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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.21 No.4

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Central limit theorem on Chebyshev polynomials

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2014, v.21 no.4, pp.271-279
https://doi.org/10.7468/jksmeb.2014.21.4.271
Ahn, Young-Ho
Ahn,, Y. (2014). Central limit theorem on Chebyshev polynomials. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 21(4), 271-279, https://doi.org/10.7468/jksmeb.2014.21.4.271

Abstract

Let <TEX>$T_l$</TEX> be a transformation on the interval [-1, 1] defined by Chebyshev polynomial of degree <TEX>$l(l{\geq}2)$</TEX>, i.e., <TEX>$T_l(cos{\theta})=cos(l{\theta})$</TEX>. In this paper, we consider <TEX>$T_l$</TEX> as a measure preserving transformation on [-1, 1] with an invariant measure <TEX>$\frac{1}{\sqrt[\pi]{1-x^2}}dx$</TEX>. We show that If f(x) is a nonconstant step function with finite k-discontinuity points with k < l-1, then it satisfies the Central Limit Theorem. We also give an explicit method how to check whether it satisfies the Central Limit Theorem or not in the cases of general step functions with finite discontinuity points.

keywords
Chebyshev polynomials, the central limit theorem, measure preserving, ergodic, weakly mixing, bounded variation function

Reference

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R.L. Adler & M.H. McAndrew. (1966). The entropy of Chebyshev polynomials. Trans. Amer. Math. Soc., 121, 236-241. 10.1090/S0002-9947-1966-0189005-0.

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A. Boyarsky & P. Gora. Laws of Chaos.

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G.H. Choe. Computational Ergodic Theory.

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W. Rudin. Real and Complex Analysis.

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P. Walters. An Introduction to Ergodic Theory.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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