TY - JOUR

T1 - Wavelet decompositions of L2-Functionals

AU - Hochmuth, Reinhard

AU - Haf, Herbert

PY - 2004/12/1

Y1 - 2004/12/1

N2 - 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.

AB - 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.

KW - Mathematics

KW - 41A65

KW - 42C40

KW - AMS Subject Classifications: 46E35

KW - Approximation spaces

KW - Besov spaces

KW - Distributions

KW - Interpolation spaces

KW - Moduli of smoothness

KW - Sobolev spaces

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=85064784969&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/6e3fbb78-f500-3e05-a877-846dbea44555/

U2 - 10.1080/00036810410001724698

DO - 10.1080/00036810410001724698

M3 - Journal articles

VL - 83

SP - 1187

EP - 1209

JO - Applicable Analysis

JF - Applicable Analysis

SN - 0003-6811

IS - 12

ER -