Wavelet decompositions of L2-Functionals

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Based on distribution-theoretical definitions of L 2 and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L 2-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L 2-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.
Original languageEnglish
JournalApplicable analysis. An international journal
Volume83
Issue number12
Pages (from-to)1187-1209
Number of pages23
ISSN0003-6811
DOIs
Publication statusPublished - 01.12.2004
Externally publishedYes

    Research areas

  • Mathematics
  • 41A65, 42C40, AMS Subject Classifications: 46E35, Approximation spaces, Besov spaces, Distributions, Interpolation spaces, Moduli of smoothness, Sobolev spaces, Wavelets

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