1 Algebraic and Analytic Methods Topics of Discussion 1.7 Inequalities 1.9 Calculus of a Complex Variable

§1.8 Fourier Series

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Keywords:: Fourier series
Referenced by:: Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8
See also:: Annotations for Ch.1

§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

§1.8(i) Definitions and Elementary Properties

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Keywords:: Fourier series, coefficients, definition, properties
Notes:: See Protter and Morrey (1991, Chapter 10), Tolstov (1962, Chapter 1), or Titchmarsh (1962b, Chapter 13). For the Riemann–Lebesgue lemma, see Olver (1997b, p. 73).
Referenced by:: §2.10(iv), §2.4(ii), §2.5(ii), §3.11(ii), §3.11(ii), Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8.i
Modification (effective with 1.2.0):: The first sentence of this subsection was changed from “Formally,” to “Formally, if $f(x)$ is a real- or complex-valued $2\pi$ -periodic function”.
See also:: Annotations for §1.8 and Ch.1

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

1.8.1		$f(x)=\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}\cos\left(nx\right)+b_{n}\sin% \left(nx\right)),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Referenced by: §1.18(v), §1.8(i), §1.8(i), §1.8(ii), §18.2(viii) Permalink: http://dlmf.nist.gov/1.8.E1 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8 and Ch.1

1.8.2		$\displaystyle a_{n}$	$\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\left(nx\right)\,\mathrm{d% }x,$
	$n=0,1,2,\dots$ ,
		$\displaystyle b_{n}$	$\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\left(nx\right)\,\mathrm{d% }x,$
	$n=1,2,\dots$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Referenced by: §1.18(v), §1.8(i), §1.8(i), §18.2(viii) Permalink: http://dlmf.nist.gov/1.8.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.8(i), §1.8 and Ch.1

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

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Modification (effective with 1.2.0):: A sentence was added at the end of this paragraph.
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.3		$f(x)=\sum^{\infty}_{n=-\infty}c_{n}{\mathrm{e}}^{\mathrm{i}nx},$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $n$ : nonnegative integer Referenced by: §1.17(iii), §1.18(v) Permalink: http://dlmf.nist.gov/1.8.E3 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

1.8.4		$c_{n}=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x){\mathrm{e}}^{-\mathrm{i}nx}\,\mathrm% {d}x.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $n$ : nonnegative integer Referenced by: §1.17(iii), §1.18(v), §1.8(i) Permalink: http://dlmf.nist.gov/1.8.E4 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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 $c_{0}=\frac{1}{2}a_{0}$ .

Parseval’s Formula

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Keywords:: Fourier series, Parseval’s Formula
Modification (effective with 1.2.0):: Previously this paragraph was named Bessel’s Inequality. The equation which was originally (1.8.13) was moved to (1.8.6_1).
Addition (effective with 1.2.0):: Some text was added just after (1.8.6). The equation (1.8.6_2) was added.
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.5		$\frac{1}{\pi}\int^{\pi}_{-\pi}{\left\|f(x)\right\|}^{2}\,\mathrm{d}x=\tfrac{1}{2% }{\left\|a_{0}\right\|}^{2}+\sum^{\infty}_{n=1}({\left\|a_{n}\right\|}^{2}+{\left\|% b_{n}\right\|}^{2}),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Referenced by: Erratum (V1.2.0) for Equations (1.8.5), (1.8.6) Permalink: http://dlmf.nist.gov/1.8.E5 Encodings: TeX, pMML, png Modification (effective with 1.2.0): Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$ . See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

1.8.6		$\frac{1}{2\pi}\int^{\pi}_{-\pi}{\left\|f(x)\right\|}^{2}\,\mathrm{d}x=\sum^{% \infty}_{n=-\infty}{\left\|c_{n}\right\|}^{2},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Referenced by: §1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6) Permalink: http://dlmf.nist.gov/1.8.E6 Encodings: TeX, pMML, png Modification (effective with 1.2.0): Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$ . See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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		$\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\tfrac{1}{2}a_{% 0}\overline{a_{0}^{\prime}}+\sum^{\infty}_{n=1}(a_{n}\overline{a^{\prime}_{n}}% +b_{n}\overline{b^{\prime}_{n}}),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer Referenced by: (1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) §1.8 Permalink: http://dlmf.nist.gov/1.8.E6_1 Encodings: TeX, pMML, png Rearrangement (effective with 1.2.0): This equation, which was originally (1.8.13), was moved here. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

1.8.6_2		$\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\sum^{\infty}_% {n=-\infty}c_{n}\overline{c_{n}^{\prime}}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer Referenced by: §1.8(i), Erratum (V1.2.0) §1.8 Permalink: http://dlmf.nist.gov/1.8.E6_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

Asymptotic Estimates of Coefficients

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See also:: Annotations for §1.8(i), §1.8 and Ch.1

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

1.8.7		$a_{n},b_{n},c_{n}=o\left(n^{-m}\right),$
$n\to\infty$ .
ⓘ Symbols: $o\left(\NVar{x}\right)$ : 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), §1.8(i), §1.8 and Ch.1

Uniqueness of Fourier Series

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Keywords:: Fourier series, uniqueness
See also:: Annotations for §1.8(i), §1.8 and Ch.1

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

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Keywords:: Lebesgue constants, asymptotic behavior
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.8		$L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left\|\sin\left(n+\frac{1}{2}\right)t% \right\|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,$
$n=0,1,\dots$ .
ⓘ 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, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.8.E8 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

As $n\to\infty$

1.8.9		$L_{n}\sim(4/{\pi}^{2})\ln n;$
ⓘ Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $L_{n}$ : Lebesgue constants Permalink: http://dlmf.nist.gov/1.8.E9 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

see Frenzen and Wong (1986).

Riemann–Lebesgue Lemma

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Keywords:: Riemann–Lebesgue lemma
See also:: Annotations for §1.8(i), §1.8 and Ch.1

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

1.8.10		$\int^{b}_{a}f(x){\mathrm{e}}^{\mathrm{i}\lambda x}\,\mathrm{d}x\to 0,$
as $\lambda\to\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : 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(i), §1.8 and Ch.1

(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

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Keywords:: Fourier series, convergence, derivatives, left-hand, right-hand
Notes:: See Tolstov (1962, p. 77).
Referenced by:: §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.8.ii
See also:: Annotations for §1.8 and Ch.1

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), §1.8 and Ch.1

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 (1962b, pp. 405–410).

§1.8(iii) Integration and Differentiation

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Keywords:: Fourier series, differentiation, integration
Notes:: See Titchmarsh (1962b, p. 419).
Permalink:: http://dlmf.nist.gov/1.8.iii
See also:: Annotations for §1.8 and Ch.1

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		$\int^{x}_{0}(f(t)-\tfrac{1}{2}a_{0})\,\mathrm{d}t=\sum^{\infty}_{n=1}\frac{a_{% n}\sin\left(nx\right)+b_{n}(1-\cos\left(nx\right))}{n},$
$0\leq x\leq 2\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.8.E12 Encodings: TeX, pMML, png See also: Annotations for §1.8(iii), §1.8 and Ch.1

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(iv) Poisson’s Summation Formula

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Keywords:: Fourier series, Poisson’s summation formula
Notes:: For Poisson’s summation formula see Rademacher (1973, pp. 71–75) and Titchmarsh (1986a, p. 61). For (1.8.16) set $f(x)={\mathrm{e}}^{-\omega x^{2}}$ in (1.8.14).
Referenced by:: §1.14(i), §20.11(i), §3.5(i), Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8.iv
Modification (effective with 1.2.0):: This subsection was previously called “Transformations”.
Rearrangement (effective with 1.2.0):: The equation which was originally (1.8.13) has been moved to (1.8.6_1). A sentence was added just after (1.8.14).
See also:: Annotations for §1.8 and Ch.1

1.8.13		Moved to (1.8.6_1).
ⓘ Referenced by: (1.8.6_1), §1.8(i), §1.8(iv) Permalink: http://dlmf.nist.gov/1.8.E13 Rearrangement (effective with 1.2.0): This equation has been moved to (1.8.6_1). See also: Annotations for §1.8(iv), §1.8 and Ch.1

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

1.8.14		$\sum^{\infty}_{n=-\infty}f(x+n)=\sum^{\infty}_{n=-\infty}{\mathrm{e}}^{2\pi% \mathrm{i}nx}\int^{\infty}_{-\infty}f(t){\mathrm{e}}^{-2\pi\mathrm{i}nt}\,% \mathrm{d}t.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $n$ : nonnegative integer Referenced by: §1.14(i), §1.8(iv), §1.8(iv), Erratum (V1.2.0) §1.8 Permalink: http://dlmf.nist.gov/1.8.E14 Encodings: TeX, pMML, png See also: Annotations for §1.8(iv), §1.8 and Ch.1

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

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

1.8.15		$\tfrac{1}{2}f(0)+\sum^{\infty}_{n=1}f(n)=\int^{\infty}_{0}f(x)\,\mathrm{d}x+2% \sum^{\infty}_{n=1}\int^{\infty}_{0}f(x)\cos\left(2\pi nx\right)\,\mathrm{d}x.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.8.E15 Encodings: TeX, pMML, png See also: Annotations for §1.8(iv), §1.8 and Ch.1

As a special case

1.8.16		$\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{-(n+x)^{2}\omega}={\sqrt{\frac{\pi}{% \omega}}\*\left(1+2\sum_{n=1}^{\infty}{\mathrm{e}}^{-n^{2}{\pi}^{2}/\omega}% \cos\left(2n\pi x\right)\right)},$
$\Re\omega>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Re$ : real part and $n$ : nonnegative integer Referenced by: §1.8(iv) Permalink: http://dlmf.nist.gov/1.8.E16 Encodings: TeX, pMML, png See also: Annotations for §1.8(iv), §1.8 and Ch.1

§1.8(v) Examples

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Keywords:: Fourier series, compendia
Permalink:: http://dlmf.nist.gov/1.8.v
See also:: Annotations for §1.8 and Ch.1

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2015, §§1.44–1.45), and Oberhettinger (1973).

1.7 Inequalities 1.9 Calculus of a Complex Variable

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