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Harmonic Analysis and Extremal Problems

Bilateral DAAD-Tempus Project

with the Rényi Alfréd Institute of Mathematics, Hungarian Academy of Science

Participants

  • Prof. Dr. Balint Farkas (Wuppertal)
  • Prof. Dr. Szilard Revesz (Budapest)
  • Christian Budde (PhD student, Wuppertal)
  • Marcell Gaál (PhD student, Budapest)
  • Dr. Zsigmond Tarcsay (PostDoc, Budapest)
  • Prof. Dr. Béla Nagy (Szeged)
  • NN. (graduate student, Budapest)
  • NN. (graduate student, Wuppertal)
  • NN. (graduate student, Budapest)

Project description

 

We carry out research in four related areas of mathematical analysis, with the common feature that the problems are of extremal nature and they come from harmonic analysis or can be attacked by its tools. Now harmonic analysis is interpreted in the wide sense including potential theory and Fourier analysis:

1. We extend the notion of asymptotic uniform upper density from LCA groups to more general topological groups (with growth conditions). We study the relations between the various notions of positive definiteness.

2. We study packing density via Turán type extremal problems and utilizing estimates obtained from Delsarte problems. This research should be carried out also on groups, where again the settling of the questions in part 1 is decisive. We explore the linear programming duality of extremal problems to answer a question of S. Konyagin in the context of LCA groups. Positive definite functions in the unitary group are important in the study if mutually unbiased bases. In this area there are more really deep conjectures than answers. We apply the result to estimating packing density.

3. We study a potential theoretic minimax problem involving sums of translates of different functions, thus extending our previous work which concerned the unit circle, with application to generalized trigonometric polynomials. Related problems like the linear polarization problem will be also studied.

4. We study extremal sequences for the Fourier transform of measures on the unit circle, and applications in operator theory. We extend the results to more general group, like R, or even to some LCA groups, and apply the results to semigroup representations.