Mean Value Theorems for Chebyshev Polynomials T_n (x) and U_n (x)

Abstract

In this paper, we explore mean value theorems for Chebyshev polynomials of the first kind  and the second kind . These theorems provide key insights into the average behavior of the Chebyshev polynomials over the interval . Additionally, we present numerical examples to demonstrate the results derived from the mean value theorems. The findings have important applications in approximation theory and numerical analysis.