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Analysis and Geometry of Wave Packet Systems


The goal of this proposal is a systematic study of wave packets - systems generated by countable families of dilations, modulations, and translations of a single function or a finite set of functions. Wavelets and (Multi-) Gabon systems are special examples of wave packets. Using the concept of Burling density we introduce notions of dimensions through which we shall characterize the systems of wave packets forming frames (rasp. Rises bases) for closed subspaces of a Hilbert space. This characterization will take the form of bounds for possible values of the dimensions of the sets of parameters of discrete wave packets. The method of the proof will be based on an adaptation of the Feichtinger-Groechenig theory of atomic decompositions of function spaces related to integral representations, combined with the theory of localization of Banish frames with respect to Rises bases, as presented recently by K. Grouching, Chill, P.Casazza and others recently. Further, the notion of equivalence of wave packets will be defined through these atomic and molecular decompositions of function spaces. It is expected that the geometric properties of the sets of parameters of wave packets will be reflected in these equivalence classes. It is known, for example, that Besot or Triebel-Lizorkin spaces admit unconditional wavelet bases. On the other hand, modulation spaces are characterized by Gabon frames or by localized cosine bases. We expect to obtain a method of verification of usefulness of wave packets for representations of specific function spaces based on the geometric properties of the sets of parameters of wave packets. In addition to the analysis of wave packets, we shall construct libraries of non-standard examples of wave packets with good regularity properties and with different geometrical structures. Such examples can then be used for the numerical implementation of fast and efficient algorithms.

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Dr. Karl Lueger-Ring 1

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