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§1.15 Summability Methods

Contents

§1.15(i) Definitions for Series

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Referenced by:
§1.15(ii)
Permalink:
http://dlmf.nist.gov/1.15.i
1.15.1 s_{n}=\sum _{{k=0}}^{n}a_{k}.

Abel Summability

1.15.2 \sum^{\infty}_{{n=0}}a_{n}=s\;\;\;\textit{(A)},

if

1.15.3 \lim _{{x\to 1-}}\sum^{\infty}_{{n=0}}a_{n}x^{n}=s.

Cesàro Summability

1.15.4 \sum^{\infty}_{{n=0}}a_{n}=s\;\;\;\textit{(C,1)},

if

1.15.5 \lim _{{n\to\infty}}\frac{s_{0}+s_{1}+\dots+s_{n}}{n+1}=s.

General Cesàro Summability

For \alpha>-1,

1.15.6 \sum^{\infty}_{{n=0}}a_{n}=s\;\;\;\textit{(C,$\alpha$)},

if

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.

§1.15(ii) Regularity

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Notes:
See Hardy (1949, p. 10).
Keywords:
convergence, summability methods for series
Permalink:
http://dlmf.nist.gov/1.15.ii

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,

then

1.15.11 \sum^{\infty}_{{n=0}}a_{n}=s\;\;\;\textit{(A)}.

§1.15(iii) Summability of Fourier Series

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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).
Keywords:
Fourier series
Permalink:
http://dlmf.nist.gov/1.15.iii

Poisson Kernel

1.15.12 P(r,\theta)=\frac{1-r^{2}}{1-2r\mathop{\cos\/}\nolimits\theta+r^{2}}=\sum^{\infty}_{{n=-\infty}}r^{{|n|}}e^{{in\theta}}, 0\leq r<1,
1.15.13 \frac{1}{2\pi}\int^{{2\pi}}_{0}P(r,\theta)d\theta=1.

As r\to 1-

1.15.14 P(r,\theta)\to 0,
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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

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

1.15.15 K_{n}(\theta)=\frac{1}{n+1}\left(\frac{\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}(n+1)\theta\right)}{\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)}\right)^{2},
1.15.16 \frac{1}{2\pi}\int^{{2\pi}}_{0}K_{n}(\theta)d\theta=1.

As n\to\infty

1.15.17 K_{n}(\theta)\to 0,

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

Abel Means

1.15.18 A(r,\theta)=\sum^{\infty}_{{n=-\infty}}r^{{|n|}}F(n)e^{{in\theta}},
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Defines:
A: Abel mean
Symbols:
e: base of exponential function, n: nonnegative integer, F(n), r: radius and \theta: angle
Permalink:
http://dlmf.nist.gov/1.15.E18
Encodings:
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where

1.15.19 F(n)=\frac{1}{2\pi}\int^{{2\pi}}_{0}f(t)e^{{-int}}dt.

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)dt.

Cesàro (or (C,1)) Means

Let

1.15.21 \sigma _{n}(\theta)=\frac{s_{0}(\theta)+s_{1}(\theta)+\dots+s_{n}(\theta)}{n+1},
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Defines:
\sigma _{n}: Cesàro mean
Symbols:
n: nonnegative integer
Permalink:
http://dlmf.nist.gov/1.15.E21
Encodings:
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n=0,1,2,\dots, where

1.15.22 s_{n}(\theta)=\sum^{n}_{{k=-n}}F(k)e^{{ik\theta}}.

Then

1.15.23 \sigma _{n}(\theta)=\frac{1}{2\pi}\int^{{2\pi}}_{0}K_{n}(\theta-t)f(t)dt.

Convergence

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Keywords:
Fourier series

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-))
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Referenced by:
§1.15(iii)
Permalink:
http://dlmf.nist.gov/1.15.E24
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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)e^{{in\theta}}

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

1.15.26 F(0)+2\sum^{\infty}_{{n=1}}F(n)e^{{in\theta}}

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

1.15.27 G(z)=G(x+iy)=u(x,y)+iv(x,y)=F(0)+2\sum^{\infty}_{{n=1}}F(n)z^{n}.

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}}i(\mathop{\mathrm{sign}\/}\nolimits n)F(n)r^{{|n|}}e^{{in\theta}};

compare §1.15(v).

§1.15(iv) Definitions for Integrals

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Notes:
See Weiss (1965, pp. 143–147) and Wong (1989, pp. 197–198).
Keywords:
integrals
Permalink:
http://dlmf.nist.gov/1.15.iv

Abel Summability

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

1.15.29 \int^{\infty}_{{-\infty}}f(t)dt=L\;\;\;\textit{(A)},

when

1.15.30 \lim _{{\epsilon\to 0+}}\int^{\infty}_{{-\infty}}e^{{-\epsilon|t|}}f(t)dt=L.

Cesàro Summability

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

1.15.31 \int^{\infty}_{{-\infty}}f(t)dt=L\;\;\;\textit{(C,1)},

when

1.15.32 \lim _{{R\to\infty}}\int^{R}_{{-R}}\left(1-\frac{|t|}{R}\right)f(t)dt=L.

If \int^{\infty}_{{-\infty}}f(t)dt 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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Notes:
See Weiss (1965, pp. 143–148).
Keywords:
Fourier integral
Referenced by:
‣ §1.15(iii)
Permalink:
http://dlmf.nist.gov/1.15.v

Poisson Kernel

1.15.33 P(x,y)=\frac{2y}{x^{2}+y^{2}}, y>0, -\infty<x<\infty.
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Symbols:
P(x,y): Poisson kernel
Permalink:
http://dlmf.nist.gov/1.15.E33
Encodings:
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1.15.34 \frac{1}{2\pi}\int^{\infty}_{{-\infty}}P(x,y)dx=1.

For each \delta>0,

1.15.35 \int _{{|x|\geq\delta}}P(x,y)dx\to 0, as y\to 0.

Let

1.15.36 h(x,y)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{{-\infty}}e^{{-y|t|}}e^{{-ixt}}F(t)dt,

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)dt

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}}|h(x,y)-f(x)|dx=0.

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

1.15.39 \Phi(z)=\Phi(x+iy)=\frac{i}{\pi}\int^{\infty}_{{-\infty}}f(t)\frac{1}{(x-t)+iy}dt,

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 \imagpart{\Phi(x+iy)}=\frac{1}{\pi}\int^{\infty}_{{-\infty}}f(t)\frac{x-t}{(x-t)^{2}+y^{2}}dt

is the conjugate Poisson integral of f(x). Moreover, \lim _{{y\to 0+}}\imagpart{\Phi(x+iy)} is the Hilbert transform of f(x) (§1.14(v)).

Fejér Kernel

1.15.41 K_{R}(s)=\frac{1}{\pi R}\frac{1-\mathop{\cos\/}\nolimits\!\left(Rs\right)}{s^{2}},
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Symbols:
\mathop{\cos\/}\nolimits z: cosine function and K_{R}(s): Fejér kernel
Permalink:
http://dlmf.nist.gov/1.15.E41
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1.15.42 \int^{\infty}_{{-\infty}}K_{R}(s)ds=1.

For each \delta>0,

1.15.43 \int _{{|s|\geq\delta}}K_{R}(s)ds\to 0, as R\to\infty.

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

1.15.46 \lim _{{R\to\infty}}\int^{\infty}_{{-\infty}}|\sigma _{R}(\theta)-f(\theta)|d\theta=0.

§1.15(vi) Fractional Integrals

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Notes:
See Andrews et al. (1999, pp. 111-114, 604–607).
Keywords:
fractional integrals, summability methods for integrals
Referenced by:
‣ §10.22(ii), §2.6(iii), Version 1.0.1 (June 27, 2011)
Permalink:
http://dlmf.nist.gov/1.15.vi
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

For \realpart{\alpha}>0 and x\geq 0, the fractional integral operator of order \alpha is defined by

1.15.47 I^{\alpha}f(x)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}\int^{x}_{0}(x-t)^{{\alpha-1}}f(t)dt.
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Defines:
I^{{\alpha}}: fractional integral
Symbols:
\mathop{\Gamma\/}\nolimits\!\left(z\right): gamma function, dx: differential of x and \int: integral
Permalink:
http://dlmf.nist.gov/1.15.E47
Encodings:
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For \mathop{\Gamma\/}\nolimits\!\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}}, \realpart{\alpha}>0, \realpart{\beta}>0.
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Symbols:
I^{{\alpha}}: fractional integral and \realpart{}: real part
Referenced by:
§1.15(vi)
Permalink:
http://dlmf.nist.gov/1.15.E48
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For extensions of (1.15.48) see Love (1972b).

If

1.15.49 f(x)=\sum^{\infty}_{{k=0}}a_{k}x^{k},
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Symbols:
k: integer
Permalink:
http://dlmf.nist.gov/1.15.E49
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then

1.15.50 I^{\alpha}f(x)=\sum^{\infty}_{{k=0}}\frac{k!}{\mathop{\Gamma\/}\nolimits\!\left(k+\alpha+1\right)}a_{k}x^{{k+\alpha}}.

§1.15(vii) Fractional Derivatives

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Notes:
See Andrews et al. (1999, pp. 606–607). (The notation for the fractional derivative is slightly different.)
Keywords:
fractional derivatives, summability methods for integrals
Referenced by:
Version 1.0.1 (June 27, 2011)
Permalink:
http://dlmf.nist.gov/1.15.vii
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

For 0<\realpart{\alpha}<n, n an integer, and x\geq 0,

1.15.51 D^{\alpha}f(x)=\frac{{d}^{n}}{{dx}^{n}}I^{{n-\alpha}}f(x),
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Defines:
D^{\alpha}: fractional derivative
Symbols:
\frac{df}{dx}: derivative of f with respect to x, I^{{\alpha}}: fractional integral and n: nonnegative integer
Permalink:
http://dlmf.nist.gov/1.15.E51
Encodings:
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1.15.52 D^{k}I^{\alpha}=D^{n}I^{{\alpha+n-k}}, k=1,2,\dots,n.

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

1.15.53 D^{\alpha}D^{\beta}=D^{{\alpha+\beta}}.

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

§1.15(viii) Tauberian Theorems

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

If

1.15.54
\sum^{\infty}_{{n=0}}a_{n}=s\;\;\;\textit{(A)},
a_{n}>-\frac{K}{n}, n>0, K>0,

then

1.15.55 \sum^{\infty}_{{n=0}}a_{n}=s.

If

1.15.56 \lim _{{x\to 1-}}(1-x)\sum^{\infty}_{{n=0}}a_{n}x^{n}=s,

and either |a_{n}|\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.
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