§1.15 Summability Methods

1.15.2		$$\sum _{n=0}^{\mathrm{\infty }}{a}_n=s\mathit{\text{(A)}},$$
ⓘ 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

if

1.15.3		$$\underset{x\to 1-}{lim}\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n=s.$$
ⓘ 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

1.15.4		$$\sum _{n=0}^{\mathrm{\infty }}{a}_n=s\mathit{\text{(C,1)}},$$
ⓘ 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

if

1.15.5		$$\underset{n\to \mathrm{\infty }}{lim}\frac{s_0+s_1+\mathrm{\cdots }+s_n}{n+1}=s.$$
ⓘ 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

For $\alpha >-1$,

1.15.6		$$\sum _{n=0}^{\mathrm{\infty }}{a}_n=s\mathit{\text{(C,}}\alpha \mathit{\text{)}},$$
ⓘ 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

if

1.15.7		$$\underset{n\to \mathrm{\infty }}{lim}\frac{n!}{{(\alpha +1)}_n}\sum _{k=0}^n\frac{{(\alpha +1)}_k}{k!}{a}_{n-k}=s.$$
ⓘ 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

1.15.8		$$\sum _{n=0}^{\mathrm{\infty }}{a}_n=s\mathit{\text{(B)}},$$
ⓘ 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

if

1.15.9		$$\underset{t\to \mathrm{\infty }}{lim}{\mathrm{e}}^{-t}\sum _{n=0}^{\mathrm{\infty }}\frac{s_n}{n!}{t}^n=s.$$
ⓘ 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

1.15.12		$$P(r,\theta )=\frac{1-r^2}{1-2r\cos \theta +r^2}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{r}^{|n|}{\mathrm{e}}^{\mathrm{i}n\theta },$$
,
ⓘ Defines: $P(r,\theta )$: Poisson kernel (locally) Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer 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		$$\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta )d\theta =1.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 .)

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

1.15.15		$$K_n(\theta )=\frac{1}{n+1}{\left(\frac{\sin \left(\frac{1}{2}(n+1)\theta \right)}{\sin \left(\frac{1}{2}\theta \right)}\right)}^2,$$
ⓘ Defines: $K_n(\theta )$: Fejér kernel (locally) Symbols: $\sin z$: 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		$$\frac{1}{2\pi }{\int }_0^{2\pi }{K}_n(\theta )d\theta =1.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 $n\to \mathrm{\infty }$

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

1.15.18		$$A(r,\theta )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{r}^{|n|}F(n){\mathrm{e}}^{\mathrm{i}n\theta },$$
ⓘ Defines: $A(r,\theta )$: Abel mean (locally) Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $F(n)$ 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		$$F(n)=\frac{1}{2\pi }{\int }_0^{2\pi }f(t){\mathrm{e}}^{-\mathrm{i}nt}dt.$$
ⓘ Defines: $F(n)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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		$$A(r,\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta -t)f(t)dt.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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

Let

1.15.21		$${\sigma }_n(\theta )=\frac{s_0(\theta )+s_1(\theta )+\mathrm{\cdots }+s_n(\theta )}{n+1},$$
ⓘ 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

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

1.15.22		$$s_n(\theta )=\sum _{k=-n}^{n}F(k){\mathrm{e}}^{\mathrm{i}k\theta }.$$
ⓘ 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		$${\sigma }_n(\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }{K}_n(\theta -t)f(t)dt.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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

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

1.15.24		$$\frac{1}{2}(f(\theta +)+f(\theta -))$$
ⓘ 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		$$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}F(n){\mathrm{e}}^{\mathrm{i}n\theta }$$
ⓘ 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		$$F(0)+2\sum _{n=1}^{\mathrm{\infty }}F(n){\mathrm{e}}^{\mathrm{i}n\theta }$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.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		$$G(z)=G(x+\mathrm{i}y)=u(x,y)+\mathrm{i}v(x,y)=F(0)+2\sum _{n=1}^{\mathrm{\infty }}F(n)z^n.$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $z$: variable, $n$: nonnegative integer, $F(n)$ and $u(x,y)$: 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		$$-\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{i}(\mathrm{sign}n)F(n)r^{|n|}{\mathrm{e}}^{\mathrm{i}n\theta };$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{sign}x$: sign of $x$, $n$: nonnegative integer and $F(n)$ 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).

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

1.15.29		$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=L\mathit{\text{(A)}},$$
ⓘ Symbols: $dx$: 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		$$\underset{ϵ\to 0+}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-ϵ|t|}f(t)dt=L.$$
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm and $\int$: integral 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

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

1.15.31		$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=L\mathit{\text{(C,1)}},$$
ⓘ Symbols: $dx$: 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		$$\underset{R\to \mathrm{\infty }}{lim}{\int }_{-R}^R\left(1-\frac{|t|}{R}\right)f(t)dt=L.$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral 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 ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt$ converges and equals $L$, then the integral is Abel and Cesàro summable to $L$.

1.15.33		$$P(x,y)=\frac{2y}{x^2+y^2},$$
$y>0$, .
ⓘ 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		$$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(x,y)dx=1.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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		$${\int }_{|x|\ge \delta }P(x,y)dx\to 0,$$
as $y\to 0$.
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $P(x,y)$: Poisson kernel 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		$$h(x,y)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-y|t|}{\mathrm{e}}^{-\mathrm{i}xt}F(t)dt,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ and $h(x,y)$: function 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		$$h(x,y)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)P(x-t,y)dt$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 $(-\mathrm{\infty },\mathrm{\infty })$, then

1.15.38		$$\underset{y\to 0+}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|h(x,y)-f(x)|dx=0.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $h(x,y)$: function 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 $(-\mathrm{\infty },\mathrm{\infty })$. Let

1.15.39		$$\mathrm{\Phi }(z)=\mathrm{\Phi }(x+\mathrm{i}y)=\frac{\mathrm{i}}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\frac{1}{(x-t)+\mathrm{i}y}dt,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: variable and $\mathrm{\Phi }(z)$: 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 . Then $\mathrm{\Phi }(z)$ 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		$$\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\frac{x-t}{{(x-t)}^2+y^2}dt$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Phi }(z)$: 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, ${lim}_{y\to 0+}\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)).

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 $\cos z$: 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		$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{K}_R(s)ds=1.$$
ⓘ Symbols: $dx$: 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		$${\int }_{|s|\ge \delta }{K}_R(s)ds\to 0,$$
as $R\to \mathrm{\infty }$.
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $K_R(s)$: Fejér kernel 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		${\sigma }_R(\theta )$	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-R}^R}\left(1-{\displaystyle \frac{|t|}{R}}\right){\mathrm{e}}^{-\mathrm{i}\theta t}F(t)dt,$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ and ${\sigma }_R(\theta )$: function 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		${\sigma }_R(\theta )$	$={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(t)K_R(\theta -t)dt.$
ⓘ Symbols: $dx$: 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 $(-\mathrm{\infty },\mathrm{\infty })$, then

1.15.46		$$\underset{R\to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|{\sigma }_R(\theta )-f(\theta )|d\theta =0.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and ${\sigma }_R(\theta )$: function 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