§1.8 Fourier Series

§1.8(i) Definitions and Elementary Properties

Formally,

 1.8.1 $f(x)=\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}\mathop{\cos\/}\nolimits\!% \left(nx\right)+b_{n}\mathop{\sin\/}\nolimits\!\left(nx\right)),$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $n$: nonnegative integer Referenced by: §1.8(i), §1.8(ii) Permalink: http://dlmf.nist.gov/1.8.E1 Encodings: TeX, pMML, png See also: Annotations for 1.8(i)
 1.8.2 $\displaystyle a_{n}$ $\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\mathop{\cos\/}\nolimits\!% \left(nx\right)\mathrm{d}x,$ $n=0,1,2,\dots$, $\displaystyle b_{n}$ $\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\mathop{\sin\/}\nolimits\!% \left(nx\right)\mathrm{d}x,$ $n=1,2,\dots$.

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 $n$.

Alternative Form

 1.8.3 $f(x)=\sum^{\infty}_{n=-\infty}c_{n}e^{inx},$ Symbols: $\mathrm{e}$: base of exponential function and $n$: nonnegative integer Referenced by: §1.17(iii) Permalink: http://dlmf.nist.gov/1.8.E3 Encodings: TeX, pMML, png See also: Annotations for 1.8(i)
 1.8.4 $c_{n}=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)e^{-inx}\mathrm{d}x.$

Bessel’s Inequality

 1.8.5 $\tfrac{1}{2}a_{0}^{2}+\sum^{\infty}_{n=1}(a_{n}^{2}+b_{n}^{2})\leq\frac{1}{\pi% }\int^{\pi}_{-\pi}(f(x))^{2}\mathrm{d}x.$
 1.8.6 $\sum^{\infty}_{n=-\infty}|c_{n}|^{2}\leq\frac{1}{2\pi}\int^{\pi}_{-\pi}|f(x)|^% {2}\mathrm{d}x.$

Asymptotic Estimates of Coefficients

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

 1.8.7 $a_{n},b_{n},c_{n}=\mathop{o\/}\nolimits\!\left(n^{-m}\right),$ $n\to\infty$. Symbols: $\mathop{o\/}\nolimits\!\left(\NVar{x}\right)$: order less than, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §3.11(ii) Permalink: http://dlmf.nist.gov/1.8.E7 Encodings: TeX, pMML, png See also: Annotations for 1.8(i)

Uniqueness of Fourier Series

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

Lebesgue Constants

 1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\mathop{\sin\/}\nolimits\!\left(n% +\frac{1}{2}\right)t\right|}{\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}t% \right)}\mathrm{d}t,$ $n=0,1,\dots$. Defines: $L_{n}$: Lebesgue constants (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.8.E8 Encodings: TeX, pMML, png See also: Annotations for 1.8(i)

As $n\to\infty$

 1.8.9 $L_{n}\sim(4/\pi^{2})\mathop{\ln\/}\nolimits n;$

see Frenzen and Wong (1986).

Riemann–Lebesgue Lemma

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

 1.8.10 $\int^{b}_{a}f(x)e^{i\lambda x}\mathrm{d}x\to 0,$ as $\lambda\to\infty$. Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $\lambda$: real Referenced by: §1.8(i) Permalink: http://dlmf.nist.gov/1.8.E10 Encodings: TeX, pMML, png See also: Annotations for 1.8(i)

(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 $\lambda$.

§1.8(ii) Convergence

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 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)$ Permalink: http://dlmf.nist.gov/1.8.E11 Encodings: TeX, pMML, png See also: Annotations for 1.8(ii)

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 $f(x-)\neq f(x+)$; see §6.16(i).

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

§1.8(iii) Integration and Differentiation

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 $\int^{x}_{0}(f(t)-\tfrac{1}{2}a_{0})\mathrm{d}t=\sum^{\infty}_{n=1}\frac{a_{n}% \mathop{\sin\/}\nolimits\!\left(nx\right)+b_{n}(1-\mathop{\cos\/}\nolimits\!% \left(nx\right))}{n},$ $0\leq x\leq 2\pi$.

If a function $f(x)\in\mathop{C^{2}\/}\nolimits[0,2\pi]$ 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(iv) Transformations

Parseval’s Formula

 1.8.13 $\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)g(x)\mathrm{d}x=\tfrac{1}{2}a_{0}a^{\prime}_% {0}+\sum^{\infty}_{n=1}(a_{n}a^{\prime}_{n}+b_{n}b^{\prime}_{n}),$

when $f(x)$ and $g(x)$ are square-integrable and $a_{n},b_{n}$ and $a^{\prime}_{n},b^{\prime}_{n}$ are their respective Fourier coefficients.

Poisson’s Summation Formula

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

 1.8.14 $\sum^{\infty}_{n=-\infty}f(x+n)=\sum^{\infty}_{n=-\infty}e^{2\pi inx}\int^{% \infty}_{-\infty}f(t)e^{-2\pi int}\mathrm{d}t.$

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

 1.8.15 $\tfrac{1}{2}f(0)+\sum^{\infty}_{n=1}f(n)=\int^{\infty}_{0}f(x)\mathrm{d}x+2% \sum^{\infty}_{n=1}\int^{\infty}_{0}f(x)\mathop{\cos\/}\nolimits\!\left(2\pi nx% \right)\mathrm{d}x.$

As a special case

 1.8.16 $\sum_{n=-\infty}^{\infty}e^{-(n+x)^{2}\omega}={\sqrt{\frac{\pi}{\omega}}\*% \left(1+2\sum_{n=1}^{\infty}e^{-n^{2}\pi^{2}/\omega}\mathop{\cos\/}\nolimits\!% \left(2n\pi x\right)\right)},$ $\Re{\omega}>0$.

§1.8(v) Examples

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973).