Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study
Journal of Approximation Theory
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Let {@l"j}"j"="1^~ be a strictly increasing sequence of positive numbers with @l"1=1. We find a simple explicit formula for the orthogonal Dirichlet polynomials {@f"n} formed from linear combinations of {@l"j^-^i^t}"j"="1^n, associated with the arctangent density. Thus @!"-"~^~@f"n(t)@f"m(t)@?dt@p(1+t^2)=@d"m"n. We obtain formulae for their Christoffel functions, and deduce their asymptotics, as well as universality limits, and spacing of zeros for their reproducing kernels. We also investigate the relationship between ordinary Dirichlet series, and orthogonal expansions involving the {@f"n}, and establish Markov-Bernstein inequalities.