Title of the Thesis

Ultramodulation Spaces, Wilson Bases and Pseudodifferential Operators (in Serbian)

Date and place of defense

11.12.2000, Faculty of Science, University of Novi Sad, Novi Sad, Yugoslavia

Outline of the Thesis

The first chapter contains a short review of the concepts and notations which will be used further on. The second section of the first chapter contains basic notions of the theory of group representations. 

In the first and the second section of second chapter the author recalls the construction of Wilson bases, and the general theory of coorbit spaces. The third section contains definition and the basic properties of a Gelfand - Shilov type space and its dual the space of tempered ultradistribution of Beurling type. The first original result of the thesis is a characterization of the Gelfand - Shilov type space based on symmetric decay properties of its elements and their Fourier transforms.

After the well known definition of modulation spaces, in the first section of third chapter we introduce almost exponential weights and ultramodulation spaces. We prove some basic properties of ultramodulation spaces based on the known facts on modulation spaces, e.g., behavour under the Fourier transform, relation to the Wiener amalgam spaces. Particularly, analysis and synthesis type result using Wilson bases as atoms

In the second section of third chapter we observe the projective limit of ultramodulation spaces and obtain results analogous to the known ones. The novelty is that instead of polynomial weights we have used weights of almost exponential growth. The proofs are based on the general theory of coorbit spaces developed by H. G. Feichtinger and K. Grochenig. The most important results of third chapter are decompositions of the Gelfand Shilov type space and its dual including the growth condition on the coefficients.

The first two sections of the fourth chapter are short introduction to the theory of pseudodifferential operators. In the third section of fourth chapter some known results of K. Tachizawa are rewritten.

We introduce new classes of pseudodifferential operators and show their symbols can be ultrapolynomials, that is the introduced classes contain "sufficiently many" operators.

The main original results in fourth chapter are twofold. Firstly, we obtain almost diagonalization of the pseudodifferential operators with Wilson basis functions as atoms. Based on that result we prove boundedness of the operators on ultramodulation spaces. In such a way we obtained nontrivial generalizations of known results, since we had to use thechniques from the theory of ultradistibutions to deal with elmost exponential weights.