Abstract
While the previous chapter extends the theory of the short-time Fourier transform from L 2 to the modulation spaces, this chapter extends the theory of Gabor frames from L 2 to the modulation spaces. The generalization follows the outline of Chapters 6 and 7. We first determine suitable window classes for the short-time Fourier transform on the modulation spaces and investigate their properties. Section 2 then explores the mapping properties of the analysis and synthesis operators. Then the boundedness of the Gabor frame operator on the modulation spaces proves sufficient to characterize modulation spaces via tight Gabor frames or Wilson bases. Section 4 treats a problem in data compression by means of non-linear approximation with Wilson bases.
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© 2001 Springer Science+Business Media New York
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Gröchenig, K. (2001). Basic Fourier Analysis. In: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0003-1_2
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DOI: https://doi.org/10.1007/978-1-4612-0003-1_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6568-9
Online ISBN: 978-1-4612-0003-1
eBook Packages: Springer Book Archive