Constrained Ultraspherical-weighted Orthogonal Polynomials on Triangle

under the Creative Commons Attribution License, which permits unrestricted use, distribution , and reproduction in any medium, provided the original work is properly cited. Abstract We construct Ultraspherical-weighted orthogonal polynomials C orthogonal system over the triangular domain T with respect to the Ultraspherical weight function.


Introduction
Recent years have seen a great deal in the field of orthogonal polynomials, the Ultraspherical orthogonal polynomials are Amongst these polynomials [1,2,10,13,20].Although the main definitions and properties were considered many years ago, the cases of two or more variables of orthogonal polynomials on triangular domains have been studied by few researchers [11,12,19].Proriol [15] introduced the definition of the bivariate orthogonal polynomials on the triangle, and the results were summarized by C.F. Dunkl and T. Koornwinder [5,11].
S. Waldron start the work of a generalized beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights.Also, he computed orthogonal polynomials on a triangle by degree raising.Farouki [7] defined the orthogonal polynomials with respect to the weight function W(u, v, w) = 1 on a triangular domain T.These polynomials P n,r (u, v, w) defined in [7], are orthogonal to each polynomial of degree ≤ n − 1 and also orthogonal to each polynomial P n,s (u, v, w), r = s.
Then {R nm (x, y)} are orthogonal on the square G with respect to the weight function W(x, y) = W (y).However, The construction of orthogonal polynomials over a triangular domain is not straightforward like the tensor product over the square.

Barycentric, and Bernstein Polynomials
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 = {(u, v, w) : u, v, w ≥ 0, u + v + w = 1}, can be written using the barycentric coordinates (u, v, w), as p=up 1 + vp 2 + wp 3 .The barycentric coordinates are given in the following ratios: where area(p 1 , p 2 , p 3 ) = 0, which means that p 1 , p 2 , p 3 are not collinear.
Let the notation 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 T are defined by the formula 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 over the triangular domain T, where the kth row contains k + 1 polynomials.Thus, for a basis of linearly independent polynomials of total degree n, there are a total of (1/2)(n + 1)(n + 2) linearly independent polynomials.Any polynomial p(u, v, w) of degree n can be written in the Bernstein form as with Bézier coefficients d ζ .We can also use the degree elevation algorithm for the Bernstein representation (1).This is obtained by multiplying both sides by 1 = u + v + w, and writing ζ are defined by, see [6,9] , The Bernstein polynomials b n ζ (u, v, w), |ζ| = n, on T satisfy, see [7], , where ∆ is double the area of T.
Let p(u, v, w) and q(u, v, w) be two bivariate polynomials over T, then we define their inner product over T by We say that p and q are orthogonal if p, q = 0.For m ≥ 1, we define L m = {p ∈ Π m : p ⊥ Π m−1 } to be the space of polynomials of degree m that are orthogonal to all polynomials of degree < m over a triangular domain T, and Π n is the space of all polynomials of degree n over the 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, see [4] for proof and more details.The following lemmas will be needed in the proof of the main results, see [7,14] for the proofs and more details.

Ultraspherical Polynomials
The Ultraspherical polynomials C (λ) n (x) of degree n are the orthogonal polynomials, except for a constant factor, on [−1, 1] with respect to the weight function In this paper, it is appropriate to take x ∈ [0, 1] for both Bernstein and Ultraspherical polynomials.
The following lemmas, See A. Rababah [17], will be needed in the construction of the orthogonal bivariate polynomials and the proof of the main results.For more details and the proofs, see [17].Although the Pochhammer symbol is more appropriate, the combinatorial notation will be used, Szegö [20], since it is more compact and readable formulas.
n (x) of degree ≤ n can be expressed in the Bernstein basis of fixed degree n by the following formula In addition, the following combinatorial identity, Lemma 3.3 [18], can be used for the main results simplifications.Lemma 3.3.For an integer n, we have the following combinatorial identity 2n n 2n 2k .

Ultraspherical-Weighted Polynomials
Analogous to [7], a simple closed-form representation of degree-ordered system of orthogonal polynomials is constructed on a triangular domain T. Since the Bernstein polynomials are stable [8], it is convenient to write these polynomials in Bernstein form.
For n = 0, 1, 2, . . .and r = 0, 1, . . ., n we define the bivariate polynomials where In this section, we show that the polynomials n,r (u, v, w) for 0 ≤ r ≤ n and n ≥ 0 form a degree-ordered orthogonal sequence over T.
We first rewrite these polynomials in the Ultraspherical polynomials form: −w ), and using Lemma 3.1 we get where r (t) is the univariate Ultraspherical polynomial of degree r and q n,r (w) is defined in equation (5).
For simplicity, since we are dealing with orthogonality, and the Ultraspherical polynomials C n (x) of degree n are the orthogonal except for a constant factor, we rewrite (8) as First, we show that the polynomials C (λ,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 4.1.For each n = 1, 2, . . ., r = 0, 1, . . ., n, and the weight function Proof.For each m = 0, . . ., n − 1, and s = 0, . . ., m we construct the set of bivariate polynomials The span of these polynomials includes the set of Bernstein polynomials which span Π n−1 .Thus, it is sufficient to show that for each m = 0, . . ., n − 1, s = 0, . . ., m, we have This is simplified to (12) By making the substitution t = u 1−w , we get If m < r, then we have s < r, and the first integral is zero by the orthogonality property of the Ultraspherical polynomials.If r ≤ m ≤ n − 1, we have by Lemma 3.4 the second integral equals zero, Thus the theorem follows.
Proof.For r = s, we have By making the substitution t = u 1−w , we have q n,r (w)q n,s (w)(1 − w) 2λ+γ+r+s dw, the first integral equals zero by orthogonality property of the Ultraspherical polynomials for r = s, and thus the theorem follows.

Ultraspherical in Bernstein Basis
The Bernstein-Bézier form of curves and surfaces exhibits some interesting geometric properties, see [6,9].So, we write the orthogonal polynomials C (λ,γ) n,r (u, v, w), r = 0, 1, . . ., n and n = 0, 1, 2, . . . in the following Bernstein-Bézier form: We are interested in finding a closed form for the computation of the Bernstein coefficients a n,r ζ .These are given explicitly in the following theorem: where µ n−k i,r are given in (4).
Proof.From equation ( 6), it is clear that C (λ,γ) n,r (u, v, w) has degree ≤ n − r in the variable w, and thus For 0 ≤ k ≤ n − r, the remaining coefficients are determined by equating ( 6) and ( 13) 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 Now, we get are the coefficients resulting from writing Ultraspherical polynomial of degree r in the Bernstein basis of degree n − k, as defined by expression (4).Thus, the required Bernstein-Bézier coefficients are given by: To derive a recurrence relation for the coefficients a n,r ijk of C n,r (u, v, w) = 0, i + j + k = n − 1, and thus by Lemma 2.2, we obtain (i + 1)a n,r i+1,j,k + (j + 1)a n,r i,j+1,k + (k + 1)a n,r i,j,k+1 = 0.
But, form Theorem 5.1, we have a n,r i,n−i,0 = µ n i,r for i = 0, 1, . . ., n; we can use (17) to generate a n,r i,j,k recursively on k.