUBECHIES (I.) . - Communication personnelle .
[2] DEMKO (S.) . - Inverses of band-matrices and local convergence of spline projections , SIAM J. Numer. Anal., t. 14, 1977 , p. 616-619. MR 56 #13520 | Zbl 0367.65024 · Zbl 0367.65024 · doi:10.1137/0714041
[3] FOLLAND (G. B.) . - Subelliptic estimates and function spaces on nilpotent Lie groups , Ark. Mat., t. 13, 1975 , p. 161-207. MR 58 #13215 | Zbl 0312.35026 · Zbl 0312.35026 · doi:10.1007/BF02386204
[4] KNAPP (A. W.) and STEIN (E.M.) . - Intertwining operators for semi-simple groups , Ann. of Math., t. 93, 1971 , p. 489-578. MR 57 #536 | Zbl 0257.22015 · Zbl 0257.22015 · doi:10.2307/1970887
[5] LEMARIé (P.G.) . - Algèbres d’opérateurs et semi-groupes de Poisson sur un espace de nature homogène . - Publications Math. d’Orsay, 1984 . MR 86g:42030 | Zbl 0598.58045 · Zbl 0598.58045
[6] LEMARIé (P. G.) . - Continuité des opérateurs définis par des intégrales singulières , Ann. Inst. Fourier, t. 35, 4, 1985 , p. 175-187. Numdam | MR 87j:47074 | Zbl 0555.47032 · Zbl 0555.47032 · doi:10.5802/aif.1033
[7] LEMARIé (P. G.) . - Ondelettes à localisation exponentielle , à paraître au J. Math. Pures Appl. Zbl 0758.42020 · Zbl 0758.42020
[8] LEMARIé (P. G.) . - Théorie L2 des surfaces-splines , Preprint, Ecole Normale Supérieure.
[9] LEMARIé (P. G.) . - Isomorphie des algèbres Ab , Preprint, 1987 .
[10] LEMARIé (P. G.) , MALLAT (S.) et MEYER (Y.) . - Analyse multi-échelles, ondelettes et fonctions splines , Preprint, Université Paris IX-Dauphine.
[11] LEMARIé (P. G.) et MEYER (Y.) . - Ondelettes et bases hilbertiennes , Rev. Mat. Iberoamericana, t. 2, 1, 1986 , p. 1-17. MR 864650 | Zbl 0657.42028 · Zbl 0657.42028
[12] MEYER (Y.) . - Exposé au Séminaire Bourbaki , février 1986 .
[13] MEYER (Y.) . - Ondelettes, fonctions splines et analyses graduées , Preprint Université Paris IX-Dauphine. · Zbl 0714.42022
[14] SAKA (K.) . - Besov spaces and Sobolev spaces on a nilpotent Lie group , Tôhoku Math. J., t. 31, 1979 , p. 383-437. Article | MR 82a:46035 | Zbl 0429.43004 · Zbl 0429.43004 · doi:10.2748/tmj/1178229728
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