DOCSLIB.ORG

Chebyshev Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chebyshev Polynomials Chebyshev Polynomials Reading Problems Differential Equation and Its Solution The Chebyshev differential equation is written as d2y dy (1 − x2) − x + n2 y = 0 n = 0; 1; 2; 3;::: dx2 dx If we let x = cos t we obtain d2y + n2y = 0 dt2 whose general solution is y = A cos nt + B sin nt or as y = A cos(n cos−1 x) + B sin(n cos−1 x) jxj < 1 or equivalently y = ATn(x) + BUn(x) jxj < 1 where Tn(x) and Un(x) are defined as Chebyshev polynomials of the first and second kind of degree n, respectively. 1 If we let x = cosh t we obtain d2y − n2y = 0 dt2 whose general solution is y = A cosh nt + B sinh nt or as y = A cosh(n cosh−1 x) + B sinh(n cosh−1 x) jxj > 1 or equivalently y = ATn(x) + BUn(x) jxj > 1 The function Tn(x) is a polynomial. For jxj < 1 we have n n p 2 Tn(x) + iUn(x) = (cos t + i sin t) = x + i 1 − x n n p 2 Tn(x) − iUn(x) = (cos t − i sin t) = x − i 1 − x from which we obtain 1 h n ni p 2 p 2 Tn(x) = x + i 1 − x + x − i 1 − x 2 For jxj > 1 we have n nt p 2 Tn(x) + Un(x) = e = x ± x − 1 n −nt p 2 Tn(x) − Un(x) = e = x ∓ x − 1 The sum of the last two relationships give the same result for Tn(x). 2 Chebyshev Polynomials of the First Kind of Degree n The Chebyshev polynomials Tn(x) can be obtained by means of Rodrigue's formula (−2)nn! dn p 2 2 n−1=2 Tn(x) = 1 − x (1 − x ) n = 0; 1; 2; 3;::: (2n)! dxn The first twelve Chebyshev polynomials are listed in Table 1 and then as powers of x in terms of Tn(x) in Table 2. 3 Table 1: Chebyshev Polynomials of the First Kind T0(x) = 1 T1(x) = x 2 T2(x) = 2x − 1 3 T3(x) = 4x − 3x 4 2 T4(x) = 8x − 8x + 1 5 3 T5(x) = 16x − 20x + 5x 6 4 2 T6(x) = 32x − 48x + 18x − 1 7 5 3 T7(x) = 64x − 112x + 56x − 7x 8 6 4 2 T8(x) = 128x − 256x + 160x − 32x + 1 9 7 5 3 T9(x) = 256x − 576x + 432x − 120x + 9x 10 8 6 4 2 T10(x) = 512x − 1280x + 1120x − 400x + 50x − 1 11 9 7 5 3 T11(x) = 1024x − 2816x + 2816x − 1232x + 220x − 11x 4 Table 2: Powers of x as functions of Tn(x) 1 = T0 x = T1 2 1 x = (T0 + T2) 2 3 1 x = (3T1 + T3) 4 4 1 x = (3T0 + 4T2 + T4) 8 5 1 x = (10T1 + 5T3 + T5) 16 6 1 x = (10T0 + 15T2 + 6T4 + T6) 32 7 1 x = (35T1 + 21T3 + 7T5 + T7) 64 8 1 x = (35T0 + 56T2 + 28T4 + 8T6 + T8) 128 9 1 x = (126T1 + 84T3 + 36T5 + 9T7 + T9) 256 10 1 x = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10) 512 11 1 x = (462T1 + 330T3 + 165T5 + 55T7 + 11T9 + T11) 1024 5 Generating Function for Tn(x) The Chebyshev polynomials of the first kind can be developed by means of the generating function 1 1 − tx X = T (x)tn 2 n 1 − 2tx + t n=0 Recurrence Formulas for Tn(x) When the first two Chebyshev polynomials T0(x) and T1(x) are known, all other polyno- mials Tn(x); n ≥ 2 can be obtained by means of the recurrence formula Tn+1(x) = 2xTn(x) − Tn−1(x) The derivative of Tn(x) with respect to x can be obtained from 2 0 (1 − x )Tn(x) = −nxTn(x) + nTn−1(x) Special Values of Tn(x) The following special values and properties of Tn(x) are often useful: n n Tn(−x) = (−1) Tn(x) T2n(0) = (−1) Tn(1) = 1 T2n+1(0) = 0 n Tn(−1) = (−1) 6 Orthogonality Property of Tn(x) We can determine the orthogonality properties for the Chebyshev polynomials of the first kind from our knowledge of the orthogonality of the cosine functions, namely, 8 > 0 (m 6= n) > Z π <> cos(mθ) cos(n θ) dθ = π=2 (m = n 6= 0) 0 > > :> π (m = n = 0) Then substituting Tn(x) = cos(nθ) cos θ = x to obtain the orthogonality properties of the Chebyshev polynomials: 8 0 (m 6= n) > 1 > Z Tm(x) Tn(x) dx < p = π=2 (m = n 6= 0) 2 −1 1 − x > > :> π (m = n = 0) We observe that the Chebyshev polynomials form an orthogonal set on the interval −1 ≤ x ≤ 1 with the weighting function (1 − x2)−1=2 Orthogonal Series of Chebyshev Polynomials An arbitrary function f(x) which is continuous and single-valued, defined over the interval −1 ≤ x ≤ 1, can be expanded as a series of Chebyshev polynomials: f(x) = A0T0(x) + A1T1(x) + A2T2(x) + ::: 1 X = AnTn(x) n=0 7 where the coefficients An are given by 1 Z 1 f(x) dx A0 = p n = 0 2 π −1 1 − x and 1 2 Z f(x) Tn(x) dx An = p n = 1; 2; 3;::: 2 π −1 1 − x The following definite integrals are often useful in the series expansion of f(x): Z 1 dx Z 1 x3 dx p = π p = 0 2 2 −1 1 − x −1 1 − x Z 1 x dx Z 1 x4 dx 3π p = 0 p = 2 2 −1 1 − x −1 1 − x 8 Z 1 x2 dx π Z 1 x5 dx p = p = 0 2 2 −1 1 − x 2 −1 1 − x Chebyshev Polynomials Over a Discrete Set of Points A continuous function over a continuous interval is often replaced by a set of discrete values of the function at discrete points. It can be shown that the Chebyshev polynomials Tn(x) are orthogonal over the following discrete set of N + 1 points xi, equally spaced on θ, π 2π π θi = 0; ; ;::: (N − 1) ; π N N N where xi = arccos θi We have 8 8 > 0 (m 6= n) N−1 > 1 X 1 <> Tm(−1)Tn(−1)+ Tm(xi)Tn(xi)+ Tm(1)Tn(1) = N=2 (m = n 6= 0) 2 2 > i=2 > :> N (m = n = 0) The Tm(x) are also orthogonal over the following N points ti equally spaced, π 3π 5π (2N − 1)π θi = ; ; ;:::; 2N 2N 2N 2N and ti = arccos θi 8 0 (m 6= n) > N > X < Tm(ti)Tn(ti) = N=2 (m = n 6= 0) > i=1 > :> N (m = n = 0) The set of points ti are clearly the midpoints in θ of the first case. The unequal spacing of the points in xi(Nti) compensates for the weight factor W (x) = (1 − x2)−1=2 in the continuous case. 9 Additional Identities of Chebyshev Polynomials The Chebyshev polynomials are both orthogonal polynomials and the trigonometric cos nx functions in disguise, therefore they satisfy a large number of useful relationships. The differentiation and integration properties are very important in analytical and numerical work. We begin with −1 Tn+1(x) = cos[(n + 1) cos x] and −1 Tn−1(x) = cos[(n − 1) cos x] Differentiating both expressions gives −1 1 d[Tn+1(x)] − sin[(n + 1) cos x = p (n + 1) dx − 1 − x2 and −1 1 d[Tn−1(x)] − sin[(n − 1) cos x = p (n − 1) dx − 1 − x2 Subtracting the last two expressions yields 1 d[T (x)] 1 d[T (x)] sin(n + 1)θ − sin(n − 1)θ n+1 − n−1 = (n + 1) dx (n − 1) dx sin θ or 0 0 Tn+1(x) Tn−1(x) 2 cos nθ sin θ − = = 2Tn(x) n ≥ 2 (n + 1) (n − 1) sin θ 10 Therefore 0 T2(x) = 4T1 0 T1(x) = T0 0 T0(x) = 0 We have the formulas for the differentiation of Chebyshev polynomials, therefore these for- mulas can be used to develop integration for the Chebyshev polynomials: Z 1 Tn+1(x) Tn−1(x) Tn(x)dx = − + C n ≥ 2 2 (n + 1) (n − 1) Z 1 T1(x)dx = T2(x) + C 4 Z T0(x)dx = T1(x) + C The Shifted Chebyshev Polynomials For analytical and numerical work it is often convenient to use the half interval 0 ≤ x ≤ 1 instead of the full interval −1 ≤ x ≤ 1. For this purpose the shifted Chebyshev polynomials are defined: ∗ Tn (x) = Tn ∗ (2x − 1) Thus we have for the first few polynomials ∗ T0 = 1 ∗ T1 = 2x − 1 ∗ 2 T2 = 8x − 8x + 1 ∗ 3 2 T3 = 32x − 48x + 18x − 1 ∗ 4 3 2 T4 = 128x − 256x + 160x − 32x + 1 11 ∗ and the following powers of x as functions of Tn (x); ∗ 1 = T0 1 x = (T ∗ + T ∗) 2 0 1 1 x2 = (3T ∗ + 4T ∗ + T ∗) 8 0 1 2 1 x3 = (10T ∗ + 15T ∗ + 6T ∗ + T ∗) 32 0 1 2 3 1 x4 = (35T ∗ + 56T ∗ + 28T ∗ + 8T ∗ + T ∗) 128 0 1 2 3 4 The recurrence relationship for the shifted polynomials is: ∗ ∗ ∗ ∗ Tn+1(x) = (4x − 2)Tn (x) − Tn−1(x) T0 (x) = 1 or 1 1 1 xT ∗(x) = T ∗ (x) + T ∗(x) + T ∗ (x) n 4 n+1 2 n 4 n−1 where ∗ −1 Tn (x) = cos n cos (2x − 1) = Tn(2x − 1) n Expansion of x in a Series of Tn(x) A method of expanding xn in a series of Chebyshev polynomials employes the recurrence relation written as 1 xTn(x) = [Tn+1(x) + Tn−1(x)] n = 1; 2; 3 ::: 2 xT0(x) = T1(x) 12 To illustrate the method, consider x4 2 4 2 x x x = x (xT1) = [T2 + T0] = [T1 + T3 + 2T1] 2 4 1 1 = [3xT1 + xT3] = [3T0 + 3T2 + T4 + T2] 4 8 1 1 3 = T4 + T2 + T0 8 2 8 This result is consistent with the expansion of x4 given in Table 2.
Load more

Recommended publications
  • Chebyshev Polynomials of the Second, Third and Fourth Kinds in Approximation, Indefinite Integration, and Integral Transforms *
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 49 (1993) 169-178 169 North-Holland CAM 1429 Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms * J.C. Mason Applied and Computational Mathematics Group, Royal Military College of Science, Shrivenham, Swindon, Wiltshire, United Kingdom Dedicated to Dr. D.F. Mayers on the occasion of his 60th birthday Received 18 February 1992 Revised 31 March 1992 Abstract Mason, J.C., Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms, Journal of Computational and Applied Mathematics 49 (1993) 169-178. Chebyshev polynomials of the third and fourth kinds, orthogonal with respect to (l+ x)‘/‘(l- x)-‘I* and (l- x)‘/*(l+ x)-‘/~, respectively, on [ - 1, 11, are less well known than traditional first- and second-kind polynomials. We therefore summarise basic properties of all four polynomials, and then show how some well-known properties of first-kind polynomials extend to cover second-, third- and fourth-kind polynomials. Specifically, we summarise a recent set of first-, second-, third- and fourth-kind results for near-minimax constrained approximation by series and interpolation criteria, then we give new uniform convergence results for the indefinite integration of functions weighted by (1 + x)-i/* or (1 - x)-l/* using third- or fourth-kind polynomial expansions, and finally we establish a set of logarithmically singular integral transforms for which weighted first-, second-, third- and fourth-kind polynomials are eigenfunctions.
    [Show full text]
  • Generalizations of Chebyshev Polynomials and Polynomial Mappings
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 10, October 2007, Pages 4787–4828 S 0002-9947(07)04022-6 Article electronically published on May 17, 2007 GENERALIZATIONS OF CHEBYSHEV POLYNOMIALS AND POLYNOMIAL MAPPINGS YANG CHEN, JAMES GRIFFIN, AND MOURAD E.H. ISMAIL Abstract. In this paper we show how polynomial mappings of degree K from a union of disjoint intervals onto [−1, 1] generate a countable number of spe- cial cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus g, from which the coefficients of xn can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is use- ful for specializing to polynomial mapping cases for small K where we will have explicit expressions for the recurrence coefficients in terms of the branch points. We study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an explicit expression for the discrimi- nant for the genus 1 case of our Chebyshev polynomials that is valid for any configuration of the branch point. 1. Introduction and preliminaries Akhiezer [2], [1] and, Akhiezer and Tomˇcuk [3] introduced orthogonal polynomi- als on two intervals which generalize the Chebyshev polynomials. He observed that the study of properties of these polynomials requires the use of elliptic functions. In thecaseofmorethantwointervals,Tomˇcuk [17], investigated their Bernstein-Szeg˝o asymptotics, with the theory of Hyperelliptic integrals, and found expressions in terms of a certain Abelian integral of the third kind.
    [Show full text]
  • Chebyshev and Fourier Spectral Methods 2000
    Chebyshev and Fourier Spectral Methods Second Edition John P. Boyd University of Michigan Ann Arbor, Michigan 48109-2143 email: [email protected] http://www-personal.engin.umich.edu/jpboyd/ 2000 DOVER Publications, Inc. 31 East 2nd Street Mineola, New York 11501 1 Dedication To Marilyn, Ian, and Emma “A computation is a temptation that should be resisted as long as possible.” — J. P. Boyd, paraphrasing T. S. Eliot i Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series .................................... 20 2.3 Orders of Convergence ............................... 25 2.4 Convergence Order ................................. 27 2.5 Assumption of Equal Errors ...........................
    [Show full text]
  • Dymore User's Manual Chebyshev Polynomials
    Dymore User's Manual Chebyshev polynomials Olivier A. Bauchau August 27, 2019 Contents 1 Definition 1 1.1 Zeros and extrema....................................2 1.2 Orthogonality relationships................................3 1.3 Derivatives of Chebyshev polynomials..........................5 1.4 Integral of Chebyshev polynomials............................5 1.5 Products of Chebyshev polynomials...........................5 2 Chebyshev approximation of functions of a single variable6 2.1 Expansion of a function in Chebyshev polynomials...................6 2.2 Evaluation of Chebyshev expansions: Clenshaw's recurrence.............7 2.3 Derivatives and integrals of Chebyshev expansions...................7 2.4 Products of Chebyshev expansions...........................8 2.5 Examples.........................................9 2.6 Clenshaw-Curtis quadrature............................... 10 3 Chebyshev approximation of functions of two variables 12 3.1 Expansion of a function in Chebyshev polynomials................... 12 3.2 Evaluation of Chebyshev expansions: Clenshaw's recurrence............. 13 3.3 Derivatives of Chebyshev expansions.......................... 14 4 Chebychev polynomials 15 4.1 Examples......................................... 16 1 Definition Chebyshev polynomials [1,2] form a series of orthogonal polynomials, which play an important role in the theory of approximation. The lowest polynomials are 2 3 4 2 T0(x) = 1;T1(x) = x; T2(x) = 2x − 1;T3(x) = 4x − 3x; T4(x) = 8x − 8x + 1;::: (1) 1 and are depicted in fig.1. The polynomials can be generated from the following recurrence rela- tionship Tn+1 = 2xTn − Tn−1; n ≥ 1: (2) 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 CHEBYSHEV POLYNOMIALS −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 XX Figure 1: The seven lowest order Chebyshev polynomials It is possible to give an explicit expression of Chebyshev polynomials as Tn(x) = cos(n arccos x): (3) This equation can be verified by using elementary trigonometric identities.
    [Show full text]
  • Approximation Atkinson Chapter 4, Dahlquist & Bjork Section 4.5
    Approximation Atkinson Chapter 4, Dahlquist & Bjork Section 4.5, Trefethen's book Topics marked with ∗ are not on the exam 1 In approximation theory we want to find a function p(x) that is `close' to another function f(x). We can define closeness using any metric or norm, e.g. Z 2 2 kf(x) − p(x)k2 = (f(x) − p(x)) dx or kf(x) − p(x)k1 = sup jf(x) − p(x)j or Z kf(x) − p(x)k1 = jf(x) − p(x)jdx: In order for these norms to make sense we need to restrict the functions f and p to suitable function spaces. The polynomial approximation problem takes the form: Find a polynomial of degree at most n that minimizes the norm of the error. Naturally we will consider (i) whether a solution exists and is unique, (ii) whether the approximation converges as n ! 1. In our section on approximation (loosely following Atkinson, Chapter 4), we will first focus on approximation in the infinity norm, then in the 2 norm and related norms. 2 Existence for optimal polynomial approximation. Theorem (no reference): For every n ≥ 0 and f 2 C([a; b]) there is a polynomial of degree ≤ n that minimizes kf(x) − p(x)k where k · k is some norm on C([a; b]). Proof: To show that a minimum/minimizer exists, we want to find some compact subset of the set of polynomials of degree ≤ n (which is a finite-dimensional space) and show that the inf over this subset is less than the inf over everything else.
    [Show full text]
  • Orthogonal Polynomials on the Unit Circle Associated with the Laguerre Polynomials
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 3, Pages 873{879 S 0002-9939(00)05821-4 Article electronically published on October 11, 2000 ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE ASSOCIATED WITH THE LAGUERRE POLYNOMIALS LI-CHIEN SHEN (Communicated by Hal L. Smith) Abstract. Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated. 1. Introduction This paper deals with a class of orthogonal polynomials which arise from an application of the Fourier transform on the Laguerre polynomials. We shall briefly describe the essence of our method. Let Π+ denote the upper half plane fz : z = x + iy; y > 0g and let Z 1 H(Π+)=ff : f is analytic in Π+ and sup jf(x + yi)j2 dx < 1g: 0<y<1 −∞ It is well known that, from the Paley-Wiener Theorem [4, p. 368], the Fourier transform provides a unitary isometry between the spaces L2(0; 1)andH(Π+): Since the Laguerre polynomials form a complete orthogonal basis for L2([0; 1);xαe−x dx); the application of Fourier transform to the Laguerre polynomials (with suitable weight attached) generates a class of orthogonal rational functions which are com- plete in H(Π+); and by composition of which with the fractional linear transfor- mation (which maps Π+ conformally to the unit disc) z =(2t − i)=(2t + i); we obtain a family of polynomials which are orthogonal with respect to the weight α t j j sin 2 dt on the boundary z = 1 of the unit disc.
    [Show full text]
  • Arxiv:1903.11395V3 [Math.NA] 1 Dec 2020 M
    Noname manuscript No. (will be inserted by the editor) The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization Stefano Pozza · Miroslav Prani´c Received: date / Accepted: date Abstract The concept of Gauss quadrature can be generalized to approx- imate linear functionals with complex moments. Following the existing lit- erature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with for- mal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realiza- tion problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals. Keywords Linear functionals · Matching moments · Gauss quadrature · Formal orthogonal polynomials · Minimal realization · Look-ahead Lanczos algorithm · Mismatch Theorem. 1 Introduction Let A be an N × N Hermitian positive definite matrix and v a vector so that v∗v = 1, where v∗ is the conjugate transpose of v. Consider the specific linear S. Pozza Faculty of Mathematics and Physics, Charles University, Sokolovsk´a83, 186 75 Praha 8, Czech Republic. Associated member of ISTI-CNR, Pisa, Italy, and member of INdAM- GNCS group, Italy. E-mail: [email protected]ff.cuni.cz arXiv:1903.11395v3 [math.NA] 1 Dec 2020 M.
    [Show full text]
  • 8.3 - Chebyshev Polynomials
    8.3 - Chebyshev Polynomials 8.3 - Chebyshev Polynomials Chebyshev polynomials Definition Chebyshev polynomial of degree n ≥= 0 is defined as Tn(x) = cos (n arccos x) ; x 2 [−1; 1]; or, in a more instructive form, Tn(x) = cos nθ ; x = cos θ ; θ 2 [0; π] : Recursive relation of Chebyshev polynomials T0(x) = 1 ;T1(x) = x ; Tn+1(x) = 2xTn(x) − Tn−1(x) ; n ≥ 1 : Thus 2 3 4 2 T2(x) = 2x − 1 ;T3(x) = 4x − 3x ; T4(x) = 8x − 8x + 1 ··· n−1 Tn(x) is a polynomial of degree n with leading coefficient 2 for n ≥ 1. 8.3 - Chebyshev Polynomials Orthogonality Chebyshev polynomials are orthogonal w.r.t. weight function w(x) = p 1 . 1−x2 Namely, Z 1 T (x)T (x) 0 if m 6= n np m dx = (1) 2 π −1 1 − x 2 if n = m for each n ≥ 1 Theorem (Roots of Chebyshev polynomials) The roots of Tn(x) of degree n ≥ 1 has n simple zeros in [−1; 1] at 2k−1 x¯k = cos 2n π ; for each k = 1; 2 ··· n : Moreover, Tn(x) assumes its absolute extrema at 0 kπ 0 k x¯k = cos n ; with Tn(¯xk) = (−1) ; for each k = 0; 1; ··· n : 0 −1 k For k = 0; ··· n, Tn(¯xk) = cos n cos (cos(kπ=n)) = cos(kπ) = (−1) : 8.3 - Chebyshev Polynomials Definition A monic polynomial is a polynomial with leading coefficient 1. The monic Chebyshev polynomial T~n(x) is defined by dividing Tn(x) by 2n−1; n ≥ 1.
    [Show full text]
  • Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls∗
    SIAM J. MATH. ANAL. c 1998 Society for Industrial and Applied Mathematics Vol. 29, No. 3, pp. 779{793, May 1998 015 ORTHOGONAL POLYNOMIALS AND CUBATURE FORMULAE ON SPHERES AND ON BALLS∗ YUAN XUy Abstract. Orthogonal polynomials on the unit sphere in Rd+1 and on the unit ball in Rd are shown to be closely related to each other for symmetric weight functions. Furthermore, it is shown that a large class of cubature formulae on the unit sphere can be derived from those on the unit ball and vice versa. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres. Key words. orthogonal polynomials in several variables, on spheres, on balls, spherical har- monics, cubature formulae AMS subject classifications. 33C50, 33C55, 65D32 PII. S0036141096307357 1. Introduction. We are interested in orthogonal polynomials in several vari- ables with emphasis on those orthogonal with respect to a given measure on the unit sphere Sd in Rd+1. In contrast to orthogonal polynomials with respect to measures defined on the unit ball Bd in Rd, there have been relatively few studies on the struc- ture of orthogonal polynomials on Sd beyond the ordinary spherical harmonics which are orthogonal with respect to the surface (Lebesgue) measure (cf. [2, 3, 4, 5, 6, 8]). The classical theory of spherical harmonics is primarily based on the fact that the ordinary harmonics satisfy the Laplace equation. Recently Dunkl (cf. [2, 3, 4, 5] and the references therein) opened a way to study orthogonal polynomials on the spheres with respect to measures invariant under a finite reflection group by developing a theory of spherical harmonics analogous to the classical one.
    [Show full text]
  • Algorithms for Classical Orthogonal Polynomials
    Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustr. 7, D-14195 Berlin - Dahlem Wolfram Ko epf Dieter Schmersau Algorithms for Classical Orthogonal Polynomials at Berlin Fachb ereich Mathematik und Informatik der Freien Universit Preprint SC Septemb er Algorithms for Classical Orthogonal Polynomials Wolfram Ko epf Dieter Schmersau koepfzibde Abstract In this article explicit formulas for the recurrence equation p x A x B p x C p x n+1 n n n n n1 and the derivative rules 0 x p x p x p x p x n n+1 n n n n1 n and 0 p x p x x p x x n n n n1 n n resp ectively which are valid for the orthogonal p olynomial solutions p x of the dierential n equation 00 0 x y x x y x y x n of hyp ergeometric typ e are develop ed that dep end only on the co ecients x and x which themselves are p olynomials wrt x of degrees not larger than and resp ectively Partial solutions of this problem had b een previously published by Tricomi and recently by Yanez Dehesa and Nikiforov Our formulas yield an algorithm with which it can b e decided whether a given holonomic recur rence equation ie one with p olynomial co ecients generates a family of classical orthogonal p olynomials and returns the corresp onding data density function interval including the stan dardization data in the armative case In a similar way explicit formulas for the co ecients of the recurrence equation and the dierence rule x rp x p x p x p x n n n+1 n n n n1 of the classical orthogonal p olynomials of a discrete variable are given that dep end only
    [Show full text]
  • Polynomial Interpolation and Chebyshev Polynomials
    Lecture 7: Polynomial interpolation and Chebyshev polynomials Michael S. Floater September 30, 2018 These notes are based on Section 3.1 of the book. 1 Lagrange interpolation Interpolation describes the problem of finding a function that passes through a given set of values f0,f1,...,fn at data points x0,x1,...,xn. Such a function is called an interpolant. If p is an interpolant then p(xi) = fi, i = 0, 1,...,n. Let us suppose that the points and values are in R. If the values fi are samples f(xi) from some real function f then p approximates f. One choice of p is a polynomial of degree at most n. Using the Lagrange basis functions n x − xj Lk(x)= , k =0, 1,...,n, xk − xj Yj=0 j=6 k we can express p explicitly as n p(x)= fkLk(x). Xk=0 Because Lk(xi) = 0 when i =6 k and Lk(xk) = 1, it is easy to check that p(xi) = fi for i = 0, 1,...,n. The interpolant p is also unique among all polynomial interpolants of degree ≤ n. This is because if q is another such interpolant, the difference r = p − q would be a polynomial of degree ≤ n 1 such that r(xi)=0for i =0, 1,...,n. Thus r would have at least n +1 zeros and by a result from algebra this would imply that r = 0. For this kind of interpolation there is an error formula. n+1 Theorem 1 Suppose f ∈ C [a,b] and let fi = f(xi), i =0, 1,...,n, where x0,x1,...,xn are distinct points in [a,b].
    [Show full text]
  • Asymptotics of the Chebyshev Polynomials of General Sets
    Joint Work with Jacob Christiansen and Maxim Zinchenko Part 1: Inv. Math, 208 (2017), 217-245 Part 2: Submitted (also with Peter Yuditskii) Part 3: Pavlov Memorial Volume, to appear. Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work Totik Widom Bounds Main Results Totik Widom Bounds Szegő–Widom Asymptotics Part 2: Submitted (also with Peter Yuditskii) Part 3: Pavlov Memorial Volume, to appear. Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work Totik Widom Bounds Main Results Totik Widom Joint Work with Jacob Christiansen and Maxim Zinchenko Bounds Part 1: Inv. Math, 208 (2017), 217-245 Szegő–Widom Asymptotics Part 3: Pavlov Memorial Volume, to appear. Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work Totik Widom Bounds Main Results Totik Widom Joint Work with Jacob Christiansen and Maxim Zinchenko Bounds Part 1: Inv. Math, 208 (2017), 217-245 Szegő–Widom Part 2: Submitted (also with Peter Yuditskii) Asymptotics Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A.
    [Show full text]