/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 324-331 ISSN 1307-5543 – www.ejpam.com Generalized Tschebyscheff of the Second Kind and Bernstein Polynomials Change of Bases Mohammad A. AlQudah Department of Mathematics, Northwood University, Midland, MI 48640 USA Abstract. We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis. 2010 Mathematics Subject Classifications: 42C05, 33C50, 33C45, 33C70, 05A10, 33B15 Key Words and Phrases: Generalized Tschebyscheff, Bernstein Basis, Basis Transformation, Bézier Coefficient, Gamma function 1. Introduction, Background and Motivation It is possible to approximate a complicated continuous functions defined over finite do- mains by a polynomial and make the error less than a given accuracy. On the other side, polynomials can be characterized in many different bases such as the power product, Bern- stein basis, and Tschebyscheff basis form, where every type of polynomial basis has its strength, advantages, and sometimes disadvantages. It is useful to switch bases and work with more than one basis for a given polynomial; it is of vital importance in the efficiency of mathematical calculations, since many difficulties can be solved and many problems can be removed. 1.1. Bernstein Polynomials The n+ 1 polynomials Bn k (x) of degree n, x ∈ [0,1], k = 0,1, . . . , n, defined as Bn k (x) = n! k!(n− k)! xk(1− x)n−k, k = 0,1, . . . , n, (1) are called Bernstein polynomials. Email address: alqudahm@northwood.edu http://www.ejpam.com 324 c© 2015 EJPAM All rights reserved. M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 325 There are a fair amount of literature on Bernstein polynomials, they are known for their geometric and analytical properties, see [2] for more details. Analytic and geometric proper- ties of Bernstein polynomials make them important for the development of Bézier curves and surfaces. The Bernstein polynomials are the standard basis for the Bézier representations of curves and surfaces in Computer Aided Geometric Design. However, the Bernstein polynomi- als are not orthogonal and could not be used effectively in the least-squares approximation [7]. Since then the method of least squares approximation accompanied by orthogonal polynomials has been introduced and developed. 1.2. Least-Square Approximation In the following definition, we define the continuous least-square approximations of a function f (x) by using polynomials with standard power basis, {1, x , x2, . . . , xn}. Definition 1. For a function f (x), continuous on [0,1] the least square approximation requires finding a least-squares polynomial p∗n(x) = ∑n k=0 akφk(x) that minimizes the error E(a0, a1, . . . , an) = ∫ 1 0 [ f (x)− p∗n(x)] 2d x , are called least squares approximations. A necessary condition for E(a0, a1, . . . , an) to have a minimum over all values a0, a1, . . . , an, is ∂ E ∂ ak = 0. But, ∂ E ∂ ak = −2 ∫ 1 0 [ f (x)− p∗n(x)]φk(x)d x , k = 0, . . . , n. Thus, for i = 0,1, . . . , n, ai that minimize f (x)−∑n k=0 akφk(x) 2 satisfy the system ∫ 1 0 f (x)φi(x)d x = n ∑ k=0 ak ∫ 1 0 φk(x)φi(x)d x . which gives a system of (n+ 1) equations, called normal equations, in (n+ 1) unknowns: ai , i = 0, . . . , n. Those (n+ 1) unknowns of the least-squares polynomial p∗n(x), can be found by solving the normal equations. By choosing φi(x) = x i , as a basis, then ∫ 1 0 f (x)x id x = n ∑ k=0 ak ∫ 1 0 x i+kd x = n ∑ k=0 ak i + k+ 1 . The coefficients matrix of the normal equations is Hilbert matrix which has round-off error difficulties and notoriously ill-conditioned for even modest values of n. However, such com- putations can be made effective by using orthogonal polynomials. Thus, choosing {φ0(x),φ1(x), . . . ,φn(x)} to be orthogonal simplifies the least-squares approximation prob- lem. The coefficients matrix of the normal equations will be diagonal, which gives a compact form for ai , i = 0,1, . . . , n. See [7] for more details on the least squares approximations. M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 326 1.3. Gamma Functions The gamma function Γ(n) is an extension of the factorial function, with its argument shifted down by 1. That is, if n is a positive integer: Γ(n) = (n−1)!. The Eulerian integral of the first kind is useful and will be used in main result simplifications. Definition 2. The Eulerian integral of the first kind is a function of two complex variables defined by ∫ 1 0 ux−1(1− u)y−1du= Γ(x)Γ(y) Γ(x + y) , ℜ(x),ℜ(y)> 0. (2) The double factorial of an integer n is given by ¨ (2n− 1)!!= (2n− 1)(2n− 3)(2n− 5) . . . (3)(1) if n is odd n!!= (n)(n− 2)(n− 4) . . . (4)(2) if n is even, (3) where 0!!= (−1)!!= 1. Using (3), we can derive the following relation n!!=    2 n 2 ( n 2 )! if n is even n! 2 n−1 2 ( n−1 2 )! if n is odd (4) It is easy to derive the factorial of an integer plus half as � n+ 1 2 � != p π 2n+1 (2n+ 1)!!. (5) From the relation (4) to have (2n)!!= 2nn!, and (2n)!= (2n− 1)!!2nn!. 1.4. Univariate Tschebyscheff-II and the Generalized Tschebyscheff-II Polynomials The univariate classical Tschebyscheff-II orthogonal polynomials Un(x) are special case of Jacobi polynomials P (α,β) n withα= β = 1/2, where the inter-relationship between Tschebyscheff- II and Jacobi Polynomials given as P ( 1 2 , 1 2 ) n (1)Un(x) = (n+ 1)P ( 1 2 , 1 2 ) n (x). Tschebyscheff-II poly- nomials are traditional defined on [−1,1], however, it is more convenient to use [0,1]. For the convenience we recall the following explicit expressions for univariate Tschebyscheff-II polynomials of degree n in x , using combinatorial notation that gives more compact and readable formulas, see Szegö [8]: Un(x) := (n+ 1)(2n)!! (2n+ 1)!! n ∑ k=0 � n+ 1 2 n− k �� n+ 1 2 k �� x + 1 2 �n−k � x − 1 2 �k , (6) which it can be transformed in terms of Bernstein basis on x ∈ [0,1], Un(2x − 1) := (n+ 1)(2n)!! (2n+ 1)!! n ∑ k=0 (−1)n+1 �n+ 1 2 k ��n+ 1 2 n−k � � n k � Bn k (x). (7) M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 327 The Tschebyscheff-II polynomials Un(x) of degree n are the orthogonal polynomials, except for a constant factor, with respect to the weight function W(x) = p 1− x2. Also, the Tschebyscheff- II polynomials satisfy the orthogonality relation [4] ∫ 1 0 x 1 2 (1− x) 1 2 Un(x)Um(x)d x = ¨ 0 if m 6= n π 8 if m= n (8) The generalized Tschebyscheff-II polynomials been characterization in [1], for M , N ≥ 0, the generalized Tschebyscheff-II polynomials � U (M ,N) n (x) ∞ n=0 are orthogonal on [−1,1] with respect to the generalized weight function [5], 2 π (1− x) 1 2 (1+ x) 1 2 +Mδ(x + 1) + Nδ(x − 1). (9) and defined in [1] as U (M ,N) n (x) = (2n+ 1)!! 2n(n+ 1)! Un(x) + n ∑ k=0 λk (2k+ 1)!! 2k(k+ 1)! Uk(x), (10) where λk = k(k+ 1)(2k+ 1)(M + N) 6 + (k+ 2)(k+ 1)2k2(k− 1)MN 9 . (11) 2. Main Results In this section we provide a closed form for the matrix transformation of the generalized Tschebyscheff-II polynomial basis into Bernstein polynomial basis, and for Bernstein polyno- mial basis into generalized Tschebyscheff-II polynomial basis. 2.1. Bernstein to Generalized Tschebyscheff-II Transformation and Vice Versa Rababah [6] provided some results concerning the univariate Tschebyscheff polynomials of first kind with respect to the weight function (1 − x2)1/2. In this paper we extend the procedure in [6] to generalize the results for the generalized Tschebyscheff-II polynomials U (M ,N) r (x) with respect to the generalized weight function (9). The next theorem, see [1] for the proof, provides a closed form for generalized Tschebyscheff-II polynomial U (M ,N) r (x) of degree r as a linear combination of the Bernstein polynomials Br i (x), i = 0,1, . . . , r. Theorem 1 ([1]). For M , N ≥ 0, the generalized Tschebyscheff-II polynomials U (M ,N) r (x) of degree r have the following Bernstein representation: U (M ,N) r (x) = (2r + 1)!! 2r(r + 1)! r ∑ i=0 (−1)r−iϑi,r Br i (x) + r ∑ k=0 λk (2k+ 1)!! 2k(k+ 1)! k ∑ i=0 (−1)k−iϑi,kBk i (x) (12) M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 328 where λk defined by (11), ϑ0,r = (2r+1) 22r � 2r r � , and ϑi,r = (2r + 1)2 22r(2r − 2i + 1)(2i + 1) � 2r r �� 2r 2i � � r i � , i = 0,1, . . . , r. The coefficients ϑi,r satisfy the recurrence relation ϑi,r = (2r − 2i + 3) (2i + 1) ϑi−1,r , i = 1, . . . , r. (13) Now, the next theorem used to combine the superior performance of the least-squares of the generalized Tschebyscheff-II polynomials with the geometric properties of the Bernstein polynomials basis. Theorem 2. The entries M n i,r , i, r = 0,1, . . . , n of the matrix transformation of the generalized Tschebyscheff-II polynomial basis into Bernstein polynomial basis of degree n are given by M n i,r = Φ r i,n + r ∑ k=0 λkΦ k i,n, (14) where λk defined in (11) and Φr i,n = (2r + 1)!! 2r(r + 1)! min(i,r) ∑ k=max(0,i+r−n) (−1)r−k � n−r i−k ��r+ 1 2 k ��r+ 1 2 r−k � � n i � . Proof. A polynomial pn(x), x ∈ [0,1] of degree n, can be written as as a linear combination of the Bernstein polynomial basis pn(x) = ∑n r=0 cr Bn r (x) and the generalized Tschebyscheff-II polynomials pn(x) = ∑n i=0 diU (M ,N) i (x). We need to find the matrix M that maps the generalized Tschebyscheff-II coefficients {di}ni=0 into the Bernstein coefficients {cr}nr=0, ci = n ∑ r=0 M n i,r dr , (15) which can be written in matrix format as      c0 c1 ... cn      =      M n 0,0 M n 0,1 M n 0,2 . . . M n 0,n M n 1,0 M n 1,1 M n 1,2 . . . M n 1,n ... ... ... . . . ... M n n,0 M n n,1 M n n,2 . . . M n n,n      .      d0 d1 ... dn      . (16) But, the generalized Tschebyscheff-II polynomials (10) can be written as a linear combi- nation of the Bernstein polynomial basis as U (M ,N) r (x) = n ∑ i=0 N n r,iB n i (x), r = 0,1, . . . , n, (17) M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 329 where the the (n+1)× (n+1) basis conversion matrix N formed by the entries N n r,i . Thus, the elements of c can be written in the form ci = n ∑ r=0 dr N n r,i . (18) Comparing (15) and (18), we have M n i,r = N n r,i , for i, r = 0, . . . , n. Since each Bernstein polynomial of degree r ≤ n can be written in terms of Bernstein polynomials of degree n using the following degree elevation defined by [3]: Br k (x) = n−r+k ∑ i=k � r k �� n−r i−k � � n i � Bn i (x), k = 0,1, . . . , r. (19) Substituting (19) into (12) and rearrange the order of summations, we find the entries N n r,i = � n i �−1 (2r + 1)!! 2r(r + 1)! min(i,r) ∑ k=max(0,i+r−n) (−1)r−k � n− r i − k �� r + 1 2 k �� r + 1 2 r − k � + r ∑ k=0 λk � n i �−1 (2k+ 1)!! 2k(k+ 1)! min(i,k) ∑ j=max(0,i+k−n) (−1)k− j � n− k i − j �� k+ 1 2 j �� k+ 1 2 k− j � . (20) Therefore, the entries of the matrix M are given by M n i,r = Φr i,n + ∑r k=0λkΦ k i,n , where Φk i,n = (2k+ 1)!! 2k(k+ 1)! min(i,k) ∑ j=max(0,i+k−n) (−1)k− j � n−k i− j ��k+ 1 2 j ��k+ 1 2 k− j � � n i � . Now, we have the following corollary which enables us to write Tschebyscheff-II polyno- mials of degree r ≤ n in terms of Bernstein polynomials of degree n. Corollary 1. The generalized Tschebyscheff-II polynomials U (M ,N) 0 (x), . . . ,U (M ,N) n (x) of degree less than or equal to n can be expressed in the Bernstein basis of fixed degree n by the following formula U (M ,N) r (x) = n ∑ i=0 N n r,iB n i (x), r = 0,1, . . . , n where N n r,i = (2r + 1)!! 2r(r + 1)! min(i,r) ∑ k=max(0,i+r−n) (−1)r−k(2r + 1)2 22r(2r − 2k+ 1)(2k+ 1) � n−r i−k �� 2r r �� 2r 2k � � n i � + r ∑ k=0 λk (2k+ 1)!! 2k(k+ 1)! min(i,k) ∑ j=max(0,i+k−n) (−1)k− j(2k+ 1)2 22k(2k− 2 j + 1)(2 j + 1) � n−k i− j �� 2k k �� 2k 2 j � � n i � . M. AlQudah / Eur. J. Pure Appl. Math, 8 (2015), 324-331 330 Proof. From (17) in the proof of the previous theorem, it is clear that each Tschebyscheff-II polynomial of degree r ≤ n can be written in terms of Bernstein polynomials of degree n. Applying (5) with some simplifications, we have � r + 1 2 k �� r + 1 2 r − k � = (2r + 1) 2r(2r − 2k+ 1)(r − k)!k! (2r − 1)!! (2k− 1)!! (2r + 1) (2k+ 1) (2r − 1)!! (2(r − k)− 1)!! . Using the fact (2n)!= (2n− 1)!!2nn!, we get � r + 1 2 r − k �� r + 1 2 k � = (2r + 1)2 22r(2r − 2k+ 1)(2k+ 1) � 2r r �� 2r 2k � . Substituting the last identity into (20) we get the desired result. The following theorem introduced in [1] will be used to simplify a main result. Theorem 3 ([1]). Let Bn r (x) be the Bernstein polynomial of degree n and U (M ,N) i (x) be the generalized Tschebyscheff-II polynomial of degree i, then for i, r = 0,1, . . . , n we have ∫ 1 0 x 1 2 (1− x) 1 2 Bn r (x)U (M ,N) i (x)d x = Λi r,n + i ∑ d=0 λdΛ d r,n, where λd defined in (11), Λd r,n = � n r � (2d + 1)!! 2d(d + 1)! d ∑ j=0 (−1)d− j � d + 1 2 j �� d + 1 2 d − j � Γ(r + j + 3 2)Γ(n+ d − r − j + 3 2) Γ(n+ d + 3) , (21) and Γ(x) is the Gamma function. Finally, to write the Bernstein polynomial basis into generalized Tschebyscheff-II polyno- mial basis of degree n, invert (16) and let M n−1 i,r , N n−1 i,r , i, r = 0, . . . , n be the entries of M−1 and N−1 respectively. The transformation of Bernstein polynomial into generalized Tschebyscheff- II polynomial basis of degree n can then be written as Bn r (x) = n ∑ i=0 N n−1 r,i U (M ,N) i (x). (22) To find the explicit closed form of N n−1 r,i , i, r = 0,1, . . . , n, multiply (22) by x 1 2 (1−x) 1 2 U (M ,N) i (x) and integrate over [0,1] to have ∫ 1 0 x 1 2 (1− x) 1 2 Bn r (x)U (M ,N) i (x)d x = n ∑ i=0 N n−1 r,i ∫ 1 0 x 1 2 (1− x) 1 2 U (M ,N) i (x)U (M ,N) i (x)d x . (23) Use the orthogonality relation (8) to obtain ∫ 1 0 Bn r (x)(1− x) 1 2 x 1 2 U (M ,N) i (x)d x = π 8 � (2i + 1)!! 2i(i + 1)! �2 N n−1 r,i (1+λi) 2. (24) REFERENCES 331 Using (2), Theorem 3, the fact that M n i,r = N n r,i , and Λd r,n defined in (21) we get M n−1 i,r = 8 π(1+λi) 2 � 2i(i + 1)! (2i + 1)!! �2 Λi r,n + i ∑ d=0 λdΛ d r,n ! . (25) Hence, we have the following theorem. Theorem 4. The entries of the matrix of transformation of the Bernstein polynomial basis into the generalized Tschebyscheff-II polynomial basis of degree n are given by M n−1 i,r = 8 π(1+λi) 2 � 2i(i + 1)! (2i + 1)!! �2 Λi r,n + i ∑ d=0 λdΛ d r,n ! , i, r = 0,1, . . . , n. ACKNOWLEDGEMENTS The author thanks the anonymous referees for their fruitful sug- gestions, which immensely helped to improve the presentation of the paper. References [1] M. AlQudah. The generalized Tschebyscheff polynomials of the second kind. Turkish Journal of Mathematics, 39. http://dx.doi.org/10.3906/mat-1501-44. 2015. [2] R. Farouki. The Bernstein polynomial basis: A centennial retrospective. Computer Aided Geometric Design, 29(6), 379–419. 2012. [3] R. Farouki and V. Rajan. Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design, 5(1), 1–26. 1988. [4] I. Gradshtein and I. Ryzhik. Tables of integrals, series, and products. Academic Press, New York. 1980. [5] T. Koornwinder. Orthogonal polynomials with weight function (1− x)α(1+ x)β +Mδ(x + 1) + Nδ(x − 1). Canadian Mathematical Bulletin, 27(2), 205–214. 1984. [6] A. Rababah. Transformation of Chebyshev Bernstein polynomial basis. Computational Methods in Applied Mathematics, 3(4), 608–622. 2003. [7] J. Rice. The Approximation of Functions, Linear Theory. Vol. 1. Addison-Wesley, Reading, Mass. 1964. [8] G. Szegö. Orthogonal polynomials. American Mathematical Society Colloquium Vol. 23, 4th ed. Providence, RI: American Mathematical Society. 1975.