BIJ: Matlab programs for testing and extending Bernstein's Inequality for Jacobi polynomials

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By Walter Gautschi

Purdue University

Bernstein’s inequality for Jacobi polynomials is analyzed here analytically and computationally with regard to validity and sharpness

Version 1.0 - published on 23 Apr 2014 doi:10.4231/R7V985Z5 - cite this Archived on 25 Oct 2016

Licensed under CC0 1.0 Universal

Description

Bernstein’s inequality for Jacobi polynomials with parameters alpha, beta is analyzed here analytically and, above all, computationally with regard to validity and sharpness. Computation suggests that the inequality holds with new, somewhat larger, constants in any region R_s={-1/2<=alpha<=s, -1/2<=beta<=s}. Best constants are provided for s =1 : .5 : 4 and s =5 : 1 : 10. The work also sheds new light on the so-called Erdelyi–Magnus–Nevai conjecture for orthonormal Jacobi polynomials, adding further support for its validity and suggesting .66198126. . . as the best constant implied in the conjecture.

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This dataset contains 4 Matlab codes:

  • bernstein.m
  • fbern.m
  • f1bern.m
  • testbern.m

Serves as a companion piece to the paper "How sharp is Bernstein's inequality for Jacobi polynomials?" ETNA. Electronic Transactions on Numerical Analysis [electronic] 36 (2009), pp. 1-8. URL: http://eudml.org/doc/225564.

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