On approximation of ultraspherical polynomials in the oscillatory region

Ilia Krasikov

Research output: Contribution to journalArticlepeer-review


For k≥2 even, and α≥−(2k+1)∕4, we provide a uniform approximation of the ultraspherical polynomials Pk (α,α)(x) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g(x)=cb(x)(1−x2)(α+1)∕2Pk (α,α)(x)=cosB(x)+r(x). Here the constant c=c(k,α) is defined by the normalization of Pk (α,α)(x), B(x)=∫0 xb(x)dx, and the functions b(x) and B(x), as well as bounds on the error term r(x), are given by some rather simple elementary functions.

Original languageEnglish
Pages (from-to)143-156
Number of pages14
JournalJournal of Approximation Theory
StatePublished - Oct 2017
Externally publishedYes


  • Gegenbauer polynomials
  • Orthogonal polynomials
  • Ultraspherical polynomials
  • Uniform approximation


Dive into the research topics of 'On approximation of ultraspherical polynomials in the oscillatory region'. Together they form a unique fingerprint.

Cite this