[Paper Review] Generalized Chebyshev polynomials of the second kind

Mohammad Alqudah|DergiPark (Istanbul University)|Apr 5, 2015

Mathematical functions and polynomials9 references4 citations

TL;DR

This paper introduces a closed-form representation of generalized Chebyshev polynomials of the second kind (Chebyshev-II) using the Bernstein basis, enabling efficient approximation of continuous functions via Chebyshev interpolation and series. The key contribution is a novel expression combining generalized Chebyshev-II polynomials with Bernstein polynomials, supported by integral identities involving the Beta function.

ABSTRACT

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials.

Motivation & Objective

• To characterize generalized Chebyshev polynomials of the second kind (Chebyshev-II) $\mathscr{U}_{r}^{(M,N)}(x)$ for improved function approximation.
• To derive a closed-form expression of these polynomials in the Bernstein basis to enhance computational efficiency in approximation theory.
• To establish efficient methods for computing Chebyshev interpolation and series approximations using the Bernstein representation.
• To derive new integral identities involving the generalized Chebyshev-II and Bernstein polynomials using the Eulerian (Beta) integral.

Proposed method

• The paper derives a closed-form expression of $\mathscr{U}_{r}^{(M,N)}(x)$ by expressing it as a linear combination of Bernstein basis polynomials $B_{k}^{n}(x)$ using binomial coefficients and generalized binomial terms.
• It applies the degree elevation formula for Bernstein polynomials to express lower-degree Bernstein polynomials in terms of higher-degree ones, enabling consistent basis representation.
• The method leverages the symmetry and product properties of Bernstein polynomials, including the identity $B_{i}^{n}(x)B_{j}^{m}(x) = \frac{\binom{n}{i}\binom{m}{j}}{\binom{n+m}{i+j}}B_{i+j}^{n+m}(x)$.
• It uses the generalized binomial coefficient $\binom{r+\frac{1}{2}}{k}$ and the identity $(2n+1)!! = \frac{2^{n}(2n)!}{n!}$ to simplify expressions involving double factorials.
• The derivation involves substitution of Bernstein basis forms into the generalized Chebyshev-II polynomial expression and simplification using the Beta function $B(x,y) = \int_{0}^{1} u^{x-1}(1-u)^{y-1} du$.
• The method includes a recurrence relation for the coefficients $\vartheta_{i,r}$, derived from binomial coefficient identities and symmetry properties.

Experimental results

Research questions

• RQ1 How can generalized Chebyshev polynomials of the second kind be represented in the Bernstein basis for improved computational stability and efficiency?
• RQ2 What closed-form expression can be derived for $\mathscr{U}_{r}^{(M,N)}(x)$ using Bernstein polynomials and generalized binomial coefficients?
• RQ3 How do the integrals of products of generalized Chebyshev-II and Bernstein polynomials behave, and can they be expressed in terms of the Beta function?
• RQ4 What role does the Eulerian integral of the first kind (Beta function) play in simplifying integrals involving these generalized polynomials?

Key findings

• The generalized Chebyshev-II polynomial $\mathscr{U}_{r}^{(M,N)}(x)$ is expressed as a linear combination of Bernstein basis polynomials $B_{k}^{n}(x)$, with coefficients involving $\binom{r+\frac{1}{2}}{k}$, $\binom{r}{k}$, and $(-1)^{r-k}$.
• A closed-form expression is derived: $\mathscr{U}_{r}^{(M,N)}(x) = \frac{(2r+1)!!}{2^{r}(r+1)!}\sum_{i=0}^{r}(-1)^{r-i}\vartheta_{i,r}B_{i}^{r}(x) + \sum_{k=0}^{r}\lambda_{k}\frac{(2k+1)!!}{2^{k}(k+1)!}\sum_{j=0}^{k}(-1)^{k-j}\vartheta_{j,k}B_{j}^{k}(x)$, where $\vartheta_{i,r} = \frac{\binom{r+\frac{1}{2}}{i}\binom{r+\frac{1}{2}}{r-i}}{\binom{r}{i}}$.
• The integral $\int_{0}^{1}x^{\frac{1}{2}}(1-x)^{\frac{1}{2}}B_{r}^{n}(x)\mathscr{U}_{i}^{(M,N)}(x)dx$ is evaluated as a sum of Beta function terms involving $B(r+k+\frac{3}{2}, n+i-r-k+\frac{3}{2})$.
• The integral result is expressed as a double sum over $k$ and $j$, with coefficients involving $\binom{i+\frac{1}{2}}{k}$, $\binom{i+\frac{1}{2}}{i-k}$, and the Beta function, enabling efficient numerical evaluation.
• The recurrence relation for $\vartheta_{i-1,r}$ is derived as $\vartheta_{i-1,r} = \frac{(2i+1)(2r+1)^2\binom{2r}{r}\binom{2r}{2i}}{2^{2r}(2r-2i+1)(2i+1)(2r-2i+3)\binom{r}{i}}$ for $i=1,\dots,r$.
• The paper establishes that the product of two Bernstein polynomials is another Bernstein polynomial, a key property enabling recursive and stable computation in the proposed framework.