Orthogonal Dirichlet Polynomials

Doron S. Lubinsky ()
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Doron S. Lubinsky: Georgia Institute of Technology

A chapter in Approximation and Computation in Science and Engineering, 2022, pp 573-587 from Springer

Abstract: Abstract Let λ j j = 1 ∞ $$\left \{ \lambda _{j}\right \} _{j=1}^{\infty }$$ be a sequence of distinct positive numbers. Let w be a non-negative function, integrable on the real line. One can form orthogonal Dirichlet polynomials ϕ n $$\left \{ \phi _{n}\right \} $$ from linear combinations of λ j − i t j = 1 n $$\left \{ \lambda _{j}^{-it}\right \} _{j=1}^{n}$$ , satisfying the orthogonality relation ∫ − ∞ ∞ ϕ n t ϕ m t ¯ w t d t = δ m n . $$\displaystyle \int _{-\infty }^{\infty }\phi _{n}\left ( t\right ) \overline {\phi _{m}\left ( t\right ) }w\left ( t\right ) dt=\delta _{mn}. $$ Weights that have been considered include the arctan density w t = 1 π 1 + t 2 $$w\left ( t\right ) =\frac {1}{\pi \left ( 1+t^{2}\right ) }$$ ; rational function choices of w; w t = e − t $$w\left ( t\right ) =e^{-t}$$ ; and w t $$w\left ( t\right ) $$ constant on an interval symmetric about 0. We survey these results and discuss possible future directions.

Keywords: Dirichlet polynomials; Orthogonal polynomials; Christoffwl functions; Arctangent density (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-84122-5_30

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DOI: 10.1007/978-3-030-84122-5_30

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