A note on orthogonal polynomials described by Chebyshev polynomials

Note on Certain Orthogonal Polynomials

Bivariate Chebyshev-i Weighted Orthogonal Polynomials on Simplicial Domains

We construct a simple closed-form representation of degree-ordered system of bivariate Chebyshev-I orthogonal polynomials Tn,r(u, v, w) on simplicial domains. We show that these polynomials Tn,r(u, v, w), r = 0, 1, . . . , n; n ≥ 0 form an orthogonal system with respect to the Chebyshev-I weight function.

Discrete orthogonal polynomials on Gauss-Lobatto Chebyshev nodes

In this paper we present explicit formulas for discrete orthogonal polynomials over the so-called Gauss-Lobatto Chebyshev points. We also give the “three-term recurrence relation” to construct such polynomials. As a numerical application, we apply our formulas to the least-squares problem.

On integer Chebyshev polynomials

We are concerned with the problem of minimizing the supremum norm on [0, 1] of a nonzero polynomial of degree at most n with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and impr...

Curves Defined by Chebyshev Polynomials

Working over a field k of characteristic zero, this paper studies line embeddings of the form φ = (Ti, Tj , Tk) : A → A, where Tn denotes the degree n Chebyshev polynomial of the first kind. In Section 4, it is shown that (1) φ is an embedding if and only if the pairwise greatest common divisor of i, j, k is 1, and (2) for a fixed pair i, j of relatively prime positive integers, the embeddings ...