On $(\beta,\gamma)$-Chebyshev functions and points of the interval

TL;DR

This paper introduces a new class of generalized Chebyshev functions and points, exploring their orthogonality, polynomial properties, and implications for interpolation and Lebesgue constants.

Contribution

It defines $(eta,\, extgamma)$-Chebyshev functions, analyzes their orthogonality, and links them to classical Chebyshev points, extending understanding of polynomial interpolation.

Findings

$(eta, extgamma)$-Chebyshev functions are orthogonal in certain intervals.

Subsets of Chebyshev points are special cases of these new points.

Lebesgue constants vary with parameters, affecting interpolation stability.

Abstract

In this paper, we introduce the class of $(\beta,\gamma)$-Chebyshev functions and corresponding points, which can be seen as a family of {\it generalized} Chebyshev polynomials and points. For the $(\beta,\gamma)$-Chebyshev functions, we prove that they are orthogonal in certain subintervals of $[-1,1]$ with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of $(\beta,\gamma)$-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters $\beta$ and $\gamma$.