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arXiv · 1403.1374

Orthogonal polynomials for Minkowski's question mark function

Zoé Dresse·Walter Van Assche
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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 $(P_n)_{n \in \mathbb{N}}$ 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.

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Zoé Dresse, Walter Van Assche. 2014-03-06. Orthogonal polynomials for Minkowski's question mark function. https://doi.org/10.1016/j.cam.2014.07.013

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