Harmonic Analysis and Applications

June 4-8, 2018


"Two Ubiquitous Constants other than E and Pi"

Faulhuber, Markus

The aim of this talk is to provide evidence that the Strohmer and Beaver conjecture on Gaussian Gabor frames (2003) might root much deeper than expected. A solution might come from analytic number theory; in particular, the theory of hypergeometric functions and elliptic integrals as studied already by Gauss and Jacobi, developed further by Ramanujan and put on a solid basement by the Borweins. Starting from the hexagonal and the square lattice of density 2, we will have a closer look at the sharp frame bounds of the corresponding Gaussian Gabor system. In the square case, it is known (Faulhuber and Steinerberger 2017) that the frame bounds are extremal within the family of separable lattices. In the hexagonal case, we know that the upper frame bound is extremal among all lattices. The conjecture is that the lower frame bound is extremal if and only if the lattice is hexagonal. Afterwards, we will study the problem of finding Landau's "Weltkonstante" (universal constant). It can be seen as a packing problem for a certain family of holomorphic functions: Find the (universal) radius of a disk, fitting into any image of a function from the mentioned family. Landau's problem can also be formulated as a problem on the hyperbolic density of universal covering maps (UCM), in particular UCMs for tori. Finally, we will see that the constants appearing in the mentioned problems are the same. Furthermore, we will see examples of where else these constants appear; e.g. the arithmetic-geometric mean, the arc length of the lemniscate, the theory of hypergeometric functions and theta functions or a minimum problem for the heat kernel on a torus.

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