Orthogonal polynomials for the Minkowski question Mark function

Analysis Seminar
Wednesday, December 2, 2015 - 2:05pm
1 hour (actually 50 minutes)
Skiles 006
University of Leuven, Belgium
The Minkowski question mark function is a singular distribution function arising from Number Theory: it maps all quadratic irrationals to rational numbers and rational numbers to dyadic numbers. It generates a singular measure on [0,1]. We are interested in the behavior of the norms and recurrence coefficients of the orthonormal polynomials for this singular measure. Is the Minkowski measure a "regular" measure (in the sense of Ullman, Totik and Stahl), i.e., is the asymptotic zero distribution the equilibrium measure on [0,1] and do the n-th roots of the norm converge to the capacity (which is 1/4)? Do the recurrence coefficients converge (are the orthogonal polynomials in Nevai's class). We provide some numerical results which give some indication but which are not conclusive.