DOCSLIB.ORG

Convergence of Fourier Series Javier Canto Llorente

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Convergence of Fourier Series Javier Canto Llorente Convergence of Fourier series Final Degree Dissertation Degree in Mathematics Javier Canto Llorente Supervisor: Javier Duoandikoetxea Zuazo Leioa, June 2016 Contents Introduction v 1 Trigonometric Fourier Series 1 1.1 Basic notation . .1 1.2 Trigonometric series . .3 1.3 Properties of the coefficients . .4 1.4 Bessel inequality . .4 2 Convergence of Fourier Series 7 2.1 The Dirichlet kernel . .7 2.2 The Riemann-Lebesgue lemma and some consequences . .9 2.3 The Dirichlet theorem . 11 2.4 The Dini theorem . 14 2.5 Uniform convergence . 15 2.6 The Gibbs phenomenon . 17 3 Summability of Fourier series 21 3.1 Ces`arosummability . 21 3.2 Abel-Poisson summability . 24 3.3 Approximate identities . 27 p 4 Fourier series in L (T) 31 2 4.1 Parseval theorem: Fourier series in L (T)........... 31 4.2 An interpolation theorem . 35 p 4.3 The conjugate function and convergence in L (T)....... 38 5 Divergence of Fourier series 45 5.1 Pointwise divergence . 45 1 1 5.2 Divergence in L (T) and L (T)................. 47 A Further theory 51 A.1 A result about integration . 51 A.2 About numerical series . 51 A.3 Some results in Complex Analysis . 54 iii B Exercises 57 B.1 Chapter 1. Trigonometric Fourier series . 57 B.2 Chapter 2: Convergence . 59 B.3 Chapter 3. Summability . 62 Bibliography 65 iv Introduction Motivation The main goal of this dissertation is to study the problem of convergence of Fourier series. This problem consists on, given a function f, deciding if its Fourier series converges to f. What conditions are necessary or sufficient for that to happen? What do we actually mean with convergence? We are going to give an answer to these question in this dissertation. The problem of convergence of Fourier series was first posed in the begin- ning of the XIX century, and it has evolved with the development if math- ematical analysis. Several great mathematicians have worked on Fourier series, and this subject has influenced some of them. It would be bold to say that Fourier series have shaped the development of history of mathe- matics, but there is no doubt that it has influenced it in some way. Abstract This Final Degree Dissertation is organized as follows. In Chapter 1, we define the basic concepts about Fourier series. We also give some basic properties of the Fourier coefficients. In Chapter 2, we introduce the most classical theorems about conver- gence of the Fourier series, all of them proved in the XIX century. In Chapter 3, we study summability of Fourier series. These are alter- native techniques to obtain the \sum" of the Fourier series. In Chapter 4, we prove convergence of Fourier series on the spaces Lp, for 1 < p < 1: Finally, in Chapter 5, we explain the divergence of Fourier series. We give a continuous function whose Fourier series is divergent at a point, and we prove the existence of an L1 function whose Fourier series is divergent on the L1 norm. v vi Historical comment about Fourier series The problem of Fourier series began with partial differential equations, in the middle of the XVIII century. More precisely, with the vibrating string problem. D'Alembert, Euler and Daniel Bernouilli, amongst other math- ematicians, worked on the solution of the wave equation. Even though d'Alembert obtain his famous solution to the equation, Bernouilli wrote the solution as a trigonometric sine. Since then, the theory of Fourier series has had a huge development, and there are several applications to this theory. In 1822, Joseph Fourier published his Th´eorieanalytique de la chaleur. In this book, he studied the propagation of heat in a finite solid. He de- duced the constitutive equation of the propagation of heat, and he solved the problem of the distribution of the temperature at a given time if he knew the distribution at an initial moment. In order to do that, he invented the method of separation of variables, also known as the Fourier method. As we know, this method requires writing a given function as the sum of a trigonometric series. Fourier did not prove convergence in any form. He opened the problem of the expansion in a trigonometric series of a given function. There are a lot of concepts related to this problem, such as integral, sum of series, and even the mere concept of function. There is not doubt that this problem influenced the development of the mathematical analysis. There were a few attempts to prove convergence of the series by Cauchy and Poisson, but the first theorem was proved by Dirichlet in 1829 (Theorem 2.9). He determined some sufficient conditions that assure the convergence of the Fourier series of a function to the function itself at a given point. This is the first of many similar theorems that give sufficient conditions for convergence. In this dissertation we mention theorems by Camille Jordan (Theorem 2.11, 1881), Lipschitz (1864, we see it as a consequence of Dini's theorem) and Dini (Theorem 2.12, 1880). Riemann worked, like on most topics in mathematics developed in the XIX century, on the problem of Fourier series. He developed his theory of the integral and then applied it to the Fourier series. He realized that if a function f does not admit integration in the domain, then the Fourier series of f has not sense. He gave the famous Riemann-Lebesgue lemma 2.3, and the Riemann-localization principle 2.4. Not everything result positive, though. Heine pointed out in 1870 that, at that point, there was not proof that a function had a unique Fourier series, and that there was no evidence that the Fourier series of a continuous function had to be uniformly convergent. Even so, both statements were widely accepted until there appeared a counterexample. Paul du Bois-Reymond found a continuous function whose Fourier series is divergent at a point. Not only the series was not uniformly convergent, it was divergent at a point. After this counterexample, there appeared other Introduction vii ones, by Schwarz, Fej´er and Lebesgue. After the disappointing counterexample by du Bois-Reymond, there ap- peared a new way of summing the series. Fej´erproved in 1900 that given a continuous function f, the sequence of the arithmetic averages of the par- tial sums of the Fourier series of f converges uniformly to f. This is not the only way, we also recover the original function with the Abel-Poisson summability. In the beginning of the XX century, there appeared new theories in mathematical analysis. The theory of integration by Lebesgue, the theory of Hilbert and Banach spaces, for example. In this wider context, there was much to be done. Lebesgue applied his own theory to the Fourier series, and he proved his version of the Riemann-Lebesgue lemma 2.3. With the Lebesgue theory, there appeared the Lp spaces. The first the- orem in these spaces is called the Parseval identity, which actually was pub- lished by Parseval in 1806 (even before Fourier's publication, Theorem 4.2). Fatou proved in 1906 that it holds for every f 2 L2: After that came a negative result by Banach and Steinhaus. They proved in 1918 that there is no mean convergence, meaning no convergence in L1. The last theorem we are going to explain we owe it to Marcel Riesz in 1923; this theorem states that there is convergence in Lp for every 1 < p < 1 (Theorem 4.10). But there is further history of Fourier series. The stronger theorem yet about Fourier series was proved by Carleson in 1965. He proved that the Fourier series of an L2 function converges pointwise to the function for almost every point. This came as a surprise because, although that theorem was conjectured around 1913 by Lusin, it was expected to be false. It was later expanded by Richard Hunt to every 1 < p < 1. Chapter 1 Trigonometric Fourier Series 1.1 Basic notation There are many options to define the Fourier series. I have chosen the torus one. We define the torus T = [−π; π). We can identify T with the unit circle ix in C, using the exponential map e . We will identify the space of functions defined on the torus T with the space of the periodic functions of period 2π defined on the real line R. We will study both real valued and complex valued functions. Another way of understanding this concept is considering in R the fol- lowing equivalence relation: x ∼ y , x − y = 2kπ for some k 2 Z: In this case, the torus would be the quotient space T = R= ∼. Both ways of understanding the torus are obviously equivalent. This simplifies the notation, for example, when we talk about the left or right sided limits of the function at the points x = ±π, or when we talk about continuity at those points. We will not need to consider the periodic extension of the original function. 1 Given a function f 2 L (T), we define the Fourier series of f: 1 X f^(n)einx; (1.1) n=−∞ where f^(n) is the n-th Fourier coefficient of f, defined in the following way: 1 Z f^(n) = f(x)e−inxdx: (1.2) 2π T If there is no confusion about what function f is, we may call cn = f^(n): Since we use integrals, we may need to specify what we mean. We are going to use Lebesgue measure and integral, due to the advantages it carries 1 2 1.1.
Load more

Recommended publications
  • Genius Manual I
    Genius Manual i Genius Manual Genius Manual ii Copyright © 1997-2016 Jiríˇ (George) Lebl Copyright © 2004 Kai Willadsen Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License (GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link or in the file COPYING-DOCS distributed with this manual. This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the license. Many of the names used by companies to distinguish their products and services are claimed as trademarks. Where those names appear in any GNOME documentation, and the members of the GNOME Documentation Project are made aware of those trademarks, then the names are in capital letters or initial capital letters. DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE WITH THE FURTHER UNDERSTANDING THAT: 1. DOCUMENT IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES THAT THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE OF THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS WITH YOU.
    [Show full text]
  • CONVERGENCE of FOURIER SERIES in Lp SPACE Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. F
    CONVERGENCE OF FOURIER SERIES IN Lp SPACE JING MIAO Abstract. The convergence of Fourier series of trigonometric functions is easy to see, but the same question for general functions is not simple to answer. We study the convergence of Fourier series in Lp spaces. This result gives us a criterion that determines whether certain partial differential equations have solutions or not.We will follow closely the ideas from Schlag and Muscalu's Classical and Multilinear Harmonic Analysis. Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. Fej´erkernel and Approximate identity 6 4. Lp convergence of partial sums 9 Appendix A. Proofs of Theorems and Lemma 16 Acknowledgments 18 References 18 1. Fourier Series, Partial Sums, and Dirichlet Kernel Let T = R=Z be the one-dimensional torus (in other words, the circle). We consider various function spaces on the torus T, namely the space of continuous functions C(T), the space of H¨oldercontinuous functions Cα(T) where 0 < α ≤ 1, and the Lebesgue spaces Lp(T) where 1 ≤ p ≤ 1. Let f be an L1 function. Then the associated Fourier series of f is 1 X (1.1) f(x) ∼ f^(n)e2πinx n=−∞ and Z 1 (1.2) f^(n) = f(x)e−2πinx dx: 0 The symbol ∼ in (1.1) is formal, and simply means that the series on the right-hand side is associated with f. One interesting and important question is when f equals the right-hand side in (1.1). Note that if we start from a trigonometric polynomial 1 X 2πinx (1.3) f(x) = ane ; n=−∞ Date: DEADLINES: Draft AUGUST 18 and Final version AUGUST 29, 2013.
    [Show full text]
  • Fourier Analysis and Equidistribution on the P-Adic Integers
    AN ABSTRACT OF THE DISSERTATION OF Naveen Somasunderam for the degree of Doctor of Philosophy in Mathematics presented on June 10, 2019. Title: Fourier Analysis and Equidistribution on the p-adic Integers Abstract approved: Clayton J. Petsche In this dissertation, we use Fourier-analytic methods to study questions of equidistribution on the compact abelian group Zp of p-adic integers. In particu- lar, we prove a LeVeque-type Fourier analytic upper bound on the discrepancy of sequences. We establish p-adic analogues of the classical Dirichlet and Fejér kernels on R=Z, and investigate their properties. Finally, we compare notions of variation for functions on Zp due to Beer and Taibleson. We also prove a p-adic Fourier analytic Koksma inequality. c Copyright by Naveen Somasunderam June 10, 2019 All Rights Reserved Fourier Analysis and Equidistribution on the p-adic Integers by Naveen Somasunderam A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented June 10, 2019 Commencement June 2020 Doctor of Philosophy dissertation of Naveen Somasunderam presented on June 10, 2019 APPROVED: Major Professor, representing Mathematics Head of the Department of Mathematics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dis- sertation to any reader upon request. Naveen Somasunderam, Author ACKNOWLEDGEMENTS In my years here at Oregon State University, I have had the fortune to meet truly wonderful people among the graduate students, faculty and the office staff.
    [Show full text]
  • Series of Functions
    Chapter 3 Series of functions 3.9 The Dirichlet kernel Our arguments so far have been entirely abstract — we have not really used inx any properties of the functions en(x) = e except that they are orthonor- mal. To get better results, we need to take a closer look at these functions. In some of our arguments, we shall need to change variables in integrals, and such changes may take us outside our basic interval [−π, π], and hence outside the region where our functions are defined. To avoid these prob- lems, we extend our functions periodically outside the basic interval such that f(x + 2π) = f(x) for all x ∈ R. The figure shows the extension graph- ically; in part a) we have the original function, and in part b) (a part of ) the periodic extension. As there is no danger of confusion, we shall denote the original function and the extension by the same symbol f. a) 6 b) 6 −π π −3π −π π 3π q q - q q q q - Figure 1 Let us begin by looking at the partial sums N X sN (x) = hf, enien(x) n=−N 1 2 CHAPTER 3. SERIES OF FUNCTIONS of the Fourier series. Since Z π 1 −int αn = hf, eni = f(t)e dt 2π −π we have N N 1 X Z π 1 Z π X s (x) = f(t)e−int dt einx = f(t) ein(x−t) dt = N 2π 2π n=−N −π −π n=−N N 1 Z π X = f(x − u) einu du 2π −π n=−N where we in the last step has substituted u = x−t and used the periodicity of the functions to remain in the interval [−π, π].
    [Show full text]
  • Fourier Analysis and Related Topics J. Korevaar
    Fourier Analysis and Related Topics J. Korevaar Preface For many years, the author taught a one-year course called “Mathe- matical Methods”. It was intended for beginning graduate students in the physical sciences and engineering, as well as for mathematics students with an interest in applications. The aim was to provide mathematical tools used in applications, and a certain theoretical background that would make other parts of mathematical analysis accessible to the student of physical science. The course was taken by a large number of students at the University of Wisconsin (Madison), the University of California San Diego (La Jolla), and finally, the University of Amsterdam. At one time the author planned to turn his elaborate lecture notes into a multi-volume book, but only one vol- ume appeared [68]. The material in the present book represents a selection from the lecture notes, with emphasis on Fourier theory. Starting with the classical theory for well-behaved functions, and passing through L1 and L2 theory, it culminates in distributional theory, with applications to bounday value problems. At the International Congress of Mathematicians (Cambridge, Mass) in 1950, many people became interested in the Generalized Functions or “Dis- tributions” of field medallist Laurent Schwartz; cf. [110]. Right after the congress, Michael Golomb, Merrill Shanks and the author organized a year- long seminar at Purdue University to study Schwartz’s work. The seminar led the author to a more concrete approach to distributions [66], which he included in applied mathematics courses at the University of Wisconsin. (The innovation was recognized by a Reynolds award in 1956.) It took the mathematical community a while to agree that distributions were useful.
    [Show full text]
  • Lecture Notes 4: the Frequency Domain
    Optimization-based data analysis Fall 2017 Lecture Notes 4: The Frequency Domain 1 Fourier representations 1.1 Complex exponentials The complex exponential or complex sinusoids is a complex-valued function. Its real part is a cosine and its imaginary part is a sine. Definition 1.1 (Complex sinusoid). exp (ix) := cos (x) + i sin (x) : (1) The argument x of a complex sinusoid is known as its phase. For any value of the phase, the magnitude of the complex exponential is equal to one, exp(ix) 2 = cos(x)2 + sin(x)2 = 1: (2) j j If we set the phase of a complex sinusoid to equal ft for fixed f, then the sinusoid is periodic with period 1=f and the parameter f is known as the frequency of the sinusoid. Definition 1.2 (Frequency). A complex sinusoid with frequency f is of the form hf (t) := exp (i2πft) : (3) Lemma 1.3 (Periodicity). The complex exponential hf (t) with frequency f is periodic with period 1=f. Proof. 1 h (t + 1=f) = exp i2πf t + (4) f f = exp(i2πft) exp(i2π) (5) = hf (t): (6) Figure1 shows a complex sinusoid with frequency 1 together with its real and imaginary parts. If we consider a unit interval, complex sinusoids with integer frequencies form an orthonormal set. The choice of interval is arbitrary, since all these sinusoids are periodic with period 1=k for some integer k and hence also with period one. 1 Figure 1: Complex sinusoid h1 (dark red) plotted between 0 and 10. Its real (green) corresponds to a cosine function and its imaginary part (blue) to a sine function.
    [Show full text]
  • Harmonic Analysis: from Fourier to Haar Mar´Ia Cristina Pereyra Lesley A. Ward
    Harmonic Analysis: from Fourier to Haar Mar´ıaCristina Pereyra Lesley A. Ward Department of Mathematics and Statistics, MSC03 2150, 1 Univer- sity of New Mexico, Albuquerque, NM 87131-0001, USA E-mail address: [email protected] School of Mathematics and Statistics, University of South Aus- tralia, Mawson Lakes SA 5095, Australia E-mail address: [email protected] 2000 Mathematics Subject Classification. Primary 42-xx CHAPTER 4 Summability methods: Dirichlet kernels and Fej´er kernels Pointwise convergence of partial Fourier sums for continuous functions was ruled out by the Du Bois-Reymond example. However, in an attempt to obtain a convergence result for all continuous functions, mathematicians devised vari- ous averaging or summability methods. These methods require only knowledge of the Fourier coefficients in order to recover a continuous function as the sum of an appropriate trigonometric series. Along the way, a number of very important approximation techniques in analysis were developed, in particular convolutions (Section 4.1), and approximations of the identity, also known as good kernels (Sec- tions 4.4 and 7.5). Roughly speaking, an approximation of the identity is a sequence of functions whose mass is concentrating near the origin (very much like a delta function1); a precise definition is given below. We discuss these techniques in detail in the context of 2π-periodic functions. Later we will revisit these ideas in the context of functions defined on the line (Section 7.5). Here we describe several kernels that arise naturally in the theory of Fourier series. Some of them (the Fej´erand Poisson2 kernels) are good kernels that generate approximations of the identity; another (the Dirichlet kernel) is equally important but not good in this sense.
    [Show full text]
  • Introduction, Motivation, Basics About Sets, Functions, Counting
    Lectures on Advanced Calculus with Applications, I audrey terras Math. Dept., U.C.S.D., La Jolla, CA 92093-0112 November, 2010 Part I Introduction, Motivation, Basics About Sets, Functions, Counting 1Preface These notes come from various courses that I have taught at U.C.S.D. using Serge Lang’s Undergraduate Analysis as the basic text. My lectures are an attempt to make the subject more accessible. Recently Rami Shakarchi published Problems and Solutions for Undergraduate Analysis, which provides solutions to all the problems in Lang’s book. This caused me to collect my own exercises which are included. Exams are also to be found. The main difference between the approach of Lang and that of other similar books is the treatment of the integral which emphasizes the properties of the integral as a linear function from the set of piecewise continuous real valued functions on an interval to the real numbers. Thus the approach can be viewed as intermediate between the Riemann integral and the Lebesgue integral. Since we are interested mainly in piecewise continuous functions, we are really getting the Riemann integral. In these lectures we include more pictures and examples than the usual texts. Moreover, we include less definitions from point set topology. Our aim is to make sense to an audience of potential high school math teachers, or economists, or engineers. We did not write these lectures for potential math. grad students. We will always try to include examples, pictures and applications. Applications will include Fourier analysis, fractals, .... Warning to the reader: This course is to calculus as fixing a car is to driving a car.
    [Show full text]
  • Why the Dirichlet Kernel Is Not a Good Kernel
    The good, the bad and the ugly of kernels: why the Dirichlet kernel is not a good kernel Peter Haggstrom www.gotohaggstrom.com [email protected] May 24, 2016 1 Background Even though Dirichlet's name is often associated with number theory, he also did fun- damental work on the convergence of Fourier series. Dirichlet's rigorous insights into the many subtle issues surrounding Fourier theory laid the foundation for what students study today. In the early part of the 19th century Fourier advanced the stunning, even outrageous, idea that an arbitrary function defined on (−π; π) could be represented by an infinite trigonometric series of sines and cosines thus: 1 X f(x) = a0 + [ak cos(kx) + bk sin(kx)] (1) k=1 He did this in his theory of heat published in 1822 although in the mid 1750s Daniel Bernoulli had also conjectured that shape of a vibrating string could be represented by a trigonometric series. Fourier's insights predated electrical and magnetic theory by several years and yet one of the widest applications of Fourier theory is in electrical engineering. The core of Fourier theory is to establish the conditions under which (1) is true. It is a complex and highly subtle story requiring some sophisticated analysis. Applied users of Fourier theory will rarely descend into the depths of analytical detail devoted to rigorous convergence proofs. Indeed, in some undergraduate courses on Fourier theory, the Sampling Theorem is proved on a "faith" basis using distribution theory. In what follows I have used Professor Elias Stein and Rami Shakarchi's book [Stein- Shakarchi] as a foundation for fleshing out the motivations, properties and uses of "good" kernels.
    [Show full text]
  • Arxiv:0903.3845V2 [Math.CA] 17 May 2017 Harmonic Analysis Lecture Notes University of Illinois at Urbana–Champaign
    Harmonic Analysis Lecture Notes University of Illinois at Urbana–Champaign Richard S. Laugesen * May 18, 2017 arXiv:0903.3845v2 [math.CA] 17 May 2017 *Copyright c 2017, Richard S. Laugesen ([email protected]). This work is licensed under the Creative Commons Attribution-NonCommercial- NoDerivs 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-nd/4.0/. 2 Preface A textbook presents more than any professor can cover in class. In contrast, these lecture notes present exactly* what I covered in Harmonic Analysis (Math 545) at the University of Illinois, Urbana–Champaign, in Fall 2008. The first part of the course emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove Lp convergence of Fourier series. Maximal functions and Calder´on– Zygmund decompositions are treated in Rd, so that they can be applied again in the second part of the course, where the Fourier transform is studied. Real methods are used throughout. In particular, complex methods such as Poisson integrals and conjugate functions are not used to prove bounded- ness of the Hilbert transform. Distribution functions and interpolation are covered in the Appendices. I inserted these topics at the appropriate places in my lectures (after Chap- ters 4 and 12, respectively). The references at the beginning of each chapter provide guidance to stu- dents who wish to delve more deeply, or roam more widely, in the subject. Those references do not necessarily contain all the material in the chapter.
    [Show full text]
  • Fourier Series: Convergence and Summability
    CHAPTER 1 Fourier Series: Convergence and Summability Let T = R/Z be the one-dimensional torus (circle). We consider various function spaces on it, namely C(T),Cα(T), and Lp(T). The space of complex Borel measures on T will be denoted by M(T). Any µ ∈ M(T) has associated with it a Fourier series X∞ µ ∼ µˆ(n)e(nx) n=−∞ where we let e(x) = e2πix and Z 1 µˆ(n) = e(−nx) dµ(x) . 0 We first consider the classical question of convergence of Fourier series. The partial sums of f ∈ L1(T) are defined as XN ˆ SN f(x) = f(n)e(nx) n=−N XN Z 1 = e(−ny)f(y) dy e(nx) n=−N 0 Z 1 XN = e(n(x − y))f(y) dy 0 n=−N Z 1 = DN (x − y)f(y) dy 0 PN where DN (x) = n=−N e(nx) is the Dirichlet kernel. Exercise 1. Check that sin((2N + 1)πx) D (x) = , N sin(πx) and draw the graph of DN . One can also write S f(x) = (D ∗ f)(x) R N N where f ∗ g(x) := T f(x − y)g(y) dy is the convolution of f and g. You should think of f ∗ g as an average of translates of f. Exercise 2. Prove the following properties of the convolution: 1 p 1 a) kf ∗ gkp ≤ kfkpkgk1 for all 1 ≤ p ≤ ∞, f ∈ L , g ∈ L . This is called Young’s inequality. You should pay careful attention to the fact that the integral defining f ∗ g is not necessarily absolutely convergent for every x.
    [Show full text]
  • Introduction to Fourier Series 1. Pointwise Convergence
    (October 27, 2016) Introduction to Fourier series Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/02b Fourier intro.pdf] 1. Pointwise convergence 2. Fourier-Dirichlet kernel versus approximate identities 3. Fejer kernel 4. Density of trigonometric polynomials in Co(T) 5. Exponentials form a Hilbert-space basis for L2(T) In his 1822 treatise on heat, J. Fourier espoused the wonderful idea that any function on an interval, say [0; 1], could be represented as an superposition of functions sin 2πnx and cos 2πnx. Equivalently, as superposition of e2πinx with n 2 Z. Since these functions are eigenfunctions for the operator d=dx, such expressions facilitated solution of differential equations. The issue of convergence, which in those days could only have meant pointwise convergence, was recognized. The first publication proving pointwise convergence was by P. Dirichlet in 1829, although the device in the proof had appeared in an earlier manuscript of Fourier whose publication was delayed. B. Riemann's 1854 Habilitationschrift concerned the representability of functions by trigonometric series. In 1915 N. Luzin conjectured that Fourier series of functions in L2(T) converge almost everywhere pointwise. Decades later, in 1966, L. Carlson proved Luzin's conjecture. In 1968, R. Hunt generalized this to Lp(T) functions for p > 1. From the other side, in 1876 P. du Bois-Reymond found a continuous function whose Fourier series diverges at a single point. Via the uniform boundedness theorem, we will show later that there are continuous functions whose Fourier series diverges at any given countable collection of points.
    [Show full text]