1 Algebraic and Analytic MethodsTopics of Discussion1.7 Inequalities1.9 Calculus of a Complex Variable

- §1.8(i) Definitions and Elementary Properties
- §1.8(ii) Convergence
- §1.8(iii) Integration and Differentiation
- §1.8(iv) Poisson’s Summation Formula
- §1.8(v) Examples

Formally, if $f(x)$ is a real- or complex-valued $2\pi $-periodic function,

1.8.1 | $$f(x)=\frac{1}{2}{a}_{0}+\sum _{n=1}^{\mathrm{\infty}}({a}_{n}\mathrm{cos}\left(nx\right)+{b}_{n}\mathrm{sin}\left(nx\right)),$$ | ||

1.8.2 | ${a}_{n}$ | $={\displaystyle \frac{1}{\pi}}{\displaystyle {\int}_{-\pi}^{\pi}}f(x)\mathrm{cos}\left(nx\right)dx,$ | ||

$n=0,1,2,\mathrm{\dots}$, | ||||

${b}_{n}$ | $={\displaystyle \frac{1}{\pi}}{\displaystyle {\int}_{-\pi}^{\pi}}f(x)\mathrm{sin}\left(nx\right)dx,$ | |||

$n=1,2,\mathrm{\dots}$. | ||||

The series (1.8.1) is called the *Fourier series* of $f(x)$,
and ${a}_{n},{b}_{n}$ are the *Fourier coefficients* of $f(x)$.

If $f(-x)=f(x)$, then ${b}_{n}=0$ for all $n$.

If $f(-x)=-f(x)$, then ${a}_{n}=0$ for all $n$.

1.8.3 | $$f(x)=\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{c}_{n}{\mathrm{e}}^{\mathrm{i}nx},$$ | ||

1.8.4 | $${c}_{n}=\frac{1}{2\pi}{\int}_{-\pi}^{\pi}f(x){\mathrm{e}}^{-\mathrm{i}nx}dx.$$ | ||

Here ${c}_{n}$ is related to ${a}_{n}$ and ${b}_{n}$ in (1.8.1), (1.8.2) by ${c}_{n}=\frac{1}{2}({a}_{n}-\mathrm{i}{b}_{n})$, ${c}_{-n}=\frac{1}{2}({a}_{n}+\mathrm{i}{b}_{n})$ for $n>0$ and ${c}_{0}=\frac{1}{2}{a}_{0}$.

1.8.5 | $$\frac{1}{\pi}{\int}_{-\pi}^{\pi}{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|{a}_{0}\right|}^{2}+\sum _{n=1}^{\mathrm{\infty}}({\left|{a}_{n}\right|}^{2}+{\left|{b}_{n}\right|}^{2}),$$ | ||

1.8.6 | $$\frac{1}{2\pi}{\int}_{-\pi}^{\pi}{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{\left|{c}_{n}\right|}^{2},$$ | ||

where $f(x)$ is square-integrable on $[-\pi ,\pi ]$ and ${a}_{n},{b}_{n},{c}_{n}$ are given by (1.8.2), (1.8.4). If $g(x)$ is also square-integrable with Fourier coefficients ${a}_{n}^{\prime},{b}_{n}^{\prime}$ or ${c}_{n}^{\prime}$ then

1.8.6_1 | $$\frac{1}{\pi}{\int}_{-\pi}^{\pi}f(x)\overline{g(x)}dx=\frac{1}{2}{a}_{0}\overline{{a}_{0}^{\prime}}+\sum _{n=1}^{\mathrm{\infty}}({a}_{n}\overline{{a}_{n}^{\prime}}+{b}_{n}\overline{{b}_{n}^{\prime}}),$$ | ||

1.8.6_2 | $$\frac{1}{2\pi}{\int}_{-\pi}^{\pi}f(x)\overline{g(x)}dx=\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{c}_{n}\overline{{c}_{n}^{\prime}}.$$ | ||

If $f(x)$ is of period $2\pi $, and ${f}^{(m)}(x)$ is piecewise continuous, then

1.8.7 | $${a}_{n},{b}_{n},{c}_{n}=o\left({n}^{-m}\right),$$ | ||

$n\to \mathrm{\infty}$. | |||

If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$.

1.8.8 | $${L}_{n}=\frac{1}{\pi}{\int}_{0}^{\pi}\frac{\left|\mathrm{sin}\left(n+\frac{1}{2}\right)t\right|}{\mathrm{sin}\left(\frac{1}{2}t\right)}dt,$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

As $n\to \mathrm{\infty}$

1.8.9 | $${L}_{n}\sim (4/{\pi}^{2})\mathrm{ln}n;$$ | ||

see Frenzen and Wong (1986).

For $f(x)$ piecewise continuous on $[a,b]$ and real $\lambda $,

1.8.10 | $${\int}_{a}^{b}f(x){\mathrm{e}}^{\mathrm{i}\lambda x}dx\to 0,$$ | ||

as $\lambda \to \mathrm{\infty}$. | |||

(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda $.

Let $f(x)$ be an absolutely integrable function of period $2\pi $, and continuous except at a finite number of points in any bounded interval. Then the series (1.8.1) converges to the sum

1.8.11 | $$\frac{1}{2}f(x-)+\frac{1}{2}f(x+)$$ | ||

at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). The convergence is non-uniform, however, at points where $f(x-)\ne f(x+)$; see §6.16(i).

For other tests for convergence see Titchmarsh (1962b, pp. 405–410).

If ${a}_{n}$ and ${b}_{n}$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then

1.8.12 | $${\int}_{0}^{x}(f(t)-\frac{1}{2}{a}_{0})dt=\sum _{n=1}^{\mathrm{\infty}}\frac{{a}_{n}\mathrm{sin}\left(nx\right)+{b}_{n}(1-\mathrm{cos}\left(nx\right))}{n},$$ | ||

$0\le x\le 2\pi $. | |||

If a function $f(x)\in {C}^{2}[0,2\pi ]$ is periodic, with period $2\pi $, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to ${f}^{\prime}(x)$.

1.8.13 | Moved to (1.8.6_1). | ||

Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|{f}^{\prime \prime}(x)\right|$ are integrable over $(-\mathrm{\infty},\mathrm{\infty})$. Then

1.8.14 | $$\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}f(x+n)=\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{2\pi \mathrm{i}nx}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(t){\mathrm{e}}^{-2\pi \mathrm{i}nt}dt.$$ | ||

It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi}\mathcal{F}\left(f\right)\left(-2\pi n\right)$.

An alternative formulation is as follows. Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty})$. Suppose also that $f(x)$ is integrable on $[0,\mathrm{\infty})$ and $f(x)\to 0$ as $x\to \mathrm{\infty}$. Then

1.8.15 | $$\frac{1}{2}f(0)+\sum _{n=1}^{\mathrm{\infty}}f(n)={\int}_{0}^{\mathrm{\infty}}f(x)dx+2\sum _{n=1}^{\mathrm{\infty}}{\int}_{0}^{\mathrm{\infty}}f(x)\mathrm{cos}\left(2\pi nx\right)dx.$$ | ||

As a special case

1.8.16 | $$\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{-{(n+x)}^{2}\omega}=\sqrt{\frac{\pi}{\omega}}\left(1+2\sum _{n=1}^{\mathrm{\infty}}{\mathrm{e}}^{-{n}^{2}{\pi}^{2}/\omega}\mathrm{cos}\left(2n\pi x\right)\right),$$ | ||

$\mathrm{\Re}\omega >0$. | |||