Orthogonal polynomials for Minkowski’s question mark function

Abstract

Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski’s question mark function since Minkowski used the notation ?(x). This function is a distribution function on [0,1] which defines a singular continuous measure with support [0,1]. Our interest is in the (monic) orthogonal polynomials (Pn)n∈N for the Minkowski measure and in particular in the behavior of the recurrence coefficients of the three term recurrence relation. We will give some numerical experiments using the discretized Stieltjes–Gautschi method with a discrete measure supported on the Minkowski sequence. We also explain how one can compute the moments of the Minkowski measure and compute the recurrence coefficients using the Chebyshev algorithm.