Numerical Harmonic Analysis Group FOURIER STANDARD SPACES and the Kernel Theorem Hans G. Feichtinger
[email protected] www.nuhag.eu . Currently Guest Prof. at TUM (with H. Boche) Garching, TUM, July 13th, 2017 Hans G. Feichtinger
[email protected] www.nuhag.euFOURIER. Currently STANDARD GuestSPACES Prof. at TUM and the (with Kernel H. Boche) Theorem OVERVIEW d We will concentrate on the setting of the LCA group G = R , although all the results are valid in the setting of general locally compact Abelian groups as promoted by A. Weil. |||||||||||||||||||||- Classical Fourier Analysis pays a lot of attention to p d L (R ); k · kp because these spaces (specifically for p 2 f1; 2; 1g) are important to set up the Fourier transform as an integral transform which also respects convolution (we have the convolution theorem) and preserving the energy (meaning that it is 2 d a unitary transform of the Hilbert space L (R ); k · k2 ). |||||||||||||||||||||- d Occasionally the Schwartz space S(R ) is used and its dual 0 d S (R ), the space of tempered distributions (e.g. for PDE and d the kernel theorem, identifying operators from S(R ) to 0 d 0 2d S (R ) with their distributional kernels in S (R )). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem OVERVIEW II S d In the last 2-3 decades the Segal algebra 0(R ); k · kS0 1 d (equal to the modulation space (M (R ); k · kM1 )) and its dual, 0 d 1 d S 0 M ( 0 (R ); k · kS0 ) or (R ) have gained importance for many questions of Gabor analysis or time-frequency analysis.