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§1.15 Summability Methods

Contents

§1.15(i) Definitions for Series

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Referenced by:
§1.15(ii)
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http://dlmf.nist.gov/1.15.i
1.15.1s_{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(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

1.15.13\frac{1}{2\pi}\int^{{2\pi}}_{0}P(r,\theta)d\theta=1.

As r\to 1-

1.15.14P(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,

1.15.15K_{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},

As n\to\infty

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

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

Abel Means

1.15.18A(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
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where

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

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

Then

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)
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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

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

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

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

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

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

1.15.33P(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
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For each \delta>0,

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

Let

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

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

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

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

1.15.41K_{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
Encodings:
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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), Subsections 1.15(vi) and 1.15(vii)
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

1.15.47I^{\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
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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.

1.15.48I^{\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

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

§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:
Subsections 1.15(vi) and 1.15(vii)
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,

1.15.51D^{\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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When none of \alpha, \beta, and \alpha+\beta is an integer

1.15.53D^{\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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