Central limit theorem on Chebyshev polynomials

한국수학교육학회지시리즈B:순수및응용수학 / 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 (Department of Mathematics, Mokpo National University)

Ahn, Young-Ho. (2014). CENTRAL LIMIT THEOREM ON CHEBYSHEV POLYNOMIALS. 한국수학교육학회지시리즈B:순수및응용수학, 21(4), 271-279, https://doi.org/10.7468/jksmeb.2014.21.4.271

복사

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

참고문헌

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.

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

G.H. Choe. Computational Ergodic Theory.

W. Rudin. Real and Complex Analysis.

P. Walters. An Introduction to Ergodic Theory.

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논문 상세

Vol.21 No.4

CENTRAL LIMIT THEOREM ON CHEBYSHEV POLYNOMIALS

Central limit theorem on Chebyshev polynomials

Abstract

참고문헌

한국수학교육학회지시리즈B:순수및응용수학