Cauchy means of Dirichlet polynomials

Cauchy means of Dirichlet polynomials

We study Cauchy means of Dirichlet polynomials ∫R|∑n=1N1nσ+ist|2qdtπ(t2+1). These integrals were investigated when q=1,σ=1,s=1/2 by Wilf, using integral operator theory and Widom’s eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for th...

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Veröffentlicht in:Journal of approximation theory 2016-04, Vol.204, p.61-79
1. Verfasser: Weber, Michel J.G.
Format: Artikel
Sprache:eng
Schlagworte:
Arctangent density
Brownian motion
Cauchy density
Classical Analysis and ODEs
Dirichlet polynomials
Homogeneous kernels
Mathematics
Mean-value
Mellin transform
Number Theory
Spectral theory
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Zusammenfassung:We study Cauchy means of Dirichlet polynomials ∫R|∑n=1N1nσ+ist|2qdtπ(t2+1). These integrals were investigated when q=1,σ=1,s=1/2 by Wilf, using integral operator theory and Widom’s eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case q≥1, σ≥0 and s>0. We complete Wilf’s approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2016.01.001