Notes on Fourier analysis

1.1. Definition

When solving heat equations with linear and homogeneous boundary conditions, we obtained product solutions in terms of infinite series of sines and cosines. Joseph Fourier developed his ideas on the convergence of these trigonometric series while studying heat flow.

We now study Fourier series of a periodic function. Suppose ${f(\theta)}$ is a function defined on the real line such that ${f(\theta+2\pi)=f(\theta)}$ for all ${\theta}$ , then we say that ${f(\theta)}$ is ${2\pi}$ -periodic. We shall assume that ${f}$ is Riemann integrable on every bounded interval; this will be the case if ${f}$ is bounded and is continuous except at finitely many points in each bounded interval. We allow ${f}$ is be complex-valued (we will see why later). We wish to obtain the following series expansion:

$\displaystyle f(\theta)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{cos}(n\theta)) \ \ \ \ \ (1)$

The series expansion of ${f(\theta)}$ can also be written in terms of complex exponential functions ${e^{i\theta}}$ . We use the following properties:

$\displaystyle \mathrm{cos}(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} \ \ \ \ \ (2)$

$\displaystyle \mathrm{sin}(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i} \ \ \ \ \ (3)$

$\displaystyle e^{i\theta}= \mathrm{cos}(\theta)+i\mathrm{sin}(\theta) \ \ \ \ \ (4)$

Equation (1) can be rewritten as

$\displaystyle f(\theta)=\sum_{-\infty}^{\infty}c_ne^{in\theta} \ \ \ \ \ (5)$

where

$\displaystyle c_0=\frac{1}{2}a_0 {\hspace{0.5cm}} c_n=\frac{1}{2}(a_n-ib_n) {\hspace{0.5cm}} c_{-n}=\frac{1}{2}(a_n+ib_n) {\hspace{0.5cm}} n=1,2,3... \ \ \ \ \ (6)$

We can consider (5) as the sum of two infinity series, one going from ${n=0}$ to ${+\infty}$ and one going from ${n=-1}$ to ${-\infty}$ .

Alternatively, we can express the coefficients in the following way:

$\displaystyle a_0=2c_0 {\hspace{0.5cm}} a_n=c_n+c_{-n} {\hspace{0.5cm}} b_n=i(c_n-c_{-n}){\hspace{0.5cm}} {\hspace{0.5cm}} n=1,2,3... \ \ \ \ \ (7)$

How can the coefficient ${c_n}$ be expressed in terms of ${f}$ ? Let’s first assume that the series is term by term integrable, we multiply equation (5) by ${e^{-ik\theta}}$ , ${k}$ is an integer, and integrate both sides from ${-\pi}$ to ${\pi}$ . We obtain

$\displaystyle \int^{\pi}_{-\pi}f(\theta)e^{-ik\theta}\mathrm{d}\theta = \sum_{-\infty}^{\infty}c_n \int^{\pi}_{-\pi}e^{i(n-k)\theta}\mathrm{d}\theta \ \ \ \ \ (8)$

Note that

$\displaystyle \int^{\pi}_{-\pi}e^{i(n-k)\theta}\mathrm{d}\theta=\frac{1}{{i(n-k)}}e^{i(n-k)\theta}\Big|^\pi_{-\pi}=0 {\hspace{0.5cm}} n\neq k \ \ \ \ \ (9)$

$\displaystyle \int^{\pi}_{-\pi}{d}\theta = 2\pi {\hspace{0.5cm}} n= k \ \ \ \ \ (10)$

Hence, the only non-zero term after the integration is the term ${n=k}$ , and thus we have

$\displaystyle \int^{\pi}_{-\pi}f(\theta)e^{-ik\theta}\mathrm{d}\theta=c_k2\pi \ \ \ \ \ (11)$

Relabeling the term ${c_k}$ as ${c_n}$ we get

$\displaystyle c_n=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)e^{-in\theta}\mathrm{d}\theta \ \ \ \ \ (12)$

This means that we can also find ${a_0}$ , ${a_n}$ and ${b_n}$ easily

$\displaystyle a_0=2c_0=\frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{d}\theta \ \ \ \ \ (13)$

$\displaystyle a_n=c_n+c_{-n}= \frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)(e^{-in\theta}+e^{in\theta})\mathrm{d}\theta = \frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{cos}(n\theta)\mathrm{d}\theta \ \ \ \ \ (14)$

$\displaystyle b_n=i(c_n-c_{-n})=\frac{i}{2\pi}\int^{\pi}_{-\pi}f(\theta)(e^{-in\theta}-e^{in\theta})\mathrm{d}\theta = \frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{sin}(n\theta)\mathrm{d}\theta \ \ \ \ \ (15)$

for ${n=1,2,3...}$

To recapitulate: if ${f}$ has a series expansion of the form in (1), and if the series converges decently such that term by term integration is permissible, then the coefficients ${a_n, b_n}$ and ${c_n}$ are given by (14), (15), (12) respectively. But now if ${f}$ is any Riemann-integrable periodic functions, the integrals in (14), (15), and (12) make perfectly good sense, and we use them to define the coefficients ${a_n, b_n}$ and ${c_n}$ . We are able to give the formal definition as follows:

Definition 1.1 Suppose ${f}$ is periodic with period ${2\pi}$ and integrable over ${[-\pi,\pi]}$ . The numbers ${a_n}$ and ${b_n}$ defined in (14) and (15), or the number ${c_n}$ defined in (12), are called the Fourier coefficients of ${f}$ , and the corresponding series

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{sin}(n\theta)) {\hspace{0.5cm}} \mathrm{or} {\hspace{0.5cm}} \sum_{-\infty}^{\infty}c_ne^{in\theta} \ \ \ \ \ (16)$

are called the Fourier series of ${f}$ .

We also present the following Lemmas:

Lemma 1.2 With reference to the formulas (14) and (15),

if f is even,

$\displaystyle a_n=\frac{2}{\pi}\int^{\pi}_{0}f(\theta)\mathrm{cos}(n\theta)\mathrm{d}\theta {\hspace{0.5cm}}and {\hspace{0.5cm}} b_n=0 \ \ \ \ \ (17)$

if f is odd,

$\displaystyle b_n=\frac{2}{\pi}\int^{\pi}_{0}f(\theta)\mathrm{sin}(n\theta)\mathrm{d}\theta {\hspace{0.5cm}}and {\hspace{0.5cm}} a_n=0 \ \ \ \ \ (18)$

For Fourier series of a ${2\pi}$ -periodic function, the constant term of series is

$\displaystyle c_0=\frac{1}{2}a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{d}\theta \ \ \ \ \ (19)$

which is equivalent to the mean value of ${f}$ in the interval ${[-\pi,\pi]}$ . Thus,

Lemma 1.3 The constant term in the Fourier series of a ${2\pi}$ -periodic function ${f}$ is the mean value of ${f}$ on an interval of length ${2\pi}$ .

We now turn to the discussion of periodic functions with an arbitrary period ${2L}$ . Similarly, we obtain the following definition of the Fourier series.

Definition 1.4 For a function ${f}$ with period ${2L}$ the Fourier coefficients of ${f}$ are

$\displaystyle a_n=\frac{1}{L}\int^{L}_{-L}f(x)\mathrm{cos}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (20)$

$\displaystyle b_n=\frac{1}{L}\int^{L}_{-L}f(x)\mathrm{sin}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (21)$

$\displaystyle c_0=\frac{1}{2L}\int^{L}_{-L}f(x)e^{-i(\frac{n\pi x}{L})}\mathrm{d}x \ \ \ \ \ (22)$

and the corresponding series

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\frac{\pi x}{L})+b_n\mathrm{sin}(\frac{n\pi x}{L})) {\hspace{0.5cm}} \mathrm{or} {\hspace{0.5cm}} \sum_{-\infty}^{\infty}c_ne^{i\frac{n\pi x}{L}} \ \ \ \ \ (23)$

are called the Fourier series of function ${f}$ .

Figure 1. Graph of partial sums (n=10, 30, 80) of the Fourier series of a square wave.

Definition 1.5 Let ${f}$ be a function defined on the interval ${[-L,L]}$ . The periodic extension of ${f}$ is that function, denoted by ${f_p}$ , which satisfies for ${p=2L}$

$\displaystyle f_p(x+kp)=f(x) \ \ \ \ \ (24)$

for each integer k.

For an even function ${f}$ over ${[-L,L]}$ , the Fourier series for ${f}$ has the form

$\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}a_n\mathrm{cos}(n\frac{\pi x}{L}) \hspace{0.5cm} a_n=\frac{2}{L}\int^{L}_{0}f(x)\mathrm{cos}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (25)$

Formula (25) is called the cosine series of ${f}$ over ${[0,L]}$ .

For an odd function ${f}$ over ${[-L,L]}$ , the Fourier series for ${f}$ has the form

$\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}b_n\mathrm{sin}(n\frac{\pi x}{L}) \hspace{0.5cm} a_n=\frac{2}{L}\int^{L}_{0}f(x)\mathrm{sin}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (26)$

Formula (26) is called the sine series of ${f}$ over ${[0,L]}$ .

Definition 1.6 Let the function ${f}$ be defined on ${[0,L]}$ . The even extension ${f_e}$ and the odd extension ${f_o}$ of ${f}$ are the following functions

$\displaystyle f_e(x) \left\{ \begin{array}{lr} f(x) \\ f(-x) \\ \end{array} \right. \ \ \ \ \ (27)$

$\displaystyle f_o(x) \left\{ \begin{array}{lr} f(x) \\ -f(-x) \\ \end{array} \right. \ \ \ \ \ (28)$

${\square}$

1.2. A convergence theorem

Now that we have formally introduced the Fourier series. However, one important question remains unresolved: How do we know whether Fourier series converges for all function ${f(x)}$ ? Proving the convergence of the Fourier series is not a simple matter. Firstly, let’s derive an estimate of the Fourier coefficients that will be needed to establish convergence.

Theorem 2.1 (Bessel’s Inequality) If ${f}$ is ${2\pi}$ -periodic and Riemann integrable on ${[-\pi,\pi]}$ , and the Fourier coefficients are defined by (16), then

$\displaystyle \sum^{\infty}_{-\infty}|c_n|^2\leq \frac{1}{2\pi}\int^{\pi}_{-\pi}|f(\theta)|^2\mathrm{d}\theta \ \ \ \ \ (29)$

Proof We use the following property of complex numbers,

$\displaystyle \mid z \mid^2=z\overline{z} \ \ \ \ \ (30)$

and let

$\displaystyle \begin{aligned} \mid f(\theta)-\sum^{N}_{-N}c_ne^{in\theta}. \mid^2 & =( f(\theta)-\sum^{N}_{-N}c_ne^{in\theta})( \overline{f(\theta)}-\sum^{N}_{-N}\overline{c_n}e^{-in\theta}) \\ & = \mid f(\theta) \mid^2 -\sum^{N}_{-N} [f(\theta)\overline{c_n}e^{-in\theta}+\overline{f(\theta)}c_ne^{in\theta}]+\sum^{N}_{-N}c_n\overline{c_n}\\ \end{aligned} \ \ \ \ \ (31)$

Divide both sides of (31) by ${2\pi}$ and integrate from ${-\pi}$ to ${\pi}$ , we obtain the following using formula (12),

$\displaystyle \begin{aligned} \frac{1}{2\pi}\int^{\pi}_{-\pi}\mid f(\theta)-\sum^{N}_{-N}c_ne^{in\theta}. \mid^2 \mathrm{d}\theta & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2\mathrm{d}\theta -\sum^{N}_{-N} \overline{c_n}c_n+c_n\overline{c_n}+\sum^{N}_{-N}c_n\overline{c_n}\\ & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2\mathrm{d}\theta -\sum^{N}_{-N} \overline{c_n}c_n\\ & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta-\sum^{N}_{-N}\mid c_n\mid^2\\ \end{aligned} \ \ \ \ \ (32)$

It is obvious that the left-hand-side of (32) is nonnegative, thus

$\displaystyle \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta-\sum^{N}_{-N} \mid c_n\mid^2 \geq 0 \ \ \ \ \ (33)$

Letting ${n\rightarrow \infty}$ , we have

$\displaystyle \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta \geq \sum^{\infty}_{-\infty}\mid c_n\mid^2 \ \ \ \ \ (34)$

${\square}$

Bessel’s inequality can also be stated in terms of ${a_n}$ and ${b_n}$ defined by (16). By equation (7), for ${n\geq1}$ we have

$\displaystyle \begin{aligned} \mid a_n \mid^2 + \mid b_n \mid^2 & = a_n\overline{a_n}+b_n\overline{b_n}\\ & = (c_n+c_{-n})(\overline{c_n}+\overline{c_{-n}})+i(c_n-c_{-n})(-i)(\overline{c_n}-\overline{c_{-n}})\\ & = 2c_n\overline{c_n}+2c_{-n}\overline{c_{-n}} \end{aligned} \ \ \ \ \ (35)$

so that

$\displaystyle \mid a_0 \mid^2=4\mid c_0 \mid^2, {\hspace{0.5cm}} \mid a_n \mid^2+\mid b_n \mid^2 = 2(\mid c_n \mid^2 + \mid c_{-n} \mid^2) {\hspace{0.3cm}} \mathrm{for} {\hspace{0.3cm}} n\geq1 \ \ \ \ \ (36)$

Therefore,

$\displaystyle \frac{1}{4} \mid a_0 \mid^2+\frac{1}{2}\sum^{\infty}_{n=1} (\mid a_n \mid^2+\mid b_n \mid^2)= \sum^{\infty}_{-\infty} \overline\mid c_n\mid^2 \leq \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta \ \ \ \ \ (37)$

The significance of (37) is that the series ${\sum \mid a_0 \mid^2}$ , ${\sum \mid b_0 \mid^2}$ and ${\sum \mid c_0 \mid^2}$ are all convergent. As a consequence, we obtain the following result.

Corollary 2.2 The Fourier coefficients ${a_n}$ , ${b_n}$ , and ${c_n}$ all tend to ${0}$ as ${n\rightarrow\infty}$ (and as ${n\rightarrow-\infty}$ in the case of ${c_n}$ ).

Proof ${\mid a_n \mid^2}$ , ${\mid b_n \mid^2}$ and ${\mid c_n \mid^2}$ are the ${n}$ th term of convergent series, so they all tend to ${0}$ as ${n\rightarrow\infty}$ ; hence so do ${a_n}$ , ${b_n}$ , and ${c_n}$ .

In fact, corollary 2.2 is a special case of the Riemann-Lebesgue lemma. Next, we define the class of functions with which we shall be working.

Definition 2.3 Suppose ${-\infty<a<b<\infty}$ , we say that a function ${f}$ on the closed interval ${[a,b]}$ is piecewise continuous provided that (i) f is continous on ${[a,b]}$ except perhaps at finitely many points ${x_1,...,x_k}$ ;(ii) at each of the points ${x_1,...,x_k}$ the left-hand side and right-hand side limits of ${f}$ ,

$\displaystyle f(x_j-)=\lim_{h\rightarrow0, h>0}(x_j-h) \ \ \ \ \ (38)$

and

$\displaystyle f(x_j+)=\lim_{h\rightarrow0, h>0}(x_j+h) \ \ \ \ \ (39)$

exists. (If ${a}$ (or ${b}$ ) are one of the exception points ${x_j}$ , then we require only the left-hand (or the right-hand) limit to exist). Thus, we say that ${f}$ is piecewise continuous on the ${[a,b]}$ if f is continuous there except for finitely many finite jump continuities.

Figure 2. A piecewise continuous function.

Definition 2.4 A function ${f}$ on the closed interval ${[a,b]}$ is piecewise smooth if ${f}$ and its first derivative ${f'}$ are both piecewise continuous.

Pictorially, ${f}$ is piecewise smooth if its graph is a smooth curve except for finitely many jumps (where ${f}$ is discontinuous) and corners (where ${f'}$ is discontinuous). We do not allow infinite discontinuities or sharp cusps.

Figure 3. A piecewise smooth function (left) and a function that is not piecewise smooth (right).

A very useful tool in Fourier analysis is the Dirichlet’s kernel for the ${n}$ th partial sum

$\displaystyle S_N^f(\theta)=\frac{1}{2}a_0+\sum_{n=1}^{N}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{cos}(n\theta))=\sum_{-N}^{N}c_ne^{in\theta} \ \ \ \ \ (40)$

of the Fourier series for ${f}$ . Substitute ${c_n}$ as defined in (16) into (31), we get

$\displaystyle S_N^f(\theta)=\sum_{-N}^{N}\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\psi)\mathrm e^{in(\theta-\psi)} {d}\psi \ \ \ \ \ (41)$

Note that we changed the variable of integration from ${\theta}$ to ${\psi}$ for later convenience. We obtain the following

$\displaystyle S_N^f(\theta)= \frac{1}{2\pi}\sum_{-N}^{N}\int^{\pi}_{-\pi}f(\psi) e^{in(\psi-\theta)} \mathrm{d}\psi \ \ \ \ \ (42)$

by replacing ${n}$ with ${-n}$ . The sum is not affected since it ranges from ${-N}$ to ${N}$ . Now, letting ${\phi=\psi-\theta}$ , we have

$\displaystyle S_N^f(\theta)= \frac{1}{2\pi}\sum_{-N}^{N}\int^{\pi}_{-\pi}f(\phi+\theta) e^{in\phi} \mathrm{d}\phi\\ \ \ \ \ \ (43)$

We can write (34) as

$\displaystyle S_N^f(\theta)=\int^{\pi}_{-\pi}f(\phi+\theta)D_N(\phi)\mathrm{d}\phi \hspace{0.5cm} \mathrm{where} \hspace{0.5cm} D_N(\phi)= \frac{1}{2\pi}\sum_{-N}^{N}e^{in\phi} \ \ \ \ \ (44)$

The function ${D_N(\phi)}$ is called the ${N}$ th Dirichlet kernel. We introduce the following properties of the Dirichlet kernal:

${D_N(\phi)=\frac{1}{2\pi}+\frac{1}{\pi}\sum^{N}_{n=1}\mathrm{cos}(n\theta)}$
${D_N(\phi)=\frac{1}{2\pi}\frac{\mathrm{sin}(N+\frac{1}{2})\phi}{\mathrm{sin}\frac{1}{2}(\phi)}}$

Proof For (1), we use the property ${e^{ix}+e^{-ix}=\mathrm{cos}(x)}$ to obtain

$\displaystyle \begin{aligned} D_N(\phi)&=\frac{1}{2\pi}\sum^{N}_{-N}e^{in\theta}\\ &= \frac{1}{2\pi}(\sum^{N}_{1}[e^{in\theta}+e^{-in\theta}]+e^{i(0)\theta})\\ &= \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi} \end{aligned} \ \ \ \ \ (45)$

For (2), we recognize that ${D_N(\phi)}$ is the sum of a finite geometric series:

$\displaystyle D_N(\phi)= \frac{1}{2\pi}e^{-iN\theta}(1+e^{i\theta}+...+e^{i(2N+1)\theta}) \ \ \ \ \ (46)$

Since ${\sum^n_1r^{k}=\frac{r^{n+1}-1}{r-1}}$ , we have

$\displaystyle D_N(\phi)= \frac{1}{2\pi}e^{-iN\phi}\frac{e^{i(2N+1)\phi}-1}{e^{i\phi}-1}=\frac{1}{2\pi}\frac{e^{i(N+1)\phi}-e^{-iN\phi}}{e^{i\phi}-1} \ \ \ \ \ (47)$

Multiplying top and bottom by ${e^{-\frac{i\theta}{2}}}$ ,

$\displaystyle D_N(\phi)=\frac{1}{2\pi}\frac{e^{i(N+\frac{1}{2})\phi}-e^{-i(N-\frac{1}{2})\phi}}{e^{\frac{i\phi}{2}}-e^{-\frac{i\theta}{2}}} \ \ \ \ \ (48)$

using the property ${e^{ix}-e^{-ix}=\mathrm{sin}(x)}$ , we get

$\displaystyle D_N(\phi)=\frac{1}{2\pi}\frac{\mathrm{sin}(N+\frac{1}{2})\phi}{\mathrm{sin}\frac{1}{2}(\phi)} \ \ \ \ \ (49)$

${\square}$

Figure 4. Graph of the Dirichlet kernel (solid) with N=25 on the interval from -π to π .

Lemma 2.5 For any ${N}$ ,

$\displaystyle \int^{0}_{-\pi}D_N(\theta)d\theta = \int^{\pi}_{0}D_N(\theta)d\theta=\frac{1}{2} \ \ \ \ \ (50)$

Proof We integrate both sides of property (45) from ${0}$ to ${\pi}$ ,

$\displaystyle \begin{aligned} \int^{\pi}_{0}D_N(\theta)d\theta&= \int^{\pi}_{0} \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi}d\theta\\ &= \bigg[\frac{1}{\pi}\sum^{N}_{1}\frac{1}{n}\mathrm{sin}(n\theta)+\frac{\theta}{2\pi}\bigg]^\pi_0\\ &= \frac{1}{2} \end{aligned} \ \ \ \ \ (51)$

${\square}$

We are now ready to present the main convergence theorem. It says that the Fourier series of a function ${f}$ that is piecewise smooth on every bounded interval ${[a,b]}$ converges pointwise to ${f}$ , provided that we redefine ${f}$ at its points of discontinuities to be the average of its left and right hand limits.

Theorem 2.6 If ${f}$ is ${2\pi}$ periodic and piecewise smooth on ${\mathbb{R}}$ , and ${S^{f}_N}$ is defined by (40), then

$\displaystyle \lim_{n\rightarrow\infty} S^{f}_N(\theta)=\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (52)$

for every ${theta}$ . In particular, ${\lim_{N \rightarrow \infty} S^{f}_N(\theta) = f(\theta)}$ for every ${\theta}$ at which ${f}$ is continuous.

Proof By formula (50), we have

$\displaystyle \frac{1}{2}(f(\theta-))= \int^{0}_{-\pi}f(\theta-)D_N(\phi)d\phi \ \ \ \ \ (53)$

$\displaystyle \frac{1}{2}(f(\theta+))= \int^{\pi}_{0}f(\theta+)D_N(\phi)d\phi \ \ \ \ \ (54)$

and hence by equation (44)

$\displaystyle S^{f}_N(\theta)-\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (55)$

$\displaystyle \begin{aligned} &=\int^{\pi}_{0}f(\phi+\theta)D_N(\phi)d\phi-\int^{\pi}_{0}f(\theta+)D_N(\phi)d\phi\\ &+\int^{0}_{-\pi}f(\phi+\theta)D_N(\phi)d\phi-\int^{0}_{-\pi}f(\theta-)D_N(\phi)d\phi\\ &=\int^{\pi}_{0}(f(\phi+\theta)-f(\theta+))D_N(\phi)d\phi+\int^{0}_{-\pi}(f(\phi+\theta)-f(\theta-))D_N(\phi)d\phi \end{aligned} \ \ \ \ \ (56)$

We wish to show that for each fixed ${\theta}$ , equation (56) tends to 0 as ${N\rightarrow\infty}$ . By (47) we can write it as

$\displaystyle\frac{1}{2\pi} \int^{\pi}_{-\pi}g(\phi)(e^{i(N+1)\phi}-e^{-iN\phi})d\phi \ \ \ \ \ (57)$

where

$\displaystyle g(\phi)= \frac{f(\phi+\theta)-f(\theta+)}{e^{i\phi}-1}, -\pi<\phi<0 \ \ \ \ \ (58)$

$\displaystyle g(\phi)= \frac{f(\phi+\theta)-f(\theta-)}{e^{i\phi}-1}, 0<\phi<\pi \ \ \ \ \ (59)$

We see that ${g(\phi)}$ is a well-behaved function except when near ${\phi=0}$ . However, since ${\lim_{\phi\rightarrow0} f(\phi+\theta)-f(\theta-)=0}$ and ${\lim_{\phi\rightarrow0} e^{i\phi}-1=0}$ , by L’Hôpital’s rule,

$\displaystyle \lim_{\phi\rightarrow0} g(\phi) = \lim_{\phi\rightarrow0} \frac{f(\phi+\theta)-f(\theta-)}{e^{i\phi}-1} = \lim_{\phi\rightarrow0}\frac{f'(\phi+\theta)}{ie^{i\phi}}=\frac{f'(\theta)}{i} \ \ \ \ \ (60)$

Likewise, ${g(\phi)}$ approaches ${\frac{f'(\theta)}{i}}$ as ${\phi}$ approaches ${0}$ from the right. Thus, ${g(\phi)}$ is piecewise continuous on ${[-\pi,\pi]}$ . By Bessel’s inequality, the Fourier coefficient

$\displaystyle c_n=\frac{1}{2\pi}\int^{\pi}_{-\pi}g(\phi)e^{-in\theta}\mathrm{d}\phi \ \ \ \ \ (61)$

tends to 0 as ${n\rightarrow\pm\infty}$ . However, we know that formula (57) is just ${C_{-(N+1)}-C_N}$ . Thus, equation (55) tends to ${0}$ as ${n\rightarrow\pm\infty}$ . This is what we needed to show.

${\square}$

Theorem 2.6 says that the Fourier series of a ${2\pi}$ -periodic piecewise smooth function ${f}$ converges to ${f}$ everywhere, provided that ${f}$ is (re)defined at each of its points of discontinuity to be the average of its left- and right-hand limits there. With this understanding, we have the following uniqueness theorem.

Corollary 2.7 If ${f}$ and ${g}$ are 2 ${\pi}$ -periodic and piecewise smooth, and ${f}$ and ${g}$ have the same Fourier coefficients, then ${f=g}$ .

Proof ${f}$ and ${g}$ are both the sums of the same Fourier series.

${\square}$

Notes on Fourier analysis

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