Multiparameter Families of Function Spaces, Pitfalls and ChancesHans G. Feichtinger (NuHAG, Univ. Vienna, Faculty Mathematics) given at (16.09.21 11:00) id: 3705 length: 20min status: given type: LINK-Presentation: https://nuhagphp.univie.ac.at/dateien/talks/3705_FeiLubos21A.pdf ABSTRACT: Throughout my work in harmonic analysis and its applications families of function spaces have played a big role, sometimes as an technical tool, sometimes as an independent subject of interest. For me the design of families of function spaces (for me this is Banach spaces of distributions over locally compact Abelian groups, like the Euclidean space) is one of the key challenges. It is natural, that often progress is made by going from a few concrete cases to more general situations, always asking to which extent the existing function spaces are appropriate for a discussion of the continuity properties of linear operators arising in a certain context. Of course the classical Lebesgue spaces are the prototypical examples of such a “scale of spaces”, and the masters of interpolation theory (Jaak Peetre and Hans Triebel) have shown us that the well-known Besov-Triebel-Lizorkin spaces form families with similar properties. For me their work was the motivation to introduce Wiener amalgam spaces, modulation spaces and decomposition spaces. The short message of this short talk will be: If you extend or generalize a given result to a larger family, you have to do it in the right way. Just to give an example: Having a multi-parameter family of spaces (like weighed Lp-spaces with a family of weights) it is a big difference whether one claims that a given operator is bounded in any of these spaces of this family, or whether (hopefully constructively) a uniform bound can be established. I have seen various cases where the second claim can be obtained from the proofs, while the first one was stated in the results of a paper. Further examples are related to the irregular sampling problem or the theory of Banach frames. |