CONSTRUCTION OF TCHEBYSHEV-II WEIGHTED ORTHOGONAL POLYNOMIALS ON TRIANGULAR

We construct Tchebyshev-II (second kind) weighted orthogonal polynomials U (γ) n,r (u, v, w), γ > −1, on the triangular domain T. We show that U (γ) n,r (u, v, w), n = 0, 1, 2, . . . , r = 0, 1, . . . , n, form an orthogonal system over T with respect to the Tchebyshev-II weight function. AMS Subject Classification: 42C05, 33C45, 33C70


Introduction
In the last couple decades, orthogonal polynomials have been studied thoroughly [8] and [12].The Tchebyshev orthogonal polynomial of the second kind (Tchebyshev-II) is among these orthogonal polynomials.Although the main definitions and basic properties were defined many years ago, see [3] and [11], the cases of bivariate or more variables are limited.
The construction of bivariate orthogonal polynomials on the square G is straightforward [11], where G = {(x, y) : The bivariate polynomials {R nm (x, y)} are orthogonal on the square G with respect to the weight function W(x, y) = W 1 (x)W 2 (y).However, the construction of orthogonal polynomials over a triangular domains are not straightforward like the tensor product over the square G.

Bernstein and Orthogonal Polynomials over Triangular Domains
Consider a base triangle in the plane with the vertices p k = (x k , y k ), k = 1, 2, 3. Then every point p inside the triangle T can be written using the barycentric coordinates (u, v, w) as p=up 1 + vp 2 + wp 3 , where u, v, w ≥ 0, u + v + w = 1.The barycentric coordinates are the ratio of areas of subtriangles of the base triangle as follows: where p 1 , p 2 , p 3 are not collinear.
Definition 2.1.The Bernstein polynomials b n i (u), u ∈ [0, 1], i = 0, 1, . . ., n, are defined by: where n i are the binomial coefficients.Let ζ = (i, j, k) denote triples of nonnegative integers, where |ζ| = i + j + k.The generalized Bernstein polynomials of degree n on the triangular domain Note that the generalized Bernstein polynomials are nonnegative over T and form a partition of unity, These polynomials define the Bernstein basis for the space Π n , the space of all polynomials of degree n over the triangular domain T.
A basis of linearly independent and mutually orthogonal polynomials in the barycentric coordinates (u, v, w) are constructed over T.These polynomials are represented in the following triangular table The kth row of this table contains k + 1 polynomials.Thus, there are (n+1)(n+2) 2 polynomials in a basis of linearly independent polynomials of total degree n.Therefore, the sum (3) involves a total of (n+1)(n+2) 2 linearly independent polynomials.
Any polynomial p(u, v, w) of degree n can be written in the Bernstein form with Bézier coefficients d ζ .We can use the degree elevation algorithm for the Bernstein representation (4) by multiplying both sides by 1 = u + v + w and writing where the the coefficients d ζ are defined in [4] and [7] as , where ∆ is double the area of T.
Definition 2.2.Let p(u, v, w) and q(u, v, w) be two bivariate polynomials over T, then their inner product over T defined by p, q = 1 ∆ T pqdA, where p and q are orthogonal if p, q = 0.
is the space of polynomials of degree m that are orthogonal to all polynomials of degree < m over a triangular domain T.
Let f (u, v, w) be an integrable function over T and consider the operator ! is an eigenvalue of the operator S n , and L m is the corresponding eigenspace [2].The following two lemmas will be used in the proof of the main results.
For the main results simplifications, we define the double factorial of an integer n as if n is even (6) where 0!! = (−1)!! = 1.

Tchebyshev-II Weighted Orthogonal Polynomials
Tchebyshev-II polynomials U n (x) of degree n are the orthogonal polynomials except for a constant factor on [−1, 1] with respect to the weight function . For simplicity, without loss of generality, we take x ∈ [0, 1] for both Bernstein and Tchebyshev-II polynomials.
The following lemmas will be needed in the construction of the orthogonal bivariate polynomials and the proof of the main results.Lemma 3.1.[10] The Tchebyshev-II polynomials U r (x) have the Bernstein representation: Lemma 3.2.
[10] The Tchebyshev-II polynomials U 0 (x), . . ., U n (x) of degree ≤ n can be expressed in the Bernstein basis of fixed degree n by the following formula Using Pochhammer symbol is more appropriate in (3.1) and ( 8), but the combinatorial notation gives more compact and readable formulas, these have also been used by Szegö [12].
In the following lemma, let The polynomial q n,r (w) is a scalar multiple of U n−r (1 − 2w).
Rewriting (10) using Tchebyshev-II polynomials form, we obtain Using Lemma 3.1 and , and we get where U r (t) is the univariate Tchebyshev-II polynomial of degree r and q n,r (w) is defined in equation ( 9).For simplicity, we rewrite (12) as since we are dealing with orthogonality, and the Tchebyshev-II polynomials U n (x) of degree n are the orthogonal except for a constant factor.The polynomials n,s (u, v, w).In the following theorem, we show that the polynomials U (γ) n,r (u, v, w), r = 0, . . ., n, are orthogonal to all polynomials of degree less than n over the triangular domain T.
Theorem 3.1.For each r = 0, 1, . . ., n and n = 1, 2, . . ., be the set of bivariate polynomials.The span of (14) includes the set of Bernstein polynomials It is sufficient to show that for each s = 0, . . ., m; m = 0, . . ., n − 1, The integral (15) can be simplified to Using the substitution t = u 1−w in (16) we have If m < r, then s < r, the first integral is zero by the orthogonality property of the Tchebyshev-II polynomials.If r ≤ m ≤ n − 1, then by Lemma 3.3 the second integral equals zero.Thus the theorem follows.
Note that taking W (γ) (u, v, w) = u In the following theorem, we show that U n,r (u, v, w) is orthogonal to each polynomial of degree n.Proof.For r = s, we have Using the substitution t = u 1−w , we get q n,r (w)q n,s (w)(1 − w) γ+r+s+2 dw.
the first integral equals zero by orthogonality property of the Tchebyshev-II polynomials for r = s, and thus the theorem follows.

Orthogonal Polynomials in Bernstein Basis
The Bernstein-Bézier form of curves and surfaces exhibits some interesting geometric properties, see [4] and [7].Writing the orthogonal polynomials U (γ) n,r (u, v, w), r = 0, 1, . . ., n and n = 0, 1, 2, . . . in the following Bernstein-Bézier form: The following theorem provides a closed form of the Bernstein coefficients a n,r ζ .Theorem 4.1.The Bernstein coefficients a n,r ζ are given by where µ n−k i,r are given in (8).
Proof.From equation (10), it is clear that U n,r (u, v, w) has degree ≤ n − r in the variable w, thus a n,r ijk = 0 f or k > n − r.
For 0 ≤ k ≤ n − r, the remaining coefficients are determined by equating (10) and (17) as follows Comparing powers of w on both sides, we have The left hand side of the last equation can be written in the form where µ n−k i,r are the coefficients resulting from writing Tchebyshev-II polynomial of degree r in the Bernstein basis of degree n − k, as defined by expression (8).The result in (18) follows.

1 2
(1 − w) γ enables us to separate the integrand in the proof of Theorem 3.1.Also taking γ > −1 enables us to use Lemma 3.3 in the proof of Theorem 3.1.