Bernstein Polynomials and Random Walks on Hypergroups

Paolo M. Soardi
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Paolo M. Soardi: Dipartimento di Matematica dell’Università

A chapter in Probability Measures on Groups X, 1991, pp 387-393 from Springer

Abstract: Abstract The purpose of this note is to construct an analogue of Bernstein polynomials by means of random walks Z n on the hypergroups on N (the nonnegative integers) related to the Chebyshev polynomials of the second kind (see [S]). It will be shown that the expectations β n (f) = ε(f(Z n /n)) are polynomials of degree (at most) n which converge uniformly to f, as n →∞, for every continuous f on [0, 1]. We will also show that the β n(f)’s have the same relation with the Chebyshev polynomial of the second kind as the classical Bernstein polynomials have with the Chebyshev polynomials of the first kind.

Keywords: Random Walk; Markov Process; Chebyshev Polynomial; Bernstein Polynomial; Uniform Continuity (search for similar items in EconPapers)
Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-2364-6_29

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DOI: 10.1007/978-1-4899-2364-6_29

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