Math 563 Lecture Notes the Discrete Fourier Transform
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Construction of Determinantal Representation of Trigonometric Polynomials ∗ Mao-Ting Chien A, ,1, Hiroshi Nakazato B
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 435 (2011) 1277–1284 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Construction of determinantal representation of trigonometric polynomials ∗ Mao-Ting Chien a, ,1, Hiroshi Nakazato b a Department of Mathematics, Soochow University, Taipei 11102, Taiwan b Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan ARTICLE INFO ABSTRACT Article history: For a pair of n × n Hermitian matrices H and K, a real ternary homo- Received 6 December 2010 geneous polynomial defined by F(t, x, y) = det(tIn + xH + yK) is Accepted 10 March 2011 hyperbolic with respect to (1, 0, 0). The Fiedler conjecture (or Lax Available online 2 April 2011 conjecture) is recently affirmed, namely, for any real ternary hyper- SubmittedbyR.A.Brualdi bolic polynomial F(t, x, y), there exist real symmetric matrices S1 and S such that F(t, x, y) = det(tI + xS + yS ).Inthispaper,we AMS classification: 2 n 1 2 42A05 give a constructive proof of the existence of symmetric matrices for 14Q05 the ternary forms associated with trigonometric polynomials. 15A60 © 2011 Elsevier Inc. All rights reserved. Keywords: Hyperbolic polynomial Bezoutian matrix Trigonometric polynomial Numerical range 1. Introduction Let p(x) = p(x1, x2,...,xm) be a real homogeneous polynomial of degree n. The polynomial p(x) is called hyperbolic with respect to a real vector e = (e1, e2,...,em) if p(e) = 0, and for all vectors w ∈ Rm linearly independent of e, the univariate polynomial t → p(w − te) has all real roots. -
7: Inner Products, Fourier Series, Convolution
7: FOURIER SERIES STEVEN HEILMAN Contents 1. Review 1 2. Introduction 1 3. Periodic Functions 2 4. Inner Products on Periodic Functions 3 5. Trigonometric Polynomials 5 6. Periodic Convolutions 7 7. Fourier Inversion and Plancherel Theorems 10 8. Appendix: Notation 13 1. Review Exercise 1.1. Let (X; h·; ·i) be a (real or complex) inner product space. Define k·k : X ! [0; 1) by kxk := phx; xi. Then (X; k·k) is a normed linear space. Consequently, if we define d: X × X ! [0; 1) by d(x; y) := ph(x − y); (x − y)i, then (X; d) is a metric space. Exercise 1.2 (Cauchy-Schwarz Inequality). Let (X; h·; ·i) be a complex inner product space. Let x; y 2 X. Then jhx; yij ≤ hx; xi1=2hy; yi1=2: Exercise 1.3. Let (X; h·; ·i) be a complex inner product space. Let x; y 2 X. As usual, let kxk := phx; xi. Prove Pythagoras's theorem: if hx; yi = 0, then kx + yk2 = kxk2 + kyk2. 1 Theorem 1.4 (Weierstrass M-test). Let (X; d) be a metric space and let (fj)j=1 be a sequence of bounded real-valued continuous functions on X such that the series (of real num- P1 P1 bers) j=1 kfjk1 is absolutely convergent. Then the series j=1 fj converges uniformly to some continuous function f : X ! R. 2. Introduction A general problem in analysis is to approximate a general function by a series that is relatively easy to describe. With the Weierstrass Approximation theorem, we saw that it is possible to achieve this goal by approximating compactly supported continuous functions by polynomials. -
Fourier Series
Academic Press Encyclopedia of Physical Science and Technology Fourier Series James S. Walker Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI 54702–4004 Phone: 715–836–3301 Fax: 715–836–2924 e-mail: [email protected] 1 2 Encyclopedia of Physical Science and Technology I. Introduction II. Historical background III. Definition of Fourier series IV. Convergence of Fourier series V. Convergence in norm VI. Summability of Fourier series VII. Generalized Fourier series VIII. Discrete Fourier series IX. Conclusion GLOSSARY ¢¤£¦¥¨§ Bounded variation: A function has bounded variation on a closed interval ¡ ¢ if there exists a positive constant © such that, for all finite sets of points "! "! $#&% (' #*) © ¥ , the inequality is satisfied. Jordan proved that a function has bounded variation if and only if it can be expressed as the difference of two non-decreasing functions. Countably infinite set: A set is countably infinite if it can be put into one-to-one £0/"£ correspondence with the set of natural numbers ( +,£¦-.£ ). Examples: The integers and the rational numbers are countably infinite sets. "! "!;: # # 123547698 Continuous function: If , then the function is continuous at the point : . Such a point is called a continuity point for . A function which is continuous at all points is simply referred to as continuous. Lebesgue measure zero: A set < of real numbers is said to have Lebesgue measure ! $#¨CED B ¢ £¦¥ zero if, for each =?>A@ , there exists a collection of open intervals such ! ! D D J# K% $#L) ¢ £¦¥ ¥ ¢ = that <GFIH and . Examples: All finite sets, and all countably infinite sets, have Lebesgue measure zero. "! "! % # % # Odd and even functions: A function is odd if for all in its "! "! % # # domain. -
Trigonometric Interpolation and Quadrature in Perturbed Points∗
SIAM J. NUMER.ANAL. c 2017 Society for Industrial and Applied Mathematics Vol. 55, No. 5, pp. 2113{2122 TRIGONOMETRIC INTERPOLATION AND QUADRATURE IN PERTURBED POINTS∗ ANTHONY P. AUSTINy AND LLOYD N. TREFETHENz Abstract. The trigonometric interpolants to a periodic function f in equispaced points converge if f is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if f is continuous. What if the points are perturbed? With equispaced grid spacing h, let each point be perturbed by an arbitrary amount ≤ αh, where α 2 [0; 1=2) is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be trouble for α ≥ 1=4. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all α < 1=2 if f is twice continuously differentiable, with the convergence rate depending on the smoothness of f. More precisely, it is enough for f to have 4α derivatives in a certain sense, and we conjecture that 2α derivatives are enough. Connections with the Fej´er{Kalm´artheorem are discussed. Key words. trigonometric interpolation, quadrature, Lebesgue constant, Kadec 1/4 theorem, Fej´er–Kalm´artheorem, sampling theory AMS subject classifications. 42A15, 65D32, 94A20 DOI. 10.1137/16M1107760 1. Introduction and summary of results. The basic question of robustness of mathematical algorithms is \What happens if the data are perturbed?" Yet little literature exists on the effect on interpolants, or on quadratures, of perturbing the interpolation points. The questions addressed in this paper arise in two almost equivalent settings: in- terpolation by algebraic polynomials (e.g., in Gauss or Chebyshev points) and periodic interpolation by trigonometric polynomials (e.g., in equispaced points). -
Extremal Positive Trigonometric Polynomials Dimitar K
APPROXIMATION THEORY: A volume dedicated to Blagovest Sendov 2002, 1-24 Extremal Positive Trigonometric Polynomials Dimitar K. Dimitrov ∗ In this paper we review various results about nonnegative trigono- metric polynomials. The emphasis is on their applications in Fourier Series, Approximation Theory, Function Theory and Number The- ory. 1. Introduction There are various reasons for the interest in the problem of constructing nonnegative trigonometric polynomials. Among them are: Ces`aromeans and Gibbs’ phenomenon of the the Fourier series, approximation theory, univalent functions and polynomials, positive Jacobi polynomial sums, or- thogonal polynomials on the unit circle, zero-free regions for the Riemann zeta-function, just to mention a few. In this paper we summarize some of the recent results on nonnegative trigonometric polynomials. Needless to say, we shall not be able to cover all the results and applications. Because of that this short review represents our personal taste. We restrict ourselves to the results and problems we find interesting and challenging. One of the earliest examples of nonnegative trigonometric series is the Poisson kernel ∞ X 1 − ρ2 1 + 2 ρk cos kθ = , −1 < ρ < 1, (1.1) 1 − 2ρ cos θ + ρ2 k=1 which, as it was pointed out by Askey and Gasper [8], was also found but not published by Gauss [30]. The problem of constructing nonnegative trigonometric polynomials was inspired by the development of the theory ∗Research supported by the Brazilian science foundation FAPESP under Grant No. 97/06280-0 and CNPq under Grant No. 300645/95-3 2 Positive Trigonometric Polynomials of Fourier series and by the efforts for giving a simple proof of the Prime Number Theorem. -
Contentscontents 2323 Fourier Series
ContentsContents 2323 Fourier Series 23.1 Periodic Functions 2 23.2 Representing Periodic Functions by Fourier Series 9 23.3 Even and Odd Functions 30 23.4 Convergence 40 23.5 Half-range Series 46 23.6 The Complex Form 53 23.7 An Application of Fourier Series 68 Learning outcomes In this Workbook you will learn how to express a periodic signal f(t) in a series of sines and cosines. You will learn how to simplify the calculations if the signal happens to be an even or an odd function. You will learn some brief facts relating to the convergence of the Fourier series. You will learn how to approximate a non-periodic signal by a Fourier series. You will learn how to re-express a standard Fourier series in complex form which paves the way for a later examination of Fourier transforms. Finally you will learn about some simple applications of Fourier series. Periodic Functions 23.1 Introduction You should already know how to take a function of a single variable f(x) and represent it by a power series in x about any point x0 of interest. Such a series is known as a Taylor series or Taylor expansion or, if x0 = 0, as a Maclaurin series. This topic was firs met in 16. Such an expansion is only possible if the function is sufficiently smooth (that is, if it can be differentiated as often as required). Geometrically this means that there are no jumps or spikes in the curve y = f(x) near the point of expansion. -
On the Real Projections of Zeros of Almost Periodic Functions
ON THE REAL PROJECTIONS OF ZEROS OF ALMOST PERIODIC FUNCTIONS J.M. SEPULCRE AND T. VIDAL Abstract. This paper deals with the set of the real projections of the zeros of an arbitrary almost periodic function defined in a vertical strip U. It pro- vides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents {λ1, λ2, λ3,...} of an almost periodic function are linearly independent over the rational numbers, such a set has no isolated points in U. 1. Introduction The theory of almost periodic functions, which was created and developed in its main features by H. Bohr during the 1920’s, opened a way to study a wide class of trigonometric series of the general type and even exponential series. This theory shortly acquired numerous applications to various areas of mathematics, from harmonic analysis to differential equations. In the case of the functions that are defined on the real numbers, the notion of almost periodicity is a generalization of purely periodic functions and, in fact, as in classical Fourier analysis, any almost periodic function is associated with a Fourier series with real frequencies. Let us briefly recall some notions concerning the theory of the almost periodic functions of a complex variable, which was theorized in [4] (see also [3, 5, 8, 11]). A function f(s), s = σ + it, analytic in a vertical strip U = {s = σ + it ∈ C : α< σ<β} (−∞ ≤ α<β ≤ ∞), is called almost periodic in U if to any ε > 0 there exists a number l = l(ε) such that each interval t0 <t<t0 + l of length l contains a number τ satisfying |f(s + iτ) − f(s)|≤ ε ∀s ∈ U. -
Almost Periodic Sequences and Functions with Given Values
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 1–16 ALMOST PERIODIC SEQUENCES AND FUNCTIONS WITH GIVEN VALUES Michal Veselý Abstract. We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set X in a metric space, it is proved the existence of an almost periodic sequence {ψk}k∈Z such that {ψk; k ∈ Z} = X and ψk = ψk+lq(k), l ∈ Z for all k and some q(k) ∈ N which depends on k. 1. Introduction The aim of this paper is to construct almost periodic sequences and functions which attain values in a metric space. More precisely, our aim is to find almost periodic sequences and functions whose ranges contain or consist of arbitrarily given subsets of the metric space satisfying certain conditions. We are motivated by the paper [3] where a similar problem is investigated for real-valued sequences. In that paper, using an explicit construction, it is shown that, for any bounded countable set of real numbers, there exists an almost periodic sequence whose range is this set and which attains each value in this set periodically. We will extend this result to sequences attaining values in any metric space. Concerning almost periodic sequences with indices k ∈ N (or asymptotically almost periodic sequences), we refer to [4] where it is proved that, for any precompact sequence {xk}k∈N in a metric space X , there exists a permutation P of the set of positive integers such that the sequence {xP (k)}k∈N is almost periodic. -
Fourier Transform for Mean Periodic Functions
Annales Univ. Sci. Budapest., Sect. Comp. 35 (2011) 267–283 FOURIER TRANSFORM FOR MEAN PERIODIC FUNCTIONS L´aszl´oSz´ekelyhidi (Debrecen, Hungary) Dedicated to the 60th birthday of Professor Antal J´arai Abstract. Mean periodic functions are natural generalizations of periodic functions. There are different transforms - like Fourier transforms - defined for these types of functions. In this note we introduce some transforms and compare them with the usual Fourier transform. 1. Introduction In this paper C(R) denotes the locally convex topological vector space of all continuous complex valued functions on the reals, equipped with the linear operations and the topology of uniform convergence on compact sets. Any closed translation invariant subspace of C(R)iscalledavariety. The smallest variety containing a given f in C(R)iscalledthevariety generated by f and it is denoted by τ(f). If this is different from C(R), then f is called mean periodic. In other words, a function f in C(R) is mean periodic if and only if there exists a nonzero continuous linear functional μ on C(R) such that (1) f ∗ μ =0 2000 Mathematics Subject Classification: 42A16, 42A38. Key words and phrases: Mean periodic function, Fourier transform, Carleman transform. The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-68040. 268 L. Sz´ekelyhidi holds. In this case sometimes we say that f is mean periodic with respect to μ. As any continuous linear functional on C(R) can be identified with a compactly supported complex Borel measure on R, equation (1) has the form (2) f(x − y) dμ(y)=0 for each x in R. -
Stepanov and Weyl Almost Periodic Functions on Locally Compact
STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS TIMO SPINDELER Abstract. We study Stepanov and Weyl almost periodic functions on locally compact Abelian groups, which naturally appear in the theory of aperiodic order. 1. Introduction Almost periodic functions are natural generalisations of periodic functions, and they ap- pear in many different areas of mathematics. They are well studied objects on the real line because they are used in boundary value problems for differential equations [13]. A system- atic treatment of these functions can be found in [5]. Also, there is a well established theory for Bohr almost periodic functions and weakly almost periodic functions on locally compact Abelian groups [6, 8]. Bohr almost periodic functions turned out to be an important tool in harmonic analysis. Firstly, they are used to study finite-dimensional representations of locally compact Abelian groups [7, Ch. 16.2]. Secondly, they describe those autocorrelations which give rise to a pure point diffraction measure; more on that below. However, recent progress in the study of aperiodic order and dynamical systems [10, 11, 18] makes it clear that we also need to aim at a better understanding of Stepanov and Weyl almost periodic functions on locally compact Abelian groups. Let us be more precise. Meyer sets and cut and project sets are important tools in the study of aperiodic order, since they are used to model quasicrystals. If µ denotes a translation bounded measure (e.g. the Dirac comb of a Meyer set), then the diffraction measure γµ, which −1 is the Fourier transform of the autocorrelation γµ := limn An (µ A µ A ), indicates →∞ | | | n ∗ | n how (dis)ordered the system that µ models is [1]; note that (An)n∈N is a suitable averagingc sequence. -
6.436J / 15.085J Fundamentals of Probability, Homework 1 Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2018 Problem Set 1 Readings: (a) Notes from Lecture 1. (b) Handout on background material on sets and real analysis (Recitation 1). Supplementary readings: [C], Sections 1.1-1.4. [GS], Sections 1.1-1.3. [W], Sections 1.0-1.5, 1.9. Exercise 1. (a) Let N be the set of positive integers. A function f : N → {0, 1} is said to be periodic if there exists some N such that f(n + N)= f(n),for all n ∈ N. Show that the set of periodic functions is countable. (b) Does the result from part (a) remain valid if we consider rational-valued periodic functions f : N → Q? Solution: (a) For a given positive integer N,letA N denote the set of periodic functions with a period of N.Fora given N ,since thesequence, f(1), ··· ,f(N), actually defines a periodic function in AN ,wehave thateach A N contains 2N elements. For example, for N =2,there arefour functions inthe set A2: f(1)f(2)f(3)f(4) ··· =0000··· ;1111··· ;0101··· ;1010··· . The set of periodic functions from N to {0, 1}, A,can bewritten as, ∞ A = AN . N=1 ! Since the union of countably many finite sets is countable, we conclude that the set of periodic functions from N to {0, 1} is countable. (b) Still, for a given positive integer N,let AN denote the set of periodic func- tions with a period N.Fora given N ,since the sequence, f(1), ··· ,f(N), 1 actually defines a periodic function in AN ,weconclude that A N has the same cardinality as QN (the Cartesian product of N sets of rational num- bers). -
Fourier Series
Chapter 10 Fourier Series 10.1 Periodic Functions and Orthogonality Relations The differential equation ′′ y + 2y = F cos !t models a mass-spring system with natural frequency with a pure cosine forcing function of frequency !. If 2 = !2 a particular solution is easily found by undetermined coefficients (or by∕ using Laplace transforms) to be F y = cos !t. p 2 !2 − If the forcing function is a linear combination of simple cosine functions, so that the differential equation is N ′′ 2 y + y = Fn cos !nt n=1 X 2 2 where = !n for any n, then, by linearity, a particular solution is obtained as a sum ∕ N F y (t)= n cos ! t. p 2 !2 n n=1 n X − This simple procedure can be extended to any function that can be repre- sented as a sum of cosine (and sine) functions, even if that summation is not a finite sum. It turns out that the functions that can be represented as sums in this form are very general, and include most of the periodic functions that are usually encountered in applications. 723 724 10 Fourier Series Periodic Functions A function f is said to be periodic with period p> 0 if f(t + p)= f(t) for all t in the domain of f. This means that the graph of f repeats in successive intervals of length p, as can be seen in the graph in Figure 10.1. y p 2p 3p 4p 5p Fig. 10.1 An example of a periodic function with period p.