## Doctoral Dissertations

#### Title

Modulation spaces and nonlinear approximation

January 1998

Mathematics

Ph.D.

#### Abstract

It is shown that the theory of modulation spaces M\$\sbsp{p}{w}\$ can be extended to the case \$0 < p < 1\$. In particular, these spaces admit atomic decompositions similar to the case \$p \geq 1\$. It is also shown that local Fourier bases are unconditional bases for all modulation spaces \$M\sbsp{p}{w}\$ on \$\IR\$, including the Bessel potential spaces, and the Segal algebra \$S\sb0\$. The non-linear approximation procedure is used to show that the abstract spaces which are characterized by the approximation properties with respect to a local Fourier basis are exactly the modulation spaces over \$\IR\$. As a consequence, the error in approximating elements in the modulation spaces by a linear combination of N elements of a local Fourier basis is determined. Also, the error in approximating elements in the modulation spaces by a linear combination of N Gabor atoms is determined. ^

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