Padé Interpolation by F-Polynomials and Transfinite Diameter

Research output: Contribution to journalArticle

Abstract

We define F-polynomials as linear combinations of dilations by some frequencies of an entire function F. In this paper, we use Padé interpolation of holomorphic functions in the unit disk by F-polynomials to obtain explicitly approximating F-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set K⊂ℂ, then optimal choices for the frequencies of interpolating F-polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of K. In case of the Laplace transforms of measures on K, we show that the coefficients of interpolating F-polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle that ensures that the sums of the absolute values of the coefficients of interpolating F-polynomials stay bounded.

Original languageEnglish (US)
Pages (from-to)311-329
Number of pages19
JournalConstructive Approximation
Volume36
Issue number2
DOIs
StatePublished - Oct 1 2012

Keywords

  • Approximation of holomorphic functions
  • Exponential polynomials
  • Fekete points
  • Laplace transforms
  • Padé interpolation
  • Transfinite diameter

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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