# Fourier Series

Rajendra Bhatia
Edition: 1
Pages: 131
https://www.jstor.org/stable/10.4169/j.ctt5hh9w4

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-x)
4. 0 A History of Fourier Series
(pp. 1-12)

A large part of mathematics has its roots in physics. Fourier series arose in the study of two simple physical problems—the motion of a vibrating string and heat conduction in solids. Attempts to understand what these series meant and in what sense they solved these two problems in physics have contributed to the origin and growth of most of modern analysis. Among ideas and theories that owe their existence to questions arising out of the study of Fourier series are Cantor’s theory of infinite sets, the Riemann and the Lebesgue integrals and the summability of series. Even such a...

5. 1 Heat Conduction and Fourier Series
(pp. 13-26)

In this chapter we derive the mathematical equations that describe the phenomenon of heat flow in a thin plate. We explain how, and in what sense, these equations are solved using Fourier Series. Our analysis leads to several questions. Some of these are answered in Chapters 1 and 2.

The equation$\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}=0\caption {(1.1)}$is called the 2-variable Laplace equation. Hereu(x,y) is a function on${{\mathbb{R}}^{2}}$which is of classC2, i.e.,uhas continuous derivatives up to the second order.

This equation describes natural phenomena such as steady flow of heat in two dimensions. To understand this imagine a...

6. 2 Convergence of Fourier Series
(pp. 27-50)

We have defined the Fourier coefficients offas$\hat{f}(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi }{f(\theta ){{e}^{-in\theta }}d\theta .}\caption {(2.1)}$

These are well defined for each continuous function onT, or more generally, for each integrable function onT. The Fourier series offis the series$\sum\limits_{n=-\infty }^{\infty }{\hat{f}(n){{e}^{in\theta }}.}\caption {(2.2)}$

Associated with the series is the sequence of its partial sums${{S}_{N}}(f;\theta )=\sum\limits_{n=-N}^{N}{\hat{f}(n){{e}^{in\theta }},}\caption {(2.3)}$

N= 0, 1, 2, …. If at a pointθofTthe sequence (2.3) converges, we say that the Fourier series (2.2) converges atθ. It would have been nice if such convergence did take place at every pointθ. Unfortunately, this is not the case. There are...

7. 3 Odds and Ends
(pp. 51-78)

A functionfis calledoddiff(x) = −f(−x) andeveniff(x) =f(−x) for allx. An arbitrary functionfcan be decomposed asf=feven+fodd, where${{f}_{\text{even}}}(x)=\frac{1}{2}[f(x)+f(-x)]$and${{f}_{\text{odd}}}(x)=\frac{1}{2}[f(x)-f(-x)]$. The functionsfevenandfoddare called theeven partand theodd partoff, respectively. Notice that iff(x) =eix, thenfeven(x) = cosxandfodd(x) =isinx.

Exercise 3.1.1.Show that iffis an even function onT, then$\hat{f}(n)=\hat{f}(-n)$, and iffis odd, then$\hat{f}(n)=-\hat{f}(-n)$. In other words the Fourier coefficients...

8. 4 Convergence in L2 and L1
(pp. 79-94)

In Chapter 2 we saw that the Fourier series of a continuous function onTmay not converge at every point. But under weaker notions of convergence (like Cesàro convergence) the series does converge. Another notion of convergence is that of convergence in the mean square orL2convergence. However, this notion is in some sense even more natural than pointwise convergence for Fourier series. We will now study this and the relatedL1convergence. We assume that the reader is familiar with basic properties of Hilbert space and the spacesL1andL2. Some of them are quickly recalled...

9. 5 Some Applications
(pp. 95-110)

One test of the depth of a mathematical theory is the variety of its applications. We began this study with a problem of heat conduction. The solution of this problem by Fourier series gives rise to an elegant mathematical theory with several applications in diverse areas. We have already seen some of them. We have proved the Weierstrass approximation theorem in Chapters 1 and 3. This is a fundamental theorem useful in numerical analysis, in approximation theory and in other branches of analysis. In Chapter 3 and 4 the Dirichlet and Plancherel theorems were used to obtain the sums of...

10. A A Note on Normalisation
(pp. 111-112)
11. B A Brief Bibliography
(pp. 113-116)
12. Index
(pp. 117-119)
13. Back Matter
(pp. 120-120)