The Summation of Power Series and Fourier Series
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Convolution Theorem
Module 2 : Signals in Frequency Domain Lecture 18 : The Convolution Theorem Objectives In this lecture you will learn the following We shall prove the most important theorem regarding the Fourier Transform- the Convolution Theorem We are going to learn about filters. Proof of 'the Convolution theorem for the Fourier Transform'. The Dual version of the Convolution Theorem Parseval's theorem The Convolution Theorem We shall in this lecture prove the most important theorem regarding the Fourier Transform- the Convolution Theorem. It is this theorem that links the Fourier Transform to LSI systems, and opens up a wide range of applications for the Fourier Transform. We shall inspire its importance with an application example. Modulation Modulation refers to the process of embedding an information-bearing signal into a second carrier signal. Extracting the information- bearing signal is called demodulation. Modulation allows us to transmit information signals efficiently. It also makes possible the simultaneous transmission of more than one signal with overlapping spectra over the same channel. That is why we can have so many channels being broadcast on radio at the same time which would have been impossible without modulation There are several ways in which modulation is done. One technique is amplitude modulation or AM in which the information signal is used to modulate the amplitude of the carrier signal. Another important technique is frequency modulation or FM, in which the information signal is used to vary the frequency of the carrier signal. Let us consider a very simple example of AM. Consider the signal x(t) which has the spectrum X(f) as shown : Why such a spectrum ? Because it's the simplest possible multi-valued function. -
Lecture 7: Fourier Series and Complex Power Series 1 Fourier
Math 1d Instructor: Padraic Bartlett Lecture 7: Fourier Series and Complex Power Series Week 7 Caltech 2013 1 Fourier Series 1.1 Definitions and Motivation Definition 1.1. A Fourier series is a series of functions of the form 1 C X + (a sin(nx) + b cos(nx)) ; 2 n n n=1 where C; an; bn are some collection of real numbers. The first, immediate use of Fourier series is the following theorem, which tells us that they (in a sense) can approximate far more functions than power series can: Theorem 1.2. Suppose that f(x) is a real-valued function such that • f(x) is continuous with continuous derivative, except for at most finitely many points in [−π; π]. • f(x) is periodic with period 2π: i.e. f(x) = f(x ± 2π), for any x 2 R. C P1 Then there is a Fourier series 2 + n=1 (an sin(nx) + bn cos(nx)) such that 1 C X f(x) = + (a sin(nx) + b cos(nx)) : 2 n n n=1 In other words, where power series can only converge to functions that are continu- ous and infinitely differentiable everywhere the power series is defined, Fourier series can converge to far more functions! This makes them, in practice, a quite useful concept, as in science we'll often want to study functions that aren't always continuous, or infinitely differentiable. A very specific application of Fourier series is to sound and music! Specifically, re- call/observe that a musical note with frequency f is caused by the propogation of the lon- gitudinal wave sin(2πft) through some medium. -
Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
Rearrangement of Divergent Fourier Series
The Australian Journal of Mathematical Analysis and Applications AJMAA Volume 14, Issue 1, Article 3, pp. 1-9, 2017 A NOTE ON DIVERGENT FOURIER SERIES AND λ-PERMUTATIONS ANGEL CASTILLO, JOSE CHAVEZ, AND HYEJIN KIM Received 20 September, 2016; accepted 4 February, 2017; published 20 February, 2017. TUFTS UNIVERSITY,DEPARTMENT OF MATHEMATICS,MEDFORD, MA 02155, USA [email protected] TEXAS TECH UNIVERSITY,DEPARTMENT OF MATHEMATICS AND STATISTICS,LUBBOCK, TX 79409, USA [email protected] UNIVERSITY OF MICHIGAN-DEARBORN,DEPARTMENT OF MATHEMATICS AND STATISTICS,DEARBORN, MI 48128, USA [email protected] ABSTRACT. We present a continuous function on [−π, π] whose Fourier series diverges and it cannot be rearranged to converge by a λ-permutation. Key words and phrases: Fourier series, Rearrangements, λ-permutations. 2000 Mathematics Subject Classification. Primary 43A50. ISSN (electronic): 1449-5910 c 2017 Austral Internet Publishing. All rights reserved. This research was conducted during the NREUP at University of Michigan-Dearborn and it was sponsored by NSF-Grant DMS-1359016 and by NSA-Grant H98230-15-1-0020. We would like to thank Y. E. Zeytuncu for valuable discussion. We also thank the CASL and the Department of Mathematics and Statistics at the University of Michigan-Dearborn for providing a welcoming atmosphere during the summer REU program. 2 A. CASTILLO AND J. CHAVEZ AND H. KIM 1.1. Fourier series. The Fourier series associated with a continuous function f on [−π, π] is defined by ∞ X inθ fe(θ) ∼ ane , n=−∞ where Z π 1 −inθ an = f(θ)e dθ . 2π −π Here an’s are called the Fourier coefficients of f and we denote by fethe Fourier series associ- ated with f. -
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. -
Harmonic Analysis from Fourier to Wavelets
STUDENT MATHEMATICAL LIBRARY ⍀ IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society Institute for Advanced Study Harmonic Analysis From Fourier to Wavelets STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society, Providence, Rhode Island Institute for Advanced Study, Princeton, New Jersey Editorial Board of the Student Mathematical Library Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell Series Editor for the Park City Mathematics Institute John Polking 2010 Mathematics Subject Classification. Primary 42–01; Secondary 42–02, 42Axx, 42B25, 42C40. The anteater on the dedication page is by Miguel. The dragon at the back of the book is by Alexander. For additional information and updates on this book, visit www.ams.org/bookpages/stml-63 Library of Congress Cataloging-in-Publication Data Pereyra, Mar´ıa Cristina. Harmonic analysis : from Fourier to wavelets / Mar´ıa Cristina Pereyra, Lesley A. Ward. p. cm. — (Student mathematical library ; 63. IAS/Park City mathematical subseries) Includes bibliographical references and indexes. ISBN 978-0-8218-7566-7 (alk. paper) 1. Harmonic analysis—Textbooks. I. Ward, Lesley A., 1963– II. Title. QA403.P44 2012 515.2433—dc23 2012001283 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. -
Divergent and Conditionally Convergent Series Whose Product Is Absolutely Convergent
DIVERGENTAND CONDITIONALLY CONVERGENT SERIES WHOSEPRODUCT IS ABSOLUTELYCONVERGENT* BY FLORIAN CAJOKI § 1. Introduction. It has been shown by Abel that, if the product : 71—« I(¥» + V«-i + ■•• + M„«o)> 71=0 of two conditionally convergent series : 71—0 71— 0 is convergent, it converges to the product of their sums. Tests of the conver- gence of the product of conditionally convergent series have been worked out by A. Pringsheim,| A. Voss,J and myself.§ There exist certain conditionally- convergent series which yield a convergent result when they are raised to a cer- tain positive integral power, but which yield a divergent result when they are raised to a higher power. Thus, 71 = «3 Z(-l)"+13, 7>=i n where r = 7/9, is a conditionally convergent series whose fourth power is con- vergent, but whose fifth power is divergent. || These instances of conditionally * Presented to the Society April 28, 1900. Received for publication April 28, 1900. fMathematische Annalen, vol. 21 (1883), p. 327 ; vol. 2(5 (1886), p. 157. %Mathematische Annalen, vol. 24 (1884), p. 42. § American Journal of Mathematics, vol. 15 (1893), p. 339 ; vol. 18 (1896), p. 195 ; Bulletin of the American Mathematical Society, (2) vol. 1 (1895), p. 180. || This may be a convenient place to point out a slight and obvious extension of the results which I have published in the American Journal of Mathematics, vol. 18, p. 201. It was proved there that the conditionally convergent series : V(_l)M-lI (0<r5|>). Sí n when raised by Cauchy's multiplication rule to a positive integral power g , is convergent whenever (î — l)/ï <C r ! out the power of the series is divergent, if (q— 1 )¡q > r. -
Notes on Euler's Work on Divergent Factorial Series and Their Associated
Indian J. Pure Appl. Math., 41(1): 39-66, February 2010 °c Indian National Science Academy NOTES ON EULER’S WORK ON DIVERGENT FACTORIAL SERIES AND THEIR ASSOCIATED CONTINUED FRACTIONS Trond Digernes¤ and V. S. Varadarajan¤¤ ¤University of Trondheim, Trondheim, Norway e-mail: [email protected] ¤¤University of California, Los Angeles, CA, USA e-mail: [email protected] Abstract Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective. Key words Divergent series, factorial series, continued fractions, hypergeometric continued fractions, Sturmian sequences. 1. Introductory Remarks Euler was the first mathematician to develop a systematic theory of divergent se- ries. In his great 1760 paper De seriebus divergentibus [1, 2] and in his letters to Bernoulli he championed the view, which was truly revolutionary for his epoch, that one should be able to assign a numerical value to any divergent series, thus allowing the possibility of working systematically with them (see [3]). He antic- ipated by over a century the methods of summation of divergent series which are known today as the summation methods of Cesaro, Holder,¨ Abel, Euler, Borel, and so on. Eventually his views would find their proper place in the modern theory of divergent series [4]. But from the beginning Euler realized that almost none of his methods could be applied to the series X1 1 ¡ 1!x + 2!x2 ¡ 3!x3 + ::: = (¡1)nn!xn (1) n=0 40 TROND DIGERNES AND V. -
Convolution.Pdf
Convolution March 22, 2020 0.0.1 Setting up [23]: from bokeh.plotting import figure from bokeh.io import output_notebook, show,curdoc from bokeh.themes import Theme import numpy as np output_notebook() theme=Theme(json={}) curdoc().theme=theme 0.1 Convolution Convolution is a fundamental operation that appears in many different contexts in both pure and applied settings in mathematics. Polynomials: a simple case Let’s look at a simple case first. Consider the space of real valued polynomial functions P. If f 2 P, we have X1 i f(z) = aiz i=0 where ai = 0 for i sufficiently large. There is an obvious linear map a : P! c00(R) where c00(R) is the space of sequences that are eventually zero. i For each integer i, we have a projection map ai : c00(R) ! R so that ai(f) is the coefficient of z for f 2 P. Since P is a ring under addition and multiplication of functions, these operations must be reflected on the other side of the map a. 1 Clearly we can add sequences in c00(R) componentwise, and since addition of polynomials is also componentwise we have: a(f + g) = a(f) + a(g) What about multiplication? We know that X Xn X1 an(fg) = ar(f)as(g) = ar(f)an−r(g) = ar(f)an−r(g) r+s=n r=0 r=0 where we adopt the convention that ai(f) = 0 if i < 0. Given (a0; a1;:::) and (b0; b1;:::) in c00(R), define a ∗ b = (c0; c1;:::) where X1 cn = ajbn−j j=0 with the same convention that ai = bi = 0 if i < 0. -
Fourier Series of Functions of A-Bounded Variation
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 74, Number 1, April 1979 FOURIER SERIES OF FUNCTIONS OF A-BOUNDED VARIATION DANIEL WATERMAN1 Abstract. It is shown that the Fourier coefficients of functions of A- bounded variation, A = {A„}, are 0(Kn/n). This was known for \, = n&+l, — 1 < ß < 0. The classes L and HBV are shown to be complementary, but L and ABV are not complementary if ABV is not contained in HBV. The partial sums of the Fourier series of a function of harmonic bounded variation are shown to be uniformly bounded and a theorem analogous to that of Dirichlet is shown for this class of functions without recourse to the Lebesgue test. We have shown elsewhere that functions of harmonic bounded variation (HBV) satisfy the Lebesgue test for convergence of their Fourier series, but if a class of functions of A-bounded variation (ABV) is not properly contained in HBV, it contains functions whose Fourier series diverge [1]. We have also shown that Fourier series of functions of class {nB+x} — BV, -1 < ß < 0, are (C, ß) bounded, implying that the Fourier coefficients are 0(nB) [2]. Here we shall estimate the Fourier coefficients of functions in ABV. Without recourse to the Lebesgue test, we shall prove a theorem for functions of HBV analogous to that of Dirichlet and also show that the partial sums of the Fourier series of an HBV function are uniformly bounded. From this one can conclude that L and HBV are complementary classes, i.e., Parseval's formula holds (with ordinary convergence) for/ G L and g G HBV. -
Euler and His Work on Infinite Series
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. -
Divergent Series Past, Present, Future
Divergent series past, present, future. Christiane Rousseau 1 Divergent series, JMM 2015 How many students and mathematicians have never heard of divergent series? 2 Divergent series, JMM 2015 Preamble: My presentation of the subject and of some of its history is not exhaustive. Moreover, it is biased by my interest in dynamical systems. 3 Divergent series, JMM 2015 Euler: “If we get the sum 1 - 1 + 1 - 1 + 1 - 1 + ··· 1 its only reasonable value is 2.” Divergent series have been used a lot in the past by people including Fourier, Stieltjes, Euler, . 4 Divergent series, JMM 2015 Divergent series have been used a lot in the past by people including Fourier, Stieltjes, Euler, . Euler: “If we get the sum 1 - 1 + 1 - 1 + 1 - 1 + ··· 1 its only reasonable value is 2.” 5 Divergent series, JMM 2015 A hundred years before Riemann, Euler had found the functional equation of the function ζ in the form 1s-1 - 2s-1 + 3s-1 - ··· (s - 1)!(2s - 1) 1 = - cos sπ 1 1 1 (2s-1 - 1)πs 2 1s - 2s + 3s - ··· by “calculating”, for s 2 N [ f1=2;3=2g, the sums 1s - 2s + 3s - 4s + 5s - 6s + ··· and 1 1 1 1 1 1 - + - + - + ··· 1s 2s 3s 4s 5s 6s 6 Divergent series, JMM 2015 Where is the turn? 7 Divergent series, JMM 2015 Cauchy, Abel 8 Divergent series, JMM 2015 Cauchy made one exception: he justified rigorously the use of Stirling’s divergent series in numerical computations. Cauchy, Preface of “Analyse mathematique”,´ 1821 “I have been forced to admit some propositions which will seem, perhaps, hard to accept.