Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Tuesday, July 20, 19:40 ~ 20:10 UTC-3

The Tur\'an problem for a ball centered at the origin in $\mathbb{R}^d$ and its dual.

Jean-Pierre Gabardo

McMaster University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Tur\'an problem for $B_r$, the open ball of radius $r$ centered at the origin in $\mathbb{R}^d$, consists in computing the supremum of the integrals of positive definite functions supported on that ball and taking the value $1$ at the origin. Siegel proved, in the thirties, that this supremum has the value $c_r=2^{-d}\,|B_r|$, where $|\cdot|$ denotes the Lebesgue measure and is reached by the function $f_r=c_r^{-1}\,\chi_{B_{r/2}}*\chi_{B_{r/2}}$. Several proofs of this result are know and, in this talk, we will outline a new proof of it based on the explicit construction of a maximizer for the dual Tur\'an problem, which is a positive-definite distribution $T_r$ equal to the Dirac delta function $\delta_0$ on $B_r$ and which satisfies $f_r*T_r=1$ on $\mathbb{R}^d$. As an intermediary step needed for this result, we construct new families of Parseval frames, consisting of Bessel functions, on the interval $[0,r]$.

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