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Finding Chebyshev-Type Functions

Booth Id:
MATH014

Category:
Mathematics

Year:
2019

Finalist Names:
Cheng, Zong-Hong

Abstract:
This project is motivated by studying Chebyshev polynomials such as cos⁡(mx) is a polynomial of cos⁡(x). It is interesting to know if there are other non-constant continuous functions f having the similar property, i.e., f(mx) is a polynomial of f(x). In this study, we are able to characterize all non-constant continuous functions f from the positive real number set to the complex number set such that for all positive integer 𝑚, there exist polynomial P(x) satisfying f(mx)=P(f(x)). Furthermore, we generalize the problem by replacing the positive integer set with an arbitrary subset of the positive integer set.

Awards Won:
American Mathematical Society: Second Award of $1,000
Third Award of $1,000