Orthogonal Polynomial Expansions for the Riemann Xi Function Dan Romik
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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. -
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. -
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 ........................... -
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. -
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. -
And Ξ(T) Functions Associated with Riemann's Ζ(S)
Some results on the ξ(s) and Ξ(t) functions associated with Riemann’s ζ(s) function. ∗ Hisashi Kobayashi† 2016/03/04 Abstract 1 We report on some properties of the ξ(s) function and its value on the critical line, Ξ(t)= ξ 2 + it. First, we present some identities that hold for the log derivatives of a holomorphic function. We then re- examine Hadamard’s product-form representation of the ξ(s) function, and present a simple proof of the horizontal monotonicity of the modulus of ξ(s). We then show that the Ξ(t) function can be interpreted as the autocorrelation function of a weakly stationary random process, whose power spectral function S(ω) and Ξ(t) form a Fourier transform pair. We then show that ξ(s) can be formally written as the − 1 Fourier transform of S(ω) into the complex domain τ = t iλ, where s = σ + it = 2 + λ + it. We then show that the function S1(ω) studied by P´olya has g(s) as its Fourier transform, where ξ(s)= g(s)ζ(s). Finally we discuss the properties of the function g(s), including its relationships to Riemann-Siegel’s ϑ(t) function, Hardy’s Z-function, Gram’s law and the Riemann-Siegel asymptotic formula. Key words: Riemann’s ζ(s) function, ξ(s) and Ξ(t) functions, Riemann hypothesis, Monotonicity of the modulus ξ(t), Hadamard’s product formula, P´olya’s Fourier transform representation, Fourier transform to the complex domain, Riemann-Siegel’s asymptotic formula, Hardy’s Z-function. -
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. -
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 -
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]. -
Series Representations for the Stieltjes Constants
Series representations for the Stieltjes constants Mark W. Coffey Department of Physics Colorado School of Mines Golden, CO 80401 (Received 2009) September 1, 2009 Abstract The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for deriva- tives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series repre- sentation for γk = γk(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expan- arXiv:0905.1111v2 [math-ph] 13 Sep 2009 sion, Stirling numbers of the first kind, Dirichlet L functions, Lerch zeta function 2000 AMS codes 11M35, 11M06, 11Y60. secondary: 05A10 1 Introduction and statement of results The Stieltjes (or generalized Euler) constants γk(a) appear as expansion coeffi- cients in the Laurent series for the Hurwitz zeta function ζ(s, a) about its simple pole at s = 1 [5, 7, 18, 23, 27]. These constants are important in analytic number theory and elsewhere, where they appear in various estimations and as a result of asymptotic analyses. They are also of much use in developing a binomial sum Sγ(n) introduced by the author in the study of a critical subsum in application of the Li criterion for the Riemann hypothesis [9]. -
An Analysis of the Lanczos Gamma Approximation
AN ANALYSIS OF THE LANCZOS GAMMA APPROXIMATION by GLENDON RALPH PUGH B.Sc. (Mathematics) University of New Brunswick, 1992 M.Sc. (Mathematics) University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard .................................. .................................. .................................. .................................. .................................. THE UNIVERSITY OF BRITISH COLUMBIA November 2004 c Glendon Ralph Pugh, 2004 In presenting this thesis in partial fulfillment of the requirements for an ad- vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Mathematics The University of British Columbia Vancouver, Canada Date Abstract This thesis is an analysis of C. Lanczos’ approximation of the classical gamma function Γ(z+1) as given in his 1964 paper A Precision Approximation of the Gamma Function [14]. The purposes of this study are: (i) to explain the details of Lanczos’ paper, including proofs of all claims made by the author; (ii) to address the question of how best to implement the approximation method in practice; and (iii) to generalize the methods used in the derivation of the approximation. At present there are a number of algorithms for approximating the gamma function. -
Hamiltonians Arising from L-Functions in the Selberg Class
HAMILTONIANS ARISING FROM L-FUNCTIONS IN THE SELBERG CLASS MASATOSHI SUZUKI Abstract. We establish a new equivalent condition for the Grand Riemann Hypoth- esis for L-functions in a wide subclass of the Selberg class in terms of canonical systems of differential equations. A canonical system is determined by a real symmetric matrix- valued function called a Hamiltonian. To establish the equivalent condition, we use an inverse problem for canonical systems of a special type. 1. Introduction The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zeta- function ζ(s) lie on the critical line (s) = 1/2, and it has been generalized to wider classes of zeta-like functions. Especially,ℜ an analogue of RH for L-functions in the Selberg class is often called the Grand Riemann Hypothesis (GRH)1. The present paper aims to establish a new equivalent condition for GRH for L- functions in a wide subclass of the Selberg class in terms of canonical systems by using general results in the preliminary paper [45] studying an inverse problem of canonical systems of a special type. We explain the relation by dealing with the case of the Rie- mann zeta function in the introduction. The present study was mainly stimulated by the works of Lagarias [25, 26, 27] (and also Burnol [10, 11, 12, 13] as well as [45]). The Riemann xi-function 1 s ξ(s)= s(s 1)π−s/2Γ ζ(s) 2 − 2 is an entire function taking real-values on the critical line such that the zeros coincide with nontrivial zeros of ζ(s).