Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

519 Pages·2017·3.23 MB·English

by  Valery Serov (auth.)

Applied Mathematical Sciences
Valery Serov
Fourier Series,
Fourier Transform and
Their Applications to
Mathematical Physics
Applied Mathematical Sciences
Volume 197
Editors
S.S. Antman, Institute for Physical Science and Technology, University of Maryland, College Park,
MD, USA
More information about this series at http://www.springer.com/series/34
Valery Serov
Fourier Series, Fourier
Transform and Their
Applications to Mathematical
Physics
Valery Serov
Department of Mathematical Sciences
University of Oulu
Oulu
Finland
ISSN 0066-5452 ISSN 2196-968X (electronic)
Applied Mathematical Sciences
ISBN 978-3-319-65261-0 ISBN 978-3-319-65262-7 (eBook)
DOI: 10.1007/978-3-319-65262-7
Library of Congress Control Number: 2017950249
Mathematics Subject Classification (2010): 26A16, 26A45, 35A08, 35F50, 35J05, 35J08, 35J10, 35J15,
35K05, 35K08, 35L05, 35P25, 35R30
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Preface
The modern theory of analysis and differential equations in general certainly in-
cludes the Fourier transform, Fourier series, integral operators, spectral theory of
differential operators, harmonic analysis and much more. This book combines all
these subjects based on a unified approach that uses modern view on all these
themes. The book consists of four parts: Fourier series and the discrete Fourier
transform, Fourier transform and distributions, Operator theory and integral equa-
tions and Introduction to partial differential equations and it outgrew from the half-
semester courses of the same name given by the author at University of Oulu, Fin-
land during 2005–2015.
Each part forms a self-contained text (although they are linked by a common
approach) and can be read independently. The book is designed to be a modern
introduction to qualitative methods used in harmonic analysis and partial differential
equations (PDEs). It can be noted that a survey of the state of the art for all parts of
this book can be found in a very recent and fundamental work of B. Simon [35].
This book contains about 250 exercises that are an integral part of the text. Each
part contains its own collection of exercises with own numeration. They are not only
an integral part of the book, but also indispensable for the understanding of all parts
whose collection is the content of this book. It can be expected that a careful reader
will complete all these exercises.
This book is intended for graduate level students majoring in pure and applied
mathematics but even an advanced researcher can find here very useful information
which previously could only be detected in scientific articles or monographs.
Each part of the book begins with its own introduction which contains the facts
(mostly) from functional analysis used thereinafter. Some of them are proved while
the others are not.
The first part, Fourier series and the discrete Fourier transform, is devoted to
the classical one-dimensional trigonometric Fourier series with some applications
to PDEs and signal processing. This part provides a self-contained treatment of all
well known results (but not only) at the beginning graduate level. Compared with
some known texts (see [12, 18, 29, 35, 38, 44, 45]) this part uses many function
spaces such as Sobolev, Besov, Nikol’skii and Ho¨lder spaces. All these spaces are
v
vi Preface
introduced by special manner via the Fourier coefficients and they are used in the
proofs of main results. Same definition of Sobolev spaces can be found in [35]. The
advantage of such approach is that we are able to prove quite easily the precise em-
beddings for these spaces that are the same as in classical function theory (see [1, 3,
26, 42]). In the frame of this part some very delicate properties of the trigonometric
Fourier series (Chapter 10) are considered using quite elementary proofs (see also
[46]). The unified approach allows us also to consider naturally the discrete Fourier
transform and establish its deep connections with the continuous Fourier transform.
As a consequence we prove the famous Whittaker-Shannon-Boas theorem about the
reconstruction of band-limited signal via the trigonometric Fourier series (see Chap-
ter 13). Many applications of the trigonometric Fourier series to the one-dimensional
heat, wave and Laplace equation are presented in Chapter 14. It is accompanied by a
large number of very useful exercises and examples with applications in PDEs (see
also [10, 17]).
The second part, Fourier transform and distributions, probably takes a central role
in this book and it is concerned with distribution theory of L. Schwartz and its ap-
plications to the Schro¨dinger and magnetic Schro¨dinger operators (see Chapter 32).
The estimates for Laplacian and Hamiltonian that generalize well known Agmon’s
estimates on the continuous spectrum are presented in this part (see Chapter 23).
This part can be considered as one of the most important because of numerous ap-
plications in the scattering theory and inverse problems. Here we have considered
for the first time some classical direct scattering problems for the Schro¨dinger op-
erator and for the magnetic Schro¨dinger operator with singular (locally unbounded)
coefficients including the mathematical foundations of the classical approximation
of M. Born. Also, the properties of Riesz transform and Riesz potentials (see Chap-
ter 21) are investigated very carefully in this part. Before this material could only be
found in scientific journals or monographs but not in textbooks. There is a good con-
nection of this part with Operator theory and integral equations. The main technique
applied here is the Fourier transform.
The third part, Operator theory and integral equations, is devoted mostly to the
self-adjoint but unbounded operators in Hilbert spaces and their applications to in-
tegral equations in such spaces. The advantage of this part is that many important
results of J. von Neumann’s theory of symmetric operators are collected together.
J. von Neumann’s spectral theorem allows us, for example, to introduce the heat
kernel without solving the heat equation. Moreover, we show applications of the
spectral theorem of J. von Neumann (for these operators) to the spectral theory of
elliptic differential operators. In particular, the existence of Friedrichs extension for
these operators with discrete spectrum is provided. Special attention is devoted to
the Schro¨dinger and the magnetic Schro¨dinger operators. The famous diamagnetic
inequality is proved here. We follow in this consideration B. Simon [35] (slightly
different approach can be found in [28]). We recommend (in addition to this part)
the reader get acquainted with the books [4, 13, 15, 24, 41]. As a consequence of
the spectral theory of elliptic differential operators the integral equations with weak
singularities are considered in quite simple manner not only in Hilbert spaces but
also in some Banach spaces, e.g. in the space of continuous functions on closed
Preface vii
manifolds. The central point of this consideration is the Riesz theory of compact
(not necessarily self-adjoint) operators in Hilbert and Banach spaces. In order to
keep this part short, some proofs will not be given, nor will all theorems be proved
in complete generality. For many details of these integral equations we recommend
[22]. We are able to investigate in quite simple manner one-dimensional Volterra in-
∞
tegral equations with weak singularities in L (a,b) and singular integral equations
α
in the periodic Ho¨lder spaces C [−a,a]. Concerning approximation methods our
considerations use the general theory of bounded or compact operators in Hilbert
spaces and we follow mostly the monograph of Kress [22].
The fourth part, Introduction to partial differential equations, serves as an in-
troduction to modern methods for classical theory of partial differential equations.
Fourier series and Fourier transform play crucial role here too. An important (and
quite independent) segment of this part is the self-contained theory of quasi-linear
partial differential equations of order one. The main attention in this part is devoted
to elliptic boundary value problems in Sobolev and Ho¨lder spaces. In particular, the
unique solvability of direct scattering problem for Helmholtz equation is provided.
We investigate very carefully the mapping and discontinuity properties of double
and single layer potentials with continuous densities. We also refer to similar prop-
1/2
erties of double and single layer potentials with densities in Sobolev spaces H (S)
−1/2
and H (S), respectively, but will not prove any of these results, referring for their
proofs to monographs [22] and [25]. Here (and elsewhere in the book) S denotes the
n
boundary of a bounded domain in R and if the smoothness of S is not specified
explicitly then it is assumed to be such that Sobolev embedding theorem holds.
Compared with well known texts on partial differential equations some direct and
inverse scattering problems for Helmholtz, Schro¨dinger and magnetic Schro¨dinger
operators are considered in this part. As it was mentioned earlier this type of mater-
ial could not be found in textbooks. The presentation in many places of this part has
been strongly influenced by the monographs [6, 7, 11] (see also [8, 16, 24, 36, 40]).
In closing we note that this book is not as comprehensive as the fundamental
work of B. Simon [35]. But the book can be considered as a good introduction to
modern theory of analysis and differential equations and might be useful not only
to students and PhD students but also to all researchers who have applications in
mathematical physics and engineering sciences. This book could not have appeared
without the strong participation, both in content and typesetting, of my colleague
Adj. Prof. Markus Harju. Finally, a special thanks to professor David Colton from
University of Delaware (USA) who encouraged the writing of this book and who
has supported the author very much over the years.
Oulu, Finland Valery Serov
June 2017
Contents
Part I Fourier Series and the Discrete Fourier Transform
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Formulation of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Fourier Coefficients and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Convolution and Parseval’s Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Feje´r Means of Fourier Series. Uniqueness of the Fourier Series. . . . . 27
6 The Riemann–Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 The Fourier Series of a Square-Integrable Function. The
Riesz–Fischer Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Besov and Ho¨lder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9 Absolute convergence. Bernstein and Peetre Theorems. . . . . . . . . . . . . . 53
10 Dirichlet Kernel. Pointwise and Uniform Convergence. . . . . . . . . . . . . . 59
11 Formulation of the Discrete Fourier Transform and Its Properties. . . . 77
12 Connection Between the Discrete Fourier Transform
and the Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
13 Some Applications of the Discrete Fourier Transform. . . . . . . . . . . . . . . 93
14 Applications to Solving Some Model Equations . . . . . . . . . . . . . . . . . . . . 99
14.1 The One-Dimensional Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 99
14.2 The One-Dimensional Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 113
14.3 The Laplace Equation in a Rectangle and in a Disk . . . . . . . . . . . . . 121
ix
x Contents
Part II Fourier Transform and Distributions
15 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
16 The Fourier Transform in Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . 133
p n
17 The Fourier Transform in L (R ), 1 ≤ p ≤ 2. . . . . . . . . . . . . . . . . . . . . . . 143
18 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
′
19 Convolutions in S and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
20 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
20.1 Sobolev spaces on bounded domains . . . . . . . . . . . . . . . . . . . . . . . . . 188
21 Homogeneous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
22 Fundamental Solution of the Helmholtz Operator . . . . . . . . . . . . . . . . . . 207
23 Estimates for the Laplacian and Hamiltonian . . . . . . . . . . . . . . . . . . . . . 217
Part III Operator Theory and Integral Equations
24 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
25 Inner Product Spaces and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 249
26 Symmetric Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
27 John von Neumann’s spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
28 Spectra of Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
29 Quadratic Forms. Friedrichs Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . 313
30 Elliptic Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
31 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
32 The Schro¨dinger Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
33 The Magnetic Schro¨dinger Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
34 Integral Operators with Weak Singularities. Integral Equations
of the First and Second Kinds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
35 Volterra and Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . 371
36 Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Part IV Partial Differential Equations
37 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
38 Local Existence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405