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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.
