§1.8 Fourier Series

Keywords:: Fourier series
Permalink:: http://dlmf.nist.gov/1.8

§1.8(i) Definitions and Elementary Properties
§1.8(ii) Convergence
§1.8(iii) Integration and Differentiation
§1.8(iv) Transformations
§1.8(v) Examples

§1.8(i) Definitions and Elementary Properties

Notes:: See Protter and Morrey (1991, Chapter 10), Tolstov (1962, Chapter 1), or Titchmarsh (1962, Chapter 13). For the Riemann–Lebesgue lemma, see Olver (1997b, p. 73).
Keywords:: Fourier series
Referenced by:: §2.10(iv), §2.4(ii), §2.5(ii), ¶ ‣ §3.11(ii), ¶ ‣ §3.11(ii)
Permalink:: http://dlmf.nist.gov/1.8.i

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:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
Referenced by:: §1.8(i), §1.8(ii)
Permalink:: http://dlmf.nist.gov/1.8.E1
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1.8.2

$a_{n}=\frac{1}{\pi}\int^{\pi}_{{-\pi}}f(x)\mathop{\cos\/}\nolimits\!\left(nx\right)dx,$ $n=0,1,2,\dots$ ,

$b_{n}=\frac{1}{\pi}\int^{\pi}_{{-\pi}}f(x)\mathop{\sin\/}\nolimits\!\left(nx\right)dx,$ $n=1,2,\dots$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E2
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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:: $e$ : base of exponential function and $n$ : nonnegative integer
Referenced by:: §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.8.E3
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1.8.4 $c_{n}=\frac{1}{2\pi}\int^{\pi}_{{-\pi}}f(x)e^{{-inx}}dx.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $n$ : nonnegative integer
Referenced by:: §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.8.E4
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¶ Bessel’s Inequality

Keywords:: Bessel’s inequality, Fourier series

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E5
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1.8.6 $\sum^{\infty}_{{n=-\infty}}|c_{n}|^{2}\leq\frac{1}{2\pi}\int^{\pi}_{{-\pi}}|f(x)|^{2}dx.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer
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¶ 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:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $m$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §3.11(ii)
Permalink:: http://dlmf.nist.gov/1.8.E7
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¶ Uniqueness of Fourier Series

Keywords:: 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

Keywords:: Lebesgue constants

1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\mathop{\sin\/}\nolimits\!\left(n+\frac{1}{2}\right)t\right|}{\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}t\right)}dt,$ $n=0,1,\dots$ .

Defines:: $L_{n}$ : Lebesgue constant
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
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As $n\to\infty$

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

Symbols:: $L_{n}$ : Lebesgue constant, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $n$ : nonnegative integer
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see Frenzen and Wong (1986).

¶ Riemann–Lebesgue Lemma

Keywords:: Riemann–Lebesgue lemma

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

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

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\lambda$ : real
Referenced by:: ¶ ‣ §1.8(i)
Permalink:: http://dlmf.nist.gov/1.8.E10
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(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

Notes:: See Tolstov (1962, p. 77).
Keywords:: Fourier series, convergence, derivatives
Referenced by:: §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.8.ii

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
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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

Notes:: See Titchmarsh (1962, p. 419).
Keywords:: Fourier series, differentiation, integration
Permalink:: http://dlmf.nist.gov/1.8.iii

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E12
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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

Notes:: For Parseval’s formula see Titchmarsh (1962, p. 421) and for Poisson’s summation formula see Rademacher (1973, pp. 71–75) and Titchmarsh (1986a, p. 61). For (1.8.16) set $f(x)=e^{{-\omega x^{2}}}$ in (1.8.14).
Referenced by:: §20.11(i), §3.5(i)
Permalink:: http://dlmf.nist.gov/1.8.iv

¶ Parseval’s Formula

Keywords:: Fourier series, Parseval’s formula

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E13
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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

Keywords:: Fourier series, 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 $\sum^{\infty}_{{n=-\infty}}f(x+n)=\sum^{\infty}_{{n=-\infty}}e^{{2\pi inx}}\int^{\infty}_{{-\infty}}f(t)e^{{-2\pi int}}dt.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $n$ : nonnegative integer
Referenced by:: §1.8(iv)
Permalink:: http://dlmf.nist.gov/1.8.E14
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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 $\tfrac{1}{2}f(0)+\sum^{\infty}_{{n=1}}f(n)=\int^{\infty}_{0}f(x)dx+2\sum^{\infty}_{{n=1}}\int^{\infty}_{0}f(x)\mathop{\cos\/}\nolimits\!\left(2\pi nx\right)dx.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E15
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As a special case

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\realpart{}$ : real part and $n$ : nonnegative integer
Referenced by:: §1.8(iv)
Permalink:: http://dlmf.nist.gov/1.8.E16
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§1.8(v) Examples

Keywords:: Fourier series
Permalink:: http://dlmf.nist.gov/1.8.v

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).