Cubature Formula for the Generalized Chebyshev Type Polynomials

We study cubature formulas to approximate double integrals of generalized Chebyshev-type polynomials of the second type, U (U), over triangular domain.


Introduction and Motivation
Last few decades have seen a great deal in the field of orthogonal polynomials [1,13,22,23].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 [2,5,20,21].Proriol introduced the definition of the bivariate orthogonal polynomials on the triangle, and the results were summarized by C.F. Dunkl and T. Koornwinder [14,20].
In addition, recent years have seen a great deal in the field of generalized classical polynomials [3,4,8], and their applications [9,11,10], the generalized Chebyshev type polynomials of second type are amongst these polynomials.

Barycentric Coordinates
Consider a triangle T defined by its three vertices p k = (x k , y k ), k = 1, 2, 3.For each point p located inside the triangle, there is a sequence of three numbers u, v, w ≥ 0 such that p can be written uniquely as a convex combination of the three vertices, p = up 1 + vp 2 + wp 3 , where u + v + w = 1.The three numbers u = area(p,p 2 ,p 3 ) area(p 1 ,p 2 ,p 3 ) , v = area(p 1 ,p,p 3 ) area(p 1 ,p 2 ,p 3 ) , w = area(p 1 ,p 2 ,p) area(p 1 ,p 2 ,p 3 ) indicate the barycentric "area" coordinates of the point p with respect to the triangle, where area(p 1 , p 2 , p 3 ) = 0, which means that p 1 , p 2 , p 3 are not collinear.
Although there are three coordinates, there are only two degrees of freedom, since u + v + w = 1.Thus every point is uniquely defined by any two of the barycentric coordinates.That is, the triangular domain defined as (1.2)

Bernstein Polynomials
We recall a very concise overview of well-known results on Bernstein polynomials, followed by a brief summary of important properties.
Definition 1.The n + 1 Bernstein polynomials B n i (x) of degree n, x ∈ [0, 1], i = 0, 1, . . ., n, are defined by: The Bernstein polynomials have been studied thoroughly and there are a fair amount of literature on these polynomials, they are known for their geometric properties [15,19], and the Bernstein basis form is known to be optimally stable.They are all non-negative, B n k (x) ≥ 0, x ∈ [0, 1], has a single unique maximum of n i i i n −n (n − i) n−i at x = i n , i = 0, . . ., n, their roots are x = 0, 1 with multiplicities, and they form a partition of unity (normalization), satisfy symmetry relation , and the product of two Bernstein polynomials is also a Bernstein polynomial which is given by . The Bernstein polynomials of degree n can be defined by combining two Bernstein polynomials of degree n − 1, where the kth nth-degree Bernstein polynomial can be written by the known recurrence relation as Moreover, it is possible to write each Bernstein polynomials of degree r where r ≤ n in terms of Bernstein polynomials of degree n using the following degree elevation [18]: For ζ = (i, j, k) denote triples of non-negative integers such that |ζ| = i + j + k, then the generalized Bernstein polynomials of degree n are defined by the formula The generalized Bernstein polynomials have a number of useful analytical and elegant geometric properties [17].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 at most n.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 (1.5) involves a total of (n+1)(n+2) 2 linearly independent polynomials.Thus, with the revolt of computer graphics, Bernstein polynomials on [0, 1] became important in the form of Bézier curves, and the polynomials determined in the Bernstein (Bézier) basis enjoy considerable popularity in Computer Aided Geometric Design applications.
Degree elevation is a common situation in these applications, where polynomials given in the basis of degree n have to be represented in the basis of higher degree.
Any polynomial p(u, v, w) of degree n can be written in the Bernstein form With the use of degree elevation algorithm (1.4) for the Bernstein representation [18], the polynomial p(u, v, w) in (1.4) can be written as The new coefficients defined by Hoschek et al. [19] as d n+1 (id i−1,j,k + jd i,j−1,k + kd i,j,k−1 ) where |ζ| = n + 1.Moreover, the next integration is one of the interesting analytical properties of the Bernstein polynomials B n ζ (u, v, w).

Integration over Triangular Domains
The integral of a function f (u, v, w) over the triangular domain T defined in (1.2) can be expressed as where A is the area of T. We can also formulate (1.6) as a double integral over v and w or over u and w, by using u + v + w = 1.However, for integral of the generalized Bernstein polynomial, we have the following lemma.
[16] The Bernstein polynomials , where ∆ is the double the area of T and n+2 2 is the dimension of Bernstein polynomials over the triangle.
This means that the Bernstein polynomials partition the unity with equal integrals over the domain; in other words, they are equally weighted as basis functions.

The Generalized Chebyshev-II Polynomials
For M, N ≥ 0, the generalized Chebyshev-II polynomials U defined in [20], and been characterized in [8], where U n (x) is the Chebyshev-II polynomial of degree n in x, Szegö [23], and the double factorial of an integer n is given by given that 0!! = (−1)!! = 1.
The next theorem, see [8] for the proof, provides a closed form for generalized Chebyshev-II polynomial U (M,N ) r (x) of degree r as a linear combination of the Bernstein polynomials B r i (x), i = 0, 1, . . ., r of degree r.Theorem 3. [8] For M, N ≥ 0, the generalized Chebyshev-II polynomials U (M,N ) r (x) of degree r have the following Bernstein representation, where λ k defined by (2.2), and (2.4) Now, we have the following corollary which enables us to write Chebyshev-II polynomials of degree r where r ≤ n in terms of Bernstein polynomials of degree n. where

Generalized Bivariate Chebyshev-II Polynomials
In this section, we generalize the construction in [6] to formulate a simple closed-form representation of degree-ordered system of generalized orthogonal polynomials U (γ,M,N ) n,r,d (u, v, w) on a triangular domain T.
The basic idea in this construction is to make U (γ,M,N ) n,r,d (u, v, w) coincide with the univeriate Chebyshev-II polynomial along one edge of T, and to make its variation along each chord parallel to that edge a scaled version of this Chebyshev-II polynomial.The variation of U (γ,M,N ) n,r,d (u, v, w) with w can then be arranged so as to ensure its orthogonality on T with every polynomial of degree < n, and with other basis polynomials U Now, for M, N ≥ 0, γ > −1, n = 0, 1, 2, . . ., k = 0, . . ., n, r = 0, 1, . . ., n and d = 0, 1, . . ., k, we define the generalized bivariate polynomials ) where B r i (u, v) defined in (1.3), λ k defined in (2.2), and ϑ i,r defined in (2.4).The Bernstein-Bézier form of curves and surfaces exhibits some interesting geometric properties, see [17,19].So, for computational purposes, we are interested in finding a closed form of the Bernstein coefficients a n,r,d ζ , and the recursion relation that allow us to compute the coefficients efficiently.
We write the orthogonal polynomials U where M n i,r defined by where γ > −1, B r i (u, v), i = 0, 1, . . ., r, defined in equation (1.3), and λ k defined in (2.2).Comparing powers of w on both sides, we have The left hand side of the last equation can be written in the form where ∆ is the double the area of T and n+2 2 is the dimension of Bernstein polynomials over the triangle, which completes the proof of the interpolatory cubature formula.
v, w) are orthogonal to each polynomial of degree ≤ n − 1, with respect to the generalized weight function (1.1).However, for r = s, d = m, U (γ,M,N ) n,r,d (u, v, w) and U (γ,M,N ) n,s,m (u, v, w) are not orthogonal with respect to the weight function.

Corollary 4 .
[9] The generalized Chebyshev-II polynomials of degree less than or equal to n, U (M,N ) 0 (x), . . ., U (M,N ) n (x), can be expressed in the Bernstein basis of fixed degree n by the following formula

(− 1 ) i=0 (− 1 )
r−i ϑ i,r B r i (u, v)+ λ k k−d j=0 (−1) j k + d + 1 j k − d j (u + v) k−d−j × d d−i ϑ i,d B d i (u, v),With some binomial simplifications, and using Corollary 4, we get v), where M n−k i,r are the coefficients resulting from writing Chebyshev-II polynomial of degree r in the Bernstein basis of degree n − k, as defined by expression (2.7).Thus, the required Bernstein-Bézier coefficients are given bya nk ≤ n − r 0 if k > n − r.which completes the proof of the theorem.