Orthogonal Dirichlet polynomials with arctangent density

Let {λj}j=1∞ be a strictly increasing sequence of positive numbers with λ1=1. We find a simple explicit formula for the orthogonal Dirichlet polynomials {ϕn} formed from linear combinations of {λj−it}j=1n, associated with the arctangent density. Thus ∫−∞∞ϕn(t)ϕm(t)¯dtπ(1+t2)=δmn. We obtain formulae...

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Bibliographic Details
Published inJournal of approximation theory Vol. 177; pp. 43 - 56
Main Author Lubinsky, Doron S.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2014
Subjects
Dirichlet polynomials
Orthogonal polynomials
Dirichlet polynomials
Orthogonal polynomials
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Summary:Let {λj}j=1∞ be a strictly increasing sequence of positive numbers with λ1=1. We find a simple explicit formula for the orthogonal Dirichlet polynomials {ϕn} formed from linear combinations of {λj−it}j=1n, associated with the arctangent density. Thus ∫−∞∞ϕn(t)ϕm(t)¯dtπ(1+t2)=δmn. 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 {ϕn}, and establish Markov–Bernstein inequalities.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2013.10.001