# A Panaroma of Harmonic Analysis

Steven G. Krantz
Volume: 27
Edition: 1
Pages: 374
https://www.jstor.org/stable/10.4169/j.ctt5hh96g

1. Front Matter
(pp. i-viii)
(pp. ix-xii)
3. Preface
(pp. xiii-xvi)
Steven G. Krantz
4. CHAPTER 0 Overview of Measure Theory and Functional Analysis
(pp. 1-30)

We take it for granted that the reader of this book has some acquaintance with elementary real analysis. The book [KRA1], or Rudin’s classic [RUD1], will provide ample review.

However, the reader may be less acquainted with measure theory. The review that we shall now provide will give even the complete neophyte an intuitive understanding of the concept of measure, so that the remainder of the book may be appreciated. In fact, the reading of the present book will provide the reader with considerable motivation for learning more about measure theory.

By the same token, we shall follow the review...

5. CHAPTER 1 Fourier Series Basics
(pp. 31-94)

In the middle of the eighteenth century much attention was given to the problem of determining the mathematical laws governing the motion of a vibrating string with fixed endpoints at 0 and π (Figure 1). An elementary analysis of tension shows that if$y(x,t)$denotes the ordinate of the string at timetabove the pointx, then$y(x,t)$satisfies the wave equation

$\frac{{{\partial ^2}y}}{{\partial {t^2}}} = {a^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}$.

Hereais a parameter that depends on the tension of the string. A change of scale will allow us to assume thata= 1. [For completeness, we include a derivation of the wave equation...

6. CHAPTER 2 The Fourier Transform
(pp. 95-120)

A thorough treatment of the Fourier transform in Euclidean space may be found in [STG1]. Here we give a sketch of the theory. Most of the results parallel facts that we have already seen in the context of Fourier series on the circle. But some, such as the invariance properties of the Fourier transform under the groups that act on Euclidean space (Section 2.2), will be new.

If$t,\xi \in {\mathbb{R}^N}$then we let

$t \cdot \xi \equiv {t_1}{\xi _1} + ... + {t_N}{\xi _N}$.

We define the Fourier transform of a function$f \in {L^1}({\mathbb{R}^N})$by

$\hat f(\xi ) = \int_{{\mathbb{R}^N}} {f(t)} {e^{it.\xi }}dt$.

Heredtdenotes LebesgueN-dimensional measure. Many references will insert a factor of 2π in the...

7. CHAPTER 3 Multiple Fourier Series
(pp. 121-170)

In Chapter 1 we learned that a suitable method for summing the Fourier series of a functionfon the circle is to define partial sums

${S_N}f(x) \equiv \sum\limits_{j = - N}^N {\hat f(j){e^{ijx}}}$.

Recall that questions of convergence of${S_N}f$tofreduced to the study of the Hilbert transform; the Hilbert transform, in turn, corresponds (roughly speaking) to the summation operation

$f \mapsto \sum\limits_{j = 0}^\infty {\hat f(j){e^{ijx}}}$.

Running this reasoning in reverse, it is possible to prove that almostanyreasonable definition of partial sum will give a tractable theory for Fourier analysis on the circle group. More precisely, if$\alpha (N)$,$\beta (N)$are both positive, strictly increasing functions that take...

8. CHAPTER 4 Spherical Harmonics
(pp. 171-198)

We learned in Section 1.1 that each character${e^{ik\theta }}$for the circle group has a unique harmonic extension to the disc. In fact, if$k \geqslant 0$, then the extension is the function${z^k}$, and ifk< 0, then the extension is the function${\overline z ^{\left| k \right|}}$. Thus we may view the Riesz-Fischer theorem as saying that anyL² function on the circle may be expressed as a linear combination of boundary functions of the harmonic monomials$\{ {z^k}\} _{k = 0}^\infty$and$\{ {\overline z ^k}\} _{k = 1}^\infty$.

It is also natural to wish to decompose a function defined on all of${\mathbb{R}^2}$in terms of the characters$\{ {e^{ik\theta }}\} _{k = - \infty }^\infty$. For specificity,...

9. CHAPTER 5 Fractional Integrals, Singular Integrals, and Hardy Spaces
(pp. 199-234)

For$\phi \in C_c^1({\mathbb{R}^N})$we know that

$\frac{{\widehat{\partial \phi }}} {{\partial {x_j}}}(\xi ) = - i{\xi _j} \cdot \widehat\phi (\xi )$. (5.1.1)

In other words, the Fourier transform converts differentiation in thex-variable to multiplication by a monomial in the Fourier transform variable. Of course higher-order derivatives correspond to multiplication by higher-order monomials.

It is natural to wonder whether the Fourier transform can provide us with a way to think about differentiation to a fractional order. In pursuit of this goal, we begin by thinking about the Laplacian

$\Delta \phi \equiv \sum\limits_{J = 1}^N {\frac{{{\partial ^2}}} {{\partial x_j^2}}} \phi$.

of course formula (5.1.1) shows that

$\widehat{\Delta \phi }(\xi ) = - {\left| \xi \right|^2}\widehat\phi (\xi )$. (5.1.2)

In the remainder of this section, let us use the notation

${D^2}\phi (\xi ) = - \Delta \phi (\xi )$.

Then we set${D^4}\phi \equiv {D^2}o{D^2}\phi$, and so...

10. CHAPTER 6 Modern Theories of Integral Operators
(pp. 235-272)

One of the important developments in harmonic analysis in the 1960s and 1970s was a realization that, in certain important contexts, an axiomatic theory is possible. What does this assertion mean?

The Brelot potential theory gives a version of axiomatic potential theory [CHO]. That is not the focus of the current discussion. In the classical harmonic analysis of Euclidean space, we are interested in maximal operators (of Hardy-Littlewood type), in fractional integrals (of Riesz type), and in singular integral operators. What structure does a space require in order to support analytic objects like these that enjoy at least a modicum...

11. CHAPTER 7 Wavelets
(pp. 273-312)

The premise of the new versions of Fourier analysis that are being developed today is that sines and cosines are not an optimal model for some of the phenomena that we want to study. As an example, suppose that we are developing software to detect certain erratic heartbeats by analysis of an electrocardiogram. [Note that the discussion that we present here is philosophically correct but is oversimplified to facilitate the exposition.] The scheme is to have the software break down the patient’s electrocardiogram into component waves. If a wave that is known to be a telltale signal of heart disease...

12. CHAPTER 8 A Retrospective
(pp. 313-314)

We have seen that the basic questions of Fourier analysis grew out of eighteenth-century studies of the wave equation.

Decades later, J. Fourier found an algorithm for expanding an “arbitrary” function on [0, 2π) in terms of sines and cosines. Where Fourier was motivated by physical problems and was perhaps not as mathematically rigorous as we would like, P.G.L. Dirichlet began the process of putting the theory of Fourier series on a rigorous footing.

Fourier analysis has served as a pump for much of modern analysis. G. Cantor’s investigations of set theory were motivated in part by questions of sets...

13. Appendix I The Existence of Testing Functions and Their Density in ${L^P}$
(pp. 315-317)
14. Appendix II Schwartz Functions and the Fourier Transform
(pp. 317-317)
15. Appendix III The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin
(pp. 318-319)
16. Appendix IV Hausdorff Measure and Surface Measure
(pp. 320-322)
17. Appendix V Green’s Theorem
(pp. 323-323)
18. Appendix VI The Banach-Alaoglu Theorem
(pp. 323-324)
19. Appendix VII Expressing an Integral in Terms of the Distribution Function
(pp. 324-324)
20. Appendix VIII The Stone-Weierstrass Theorem
(pp. 324-325)
21. Appendix IX Landau’s O and o Notation
(pp. 325-326)
22. Table of Notation
(pp. 327-338)
23. Bibliography
(pp. 339-346)
24. Index
(pp. 347-357)