1 Algebraic and Analytic Methods Topics of Discussion 1.14 Integral Transforms 1.16 Distributions

§1.15 Summability Methods

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Keywords:: infinite series, summability methods
Permalink:: http://dlmf.nist.gov/1.15
See also:: Annotations for Ch.1

§1.15(i) Definitions for Series

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Referenced by:: §1.15(ii)
Permalink:: http://dlmf.nist.gov/1.15.i
See also:: Annotations for §1.15 and Ch.1

1.15.1		$s_{n}=\sum_{k=0}^{n}a_{k}.$
ⓘ Symbols: $k$ : integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E1 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15 and Ch.1

Abel Summability

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Keywords:: Abel, Abel summability, summability methods for series
See also:: Annotations for §1.15(i), §1.15 and Ch.1

1.15.2		$\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)},$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E2 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

1.15.3		$\lim_{x\to 1-}\sum^{\infty}_{n=0}a_{n}x^{n}=s.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E3 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

Cesàro Summability

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Keywords:: Cesàro, Cesàro summability, summability methods for series
See also:: Annotations for §1.15(i), §1.15 and Ch.1

1.15.4		$\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,1)},$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E4 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

1.15.5		$\lim_{n\to\infty}\frac{s_{0}+s_{1}+\dots+s_{n}}{n+1}=s.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E5 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

General Cesàro Summability

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Keywords:: Cesàro, general, summability methods for series
See also:: Annotations for §1.15(i), §1.15 and Ch.1

For $\alpha>-1$ ,

1.15.6		$\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,$\alpha$)},$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E6 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

1.15.7		$\lim_{n\to\infty}\frac{n!}{(\alpha+1)_{n}}\sum^{n}_{k=0}\frac{(\alpha+1)_{k}}{% k!}a_{n-k}=s.$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E7 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

Borel Summability

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Keywords:: Borel, Borel summability, summability methods for series
See also:: Annotations for §1.15(i), §1.15 and Ch.1

1.15.8		$\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(B)},$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E8 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

1.15.9		$\lim_{t\to\infty}{\mathrm{e}}^{-t}\sum^{\infty}_{n=0}\frac{s_{n}}{n!}t^{n}=s.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E9 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

§1.15(ii) Regularity

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Keywords:: convergence, regular, summability methods for series
Notes:: See Hardy (1949, p. 10).
Permalink:: http://dlmf.nist.gov/1.15.ii
See also:: Annotations for §1.15 and Ch.1

Methods of summation are regular if they are consistent with conventional summation. All of the methods described in §1.15(i) are regular. For example if

1.15.10		$\sum^{\infty}_{n=0}a_{n}=s,$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E10 Encodings: TeX, pMML, png See also: Annotations for §1.15(ii), §1.15 and Ch.1

then

1.15.11		$\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)}.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E11 Encodings: TeX, pMML, png See also: Annotations for §1.15(ii), §1.15 and Ch.1

§1.15(iii) Summability of Fourier Series

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Keywords:: Fourier series, summability
Notes:: See Weiss (1965, pp. 131–135) and Andrews et al. (1999, pp. 602–604). For (1.15.24), see Körner (1989, Chapters 2,27).
Permalink:: http://dlmf.nist.gov/1.15.iii
See also:: Annotations for §1.15 and Ch.1

Poisson Kernel

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Keywords:: Fourier series, Poisson kernel, summability methods for series
See also:: Annotations for §1.15(iii), §1.15 and Ch.1

1.15.12		$P(r,\theta)=\frac{1-r^{2}}{1-2r\cos\theta+r^{2}}=\sum^{\infty}_{n=-\infty}r^{% \left\|n\right\|}{\mathrm{e}}^{\mathrm{i}n\theta},$
$0\leq r<1$ ,
ⓘ Defines: $P(r,\theta)$ : Poisson kernel (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E12 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

1.15.13		$\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta)\,\mathrm{d}\theta=1.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $P(r,\theta)$ : Poisson kernel Permalink: http://dlmf.nist.gov/1.15.E13 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

As $r\to 1-$

1.15.14		$P(r,\theta)\to 0,$
ⓘ Symbols: $P(r,\theta)$ : Poisson kernel Permalink: http://dlmf.nist.gov/1.15.E14 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

uniformly for $\theta\in[\delta,2\pi-\delta]$ . (Here and elsewhere in this subsection $\delta$ is a constant such that $0<\delta<\pi$ .)

Fejér Kernel

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Keywords:: Fejér kernel, Fourier series, summability methods for series
See also:: Annotations for §1.15(iii), §1.15 and Ch.1

For $n=0,1,2,\dots$ ,

1.15.15		$K_{n}(\theta)=\frac{1}{n+1}\left(\frac{\sin\left(\tfrac{1}{2}(n+1)\theta\right% )}{\sin\left(\tfrac{1}{2}\theta\right)}\right)^{2},$
ⓘ Defines: $K_{n}(\theta)$ : Fejér kernel (locally) Symbols: $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E15 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

1.15.16		$\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta)\,\mathrm{d}\theta=1.$
ⓘ 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 $K_{n}(\theta)$ : Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E16 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

As $n\to\infty$

1.15.17		$K_{n}(\theta)\to 0,$
ⓘ Symbols: $n$ : nonnegative integer and $K_{n}(\theta)$ : Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E17 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

uniformly for $\theta\in[\delta,2\pi-\delta]$ .

Abel Means

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Keywords:: Abel means, Fourier series, summability methods for series
See also:: Annotations for §1.15(iii), §1.15 and Ch.1

1.15.18		$A(r,\theta)=\sum^{\infty}_{n=-\infty}r^{\left\|n\right\|}F(n){\mathrm{e}}^{% \mathrm{i}n\theta},$
ⓘ Defines: $A(r,\theta)$ : Abel mean (locally) Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer, $F(n)$ and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E18 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

where

1.15.19		$F(n)=\frac{1}{2\pi}\int^{2\pi}_{0}f(t){\mathrm{e}}^{-\mathrm{i}nt}\,\mathrm{d}t.$
ⓘ Defines: $F(n)$ (locally) 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 Permalink: http://dlmf.nist.gov/1.15.E19 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

$A(r,\theta)$ is a harmonic function in polar coordinates (1.9.27), and

1.15.20		$A(r,\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta-t)f(t)\,\mathrm{d}t.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $P(r,\theta)$ : Poisson kernel and $A(r,\theta)$ : Abel mean Permalink: http://dlmf.nist.gov/1.15.E20 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Cesàro (or (C,1)) Means

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Keywords:: Cesàro means, Fourier series, summability methods for series
See also:: Annotations for §1.15(iii), §1.15 and Ch.1

Let

1.15.21		$\sigma_{n}(\theta)=\frac{s_{0}(\theta)+s_{1}(\theta)+\dots+s_{n}(\theta)}{n+1},$
ⓘ Defines: $\sigma_{n}(\theta)$ : Cesàro mean (locally) Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E21 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

$n=0,1,2,\dots$ , where

1.15.22		$s_{n}(\theta)=\sum^{n}_{k=-n}F(k){\mathrm{e}}^{\mathrm{i}k\theta}.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $k$ : integer, $n$ : nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E22 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Then

1.15.23		$\sigma_{n}(\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta-t)f(t)\,\mathrm{d% }t.$
ⓘ 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, $K_{n}(\theta)$ : Fejér kernel and $\sigma_{n}(\theta)$ : Cesàro mean Permalink: http://dlmf.nist.gov/1.15.E23 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Convergence

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Keywords:: Fourier series, summability
See also:: Annotations for §1.15(iii), §1.15 and Ch.1

If $f(\theta)$ is periodic and integrable on $[0,2\pi]$ , then as $n\to\infty$ the Abel means $A(r,\theta)$ and the (C,1) means $\sigma_{n}(\theta)$ converge to

1.15.24		$\tfrac{1}{2}(f(\theta+)+f(\theta-))$
ⓘ Referenced by: §1.15(iii) Permalink: http://dlmf.nist.gov/1.15.E24 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

at every point $\theta$ where both limits exist. If $f(\theta)$ is also continuous, then the convergence is uniform for all $\theta$ .

For real-valued $f(\theta)$ , if

1.15.25		$\sum^{\infty}_{n=-\infty}F(n){\mathrm{e}}^{\mathrm{i}n\theta}$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E25 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

is the Fourier series of $f(\theta)$ , then the series

1.15.26		$F(0)+2\sum^{\infty}_{n=1}F(n){\mathrm{e}}^{\mathrm{i}n\theta}$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E26 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

can be extended to the interior of the unit circle as an analytic function

1.15.27		$G(z)=G(x+\mathrm{i}y)=u(x,y)+\mathrm{i}v(x,y)=F(0)+2\sum^{\infty}_{n=1}F(n)z^{% n}.$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $z$ : variable, $n$ : nonnegative integer, $F(n)$ , $u(\NVar{x},\NVar{y})$ : function and $v(\NVar{x},\NVar{y})$ : function Permalink: http://dlmf.nist.gov/1.15.E27 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Here $u(x,y)=A(r,\theta)$ is the Abel (or Poisson) sum of $f(\theta)$ , and $v(x,y)$ has the series representation

1.15.28		$-\sum^{\infty}_{n=-\infty}\mathrm{i}(\operatorname{sign}n)F(n)r^{\left\|n\right% \|}{\mathrm{e}}^{\mathrm{i}n\theta};$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{sign}\NVar{x}$ : sign of, $n$ : nonnegative integer, $F(n)$ and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E28 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

compare §1.15(v).

§1.15(iv) Definitions for Integrals

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Keywords:: integrals, summability methods
Notes:: See Weiss (1965, pp. 143–147) and Wong (1989, pp. 197–198).
Permalink:: http://dlmf.nist.gov/1.15.iv
See also:: Annotations for §1.15 and Ch.1

Abel Summability

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Keywords:: Abel, Abel summability, summability methods for integrals
See also:: Annotations for §1.15(iv), §1.15 and Ch.1

$\int^{\infty}_{-\infty}f(t)\,\mathrm{d}t$ is Abel summable to $L$ , or

1.15.29		$\int^{\infty}_{-\infty}f(t)\,\mathrm{d}t=L\;\;\;\textit{(A)},$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Permalink: http://dlmf.nist.gov/1.15.E29 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

when

1.15.30		$\lim_{\epsilon\to 0+}\int^{\infty}_{-\infty}{\mathrm{e}}^{-\epsilon\left\|t% \right\|}f(t)\,\mathrm{d}t=L.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E30 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

Cesàro Summability

ⓘ

Keywords:: Cesàro, Cesàro summability, infinite series, summability methods, summability methods for integrals
See also:: Annotations for §1.15(iv), §1.15 and Ch.1

$\int^{\infty}_{-\infty}f(t)\,\mathrm{d}t$ is (C,1) summable to $L$ , or

1.15.31		$\int^{\infty}_{-\infty}f(t)\,\mathrm{d}t=L\;\;\;\textit{(C,1)},$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Permalink: http://dlmf.nist.gov/1.15.E31 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

when

1.15.32		$\lim_{R\to\infty}\int^{R}_{-R}\left(1-\frac{\left\|t\right\|}{R}\right)f(t)\,% \mathrm{d}t=L.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E32 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

If $\int^{\infty}_{-\infty}f(t)\,\mathrm{d}t$ converges and equals $L$ , then the integral is Abel and Cesàro summable to $L$ .

§1.15(v) Summability of Fourier Integrals

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Keywords:: Fourier integral, summability
Notes:: See Weiss (1965, pp. 143–148).
Referenced by:: §1.15(iii)
Permalink:: http://dlmf.nist.gov/1.15.v
See also:: Annotations for §1.15 and Ch.1

Poisson Kernel

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Keywords:: Fourier integral, Fourier integrals, Fourier series, Poisson integral, Poisson kernel, conjugate, conjugate Poisson integral, summability methods for integrals
See also:: Annotations for §1.15(v), §1.15 and Ch.1

1.15.33		$P(x,y)=\frac{2y}{x^{2}+y^{2}},$
$y>0$ , $-\infty<x<\infty$ .
ⓘ Defines: $P(x,y)$ : Poisson kernel (locally) Permalink: http://dlmf.nist.gov/1.15.E33 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

1.15.34		$\frac{1}{2\pi}\int^{\infty}_{-\infty}P(x,y)\,\mathrm{d}x=1.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $P(x,y)$ : Poisson kernel Permalink: http://dlmf.nist.gov/1.15.E34 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

For each $\delta>0$ ,

1.15.35		$\int_{\left\|x\right\|\geq\delta}P(x,y)\,\mathrm{d}x\to 0,$
as $y\to 0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $P(x,y)$ : Poisson kernel and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E35 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Let

1.15.36		$h(x,y)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}{\mathrm{e}}^{-y\left\|t% \right\|}{\mathrm{e}}^{-\mathrm{i}xt}F(t)\,\mathrm{d}t,$
ⓘ 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, $F(x)$ : Fourier transform of $f(t)$ , $h(x,y)$ : function and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E36 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

where $F(t)$ is the Fourier transform of $f(x)$ (§1.14(i)). Then

1.15.37		$h(x,y)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(t)P(x-t,y)\,\mathrm{d}t$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $P(x,y)$ : Poisson kernel and $h(x,y)$ : function Permalink: http://dlmf.nist.gov/1.15.E37 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

is the Poisson integral of $f(t)$ .

If $f(x)$ is integrable on $(-\infty,\infty)$ , then

1.15.38		$\lim_{y\to 0+}\int^{\infty}_{-\infty}\left\|h(x,y)-f(x)\right\|\,\mathrm{d}x=0.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $h(x,y)$ : function and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E38 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Suppose now $f(x)$ is real-valued and integrable on $(-\infty,\infty)$ . Let

1.15.39		$\Phi(z)=\Phi(x+\mathrm{i}y)=\frac{\mathrm{i}}{\pi}\int^{\infty}_{-\infty}f(t)% \frac{1}{(x-t)+\mathrm{i}y}\,\mathrm{d}t,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : variable and $\Phi(z)$ : function Permalink: http://dlmf.nist.gov/1.15.E39 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

where $y>0$ and $-\infty<x<\infty$ . Then $\Phi(z)$ is an analytic function in the upper half-plane and its real part is the Poisson integral $h(x,y)$ ; compare (1.9.34). The imaginary part

1.15.40		$\Im\Phi(x+\mathrm{i}y)=\frac{1}{\pi}\int^{\infty}_{-\infty}f(t)\frac{x-t}{(x-t% )^{2}+y^{2}}\,\mathrm{d}t$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Phi(z)$ : function Permalink: http://dlmf.nist.gov/1.15.E40 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

is the conjugate Poisson integral of $f(x)$ . Moreover, $\lim_{y\to 0+}\Im\Phi(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)).

Fejér Kernel

ⓘ

Keywords:: Fejér kernel, Fourier integral, Fourier integrals, summability methods for integrals
See also:: Annotations for §1.15(v), §1.15 and Ch.1

1.15.41		$K_{R}(s)=\frac{1}{\pi R}\frac{1-\cos\left(Rs\right)}{s^{2}},$
ⓘ Defines: $K_{R}(s)$ : Fejér kernel (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter and $\cos\NVar{z}$ : cosine function Permalink: http://dlmf.nist.gov/1.15.E41 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

1.15.42		$\int^{\infty}_{-\infty}K_{R}(s)\,\mathrm{d}s=1.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $K_{R}(s)$ : Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E42 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

For each $\delta>0$ ,

1.15.43		$\int_{\left\|s\right\|\geq\delta}K_{R}(s)\,\mathrm{d}s\to 0,$
as $R\to\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{R}(s)$ : Fejér kernel and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E43 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Let

1.15.44	$\displaystyle\sigma_{R}(\theta)$	$\displaystyle=\frac{1}{\sqrt{2\pi}}\int^{R}_{-R}\left(1-\frac{\left\|t\right\|}{% R}\right){\mathrm{e}}^{-\mathrm{i}\theta t}F(t)\,\mathrm{d}t,$
ⓘ 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, $F(x)$ : Fourier transform of $f(t)$ , $\sigma_{R}(\theta)$ : function and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E44 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1
then
1.15.45	$\displaystyle\sigma_{R}(\theta)$	$\displaystyle=\int^{\infty}_{-\infty}f(t)K_{R}(\theta-t)\,\mathrm{d}t.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{R}(s)$ : Fejér kernel and $\sigma_{R}(\theta)$ : function Permalink: http://dlmf.nist.gov/1.15.E45 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

If $f(\theta)$ is integrable on $(-\infty,\infty)$ , then

1.15.46		$\lim_{R\to\infty}\int^{\infty}_{-\infty}\left\|\sigma_{R}(\theta)-f(\theta)% \right\|\,\mathrm{d}\theta=0.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sigma_{R}(\theta)$ : function and $\left\|\NVar{x}\right\|$ : absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.15.E46 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

§1.15(vi) Fractional Integrals

ⓘ

Keywords:: fractional integrals, summability methods for integrals
Notes:: See Miller and Ross (1993) and Andrews et al. (1999, pp. 111-114, 604–607).
Referenced by:: §10.22(ii), §2.6(iii), §2.6(iii), Erratum (V1.0.1) for Subsections 1.15(vi) and 1.15(vii), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)
Permalink:: http://dlmf.nist.gov/1.15.vi
Modification (effective with 1.0.15):: The word operator was removed from the definition of a fractional integral,
Suggested 2017-04-22 by Tom Koornwinder
Modification (effective with 1.0.14):: The fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$ . Also, a paragraph with a generalization of (1.15.47), a general property, and references to examples in the DLMF has been added at the end of this subsection.
Errata (effective with 1.0.1):: The formulas in this subsection are only valid for $x\geq 0$ . No conditions on $x$ were given originally.
Reported 2010-10-18 by Andreas Kurt Richter
See also:: Annotations for §1.15 and Ch.1

For $\Re\alpha>0$ and $x\geq 0$ , the Riemann-Liouville fractional integral of order $\alpha$ is defined by

1.15.47		$I^{\alpha}f(x)=\frac{1}{\Gamma\left(\alpha\right)}\int^{x}_{0}(x-t)^{\alpha-1}% f(t)\,\mathrm{d}t.$
ⓘ Defines: $I^{\alpha}$ : fractional integral (locally) Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Referenced by: §1.15(vi), §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii) Permalink: http://dlmf.nist.gov/1.15.E47 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

For $\Gamma\left(\alpha\right)$ see §5.2, and compare (1.4.31) in the case when $\alpha$ is a positive integer.

1.15.48		$I^{\alpha}I^{\beta}=I^{\alpha+\beta},$
$\Re\alpha>0$ , $\Re\beta>0$ .
ⓘ Symbols: $\Re$ : real part and $I^{\alpha}$ : fractional integral Referenced by: §1.15(vi) Permalink: http://dlmf.nist.gov/1.15.E48 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

For extensions of (1.15.48) see Love (1972b).

1.15.49		$f(x)=\sum^{\infty}_{k=0}a_{k}x^{k},$
ⓘ Symbols: $k$ : integer Permalink: http://dlmf.nist.gov/1.15.E49 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

then

1.15.50		$I^{\alpha}f(x)=\sum^{\infty}_{k=0}\frac{k!}{\Gamma\left(k+\alpha+1\right)}a_{k% }x^{k+\alpha}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : integer and $I^{\alpha}$ : fractional integral Referenced by: Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii) Permalink: http://dlmf.nist.gov/1.15.E50 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

The lower limit $0$ of the integral in (1.15.47) can be replaced by any constant $a\leq x$ . Also, we can replace the lower and upper limits of the integral by $x$ and $a$ , respectively. In that case we must also replace $(x-t)$ in the integrand by $(t-x)$ and we can even set $a=\infty$ . See (18.17.9), (18.17.11) and (18.17.13) as examples.

§1.15(vii) Fractional Derivatives

ⓘ

Keywords:: fractional derivatives, summability methods for integrals
Notes:: See Andrews et al. (1999, pp. 606–607). (The notation for the fractional derivative is slightly different.)
Referenced by:: Erratum (V1.0.1) for Subsections 1.15(vi) and 1.15(vii), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)
Permalink:: http://dlmf.nist.gov/1.15.vii
Clarification (effective with 1.0.14):: The first sentence was clarified by adding “the fractional derivative of order $\alpha$ is defined by” ahead of (1.15.51) and “and satisfies the property” ahead of (1.15.52).
Errata (effective with 1.0.1):: The formulas in this subsection are only valid for $x\geq 0$ . No conditions on $x$ were given originally.
Reported 2010-10-18 by Andreas Kurt Richter
See also:: Annotations for §1.15 and Ch.1

For $0<\Re\alpha<n$ , $n$ an integer, and $x\geq 0$ , the fractional derivative of order $\alpha$ is defined by

1.15.51		$D^{\alpha}f(x)=\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}I^{n-\alpha}f(x),$
ⓘ Defines: $D^{\alpha}$ : fractional derivative (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $I^{\alpha}$ : fractional integral Referenced by: §1.15(vii) Permalink: http://dlmf.nist.gov/1.15.E51 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

and satisfies the property

1.15.52		$D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k},$
$k=1,2,\dots,n$ .
ⓘ Symbols: $k$ : integer, $n$ : nonnegative integer, $I^{\alpha}$ : fractional integral and $D^{\alpha}$ : fractional derivative Referenced by: §1.15(vii) Permalink: http://dlmf.nist.gov/1.15.E52 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

When none of $\alpha$ , $\beta$ , and $\alpha+\beta$ is an integer

1.15.53		$D^{\alpha}D^{\beta}=D^{\alpha+\beta}.$
ⓘ Symbols: $D^{\alpha}$ : fractional derivative Permalink: http://dlmf.nist.gov/1.15.E53 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

Note that $D^{1/2}D\not=D^{3/2}$ . See also Love (1972b).

§1.15(viii) Tauberian Theorems

ⓘ

Keywords:: Tauberian theorems, integrals, summability methods, summability methods for series
Notes:: See Hardy (1949, pp. 154–155), and Widder (1941, Chapter 5).
Permalink:: http://dlmf.nist.gov/1.15.viii
See also:: Annotations for §1.15 and Ch.1

1.15.54		$\displaystyle\sum^{\infty}_{n=0}a_{n}$	$\displaystyle=s\;\;\;\textit{(A)},$
		$\displaystyle a_{n}$	$\displaystyle>-\frac{K}{n}$ ,
	$n>0$ , $K>0$ ,
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E54 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

then

1.15.55		$\sum^{\infty}_{n=0}a_{n}=s.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E55 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

1.15.56		$\lim_{x\to 1-}(1-x)\sum^{\infty}_{n=0}a_{n}x^{n}=s,$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E56 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

and either $\left|a_{n}\right|\leq K$ or $a_{n}\geq 0$ , then

1.15.57		$\lim_{n\to\infty}\frac{a_{0}+a_{1}+\dots+a_{n}}{n+1}=s.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E57 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

1.14 Integral Transforms 1.16 Distributions

NIST

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§1.15 Summability Methods

Contents

§1.15(i) Definitions for Series

Abel Summability

Cesàro Summability

General Cesàro Summability

Borel Summability

§1.15(ii) Regularity

§1.15(iii) Summability of Fourier Series

Poisson Kernel

Fejér Kernel

Abel Means

Cesàro (or (C,1)) Means

Convergence

§1.15(iv) Definitions for Integrals

Abel Summability

Cesàro Summability

§1.15(v) Summability of Fourier Integrals

Poisson Kernel

Fejér Kernel

§1.15(vi) Fractional Integrals

§1.15(vii) Fractional Derivatives

§1.15(viii) Tauberian Theorems