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§1.15 Summability Methods


§1.15(i) Definitions for Series

1.15.1 sn=k=0nak.

Abel Summability

1.15.2 n=0an=s(A),


1.15.3 limx1-n=0anxn=s.

Cesàro Summability

1.15.4 n=0an=s(C,1),


1.15.5 limns0+s1++snn+1=s.

General Cesàro Summability

For α>-1,

1.15.6 n=0an=s(C,α),


1.15.7 limnn!(α+1)nk=0n(α+1)kk!an-k=s.

Borel Summability

1.15.8 n=0an=s(B),


1.15.9 limte-tn=0snn!tn=s.

§1.15(ii) Regularity

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 n=0an=s,


1.15.11 n=0an=s(A).

§1.15(iii) Summability of Fourier Series

Poisson Kernel

1.15.12 P(r,θ)=1-r21-2rcosθ+r2=n=-r|n|einθ,
1.15.13 12π02πP(r,θ)dθ=1.

As r1-

1.15.14 P(r,θ)0,

uniformly for θ[δ,2π-δ]. (Here and elsewhere in this subsection δ is a constant such that 0<δ<π.)

Fejér Kernel

For n=0,1,2,,

1.15.15 Kn(θ)=1n+1(sin(12(n+1)θ)sin(12θ))2,
1.15.16 12π02πKn(θ)dθ=1.

As n

1.15.17 Kn(θ)0,

uniformly for θ[δ,2π-δ].

Abel Means

1.15.18 A(r,θ)=n=-r|n|F(n)einθ,


1.15.19 F(n)=12π02πf(t)e-intdt.

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

1.15.20 A(r,θ)=12π02πP(r,θ-t)f(t)dt.

Cesàro (or (C,1)) Means


1.15.21 σn(θ)=s0(θ)+s1(θ)++sn(θ)n+1,

n=0,1,2,, where

1.15.22 sn(θ)=k=-nnF(k)eikθ.


1.15.23 σn(θ)=12π02πKn(θ-t)f(t)dt.


If f(θ) is periodic and integrable on [0,2π], then as n the Abel means A(r,θ) and the (C,1) means σn(θ) converge to

1.15.24 12(f(θ+)+f(θ-))

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

For real-valued f(θ), if

1.15.25 n=-F(n)einθ

is the Fourier series of f(θ), then the series

1.15.26 F(0)+2n=1F(n)einθ

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)+2n=1F(n)zn.

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

1.15.28 -n=-i(signn)F(n)r|n|einθ;

compare §1.15(v).

§1.15(iv) Definitions for Integrals

Abel Summability

-f(t)dt is Abel summable to L, or

1.15.29 -f(t)dt=L(A),


1.15.30 limϵ0+-e-ϵ|t|f(t)dt=L.

Cesàro Summability

-f(t)dt is (C,1) summable to L, or

1.15.31 -f(t)dt=L(C,1),


1.15.32 limR-RR(1-|t|R)f(t)dt=L.

If -f(t)dt converges and equals L, then the integral is Abel and Cesàro summable to L.

§1.15(v) Summability of Fourier Integrals

Poisson Kernel

1.15.33 P(x,y)=2yx2+y2,
y>0, -<x<.
1.15.34 12π-P(x,y)dx=1.

For each δ>0,

1.15.35 |x|δP(x,y)dx0,
as y0.


1.15.36 h(x,y)=12π-e-y|t|e-ixtF(t)dt,

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

1.15.37 h(x,y)=12π-f(t)P(x-t,y)dt

is the Poisson integral of f(t).

If f(x) is integrable on (-,), then

1.15.38 limy0+-|h(x,y)-f(x)|dx=0.

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

1.15.39 Φ(z)=Φ(x+iy)=iπ-f(t)1(x-t)+iydt,

where y>0 and -<x<. Then Φ(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 Φ(x+iy)=1π-f(t)x-t(x-t)2+y2dt

is the conjugate Poisson integral of f(x). Moreover, limy0+Φ(x+iy) is the Hilbert transform of f(x)1.14(v)).

Fejér Kernel

1.15.41 KR(s)=1πR1-cos(Rs)s2,
1.15.42 -KR(s)ds=1.

For each δ>0,

1.15.43 |s|δKR(s)ds0,
as R.


1.15.44 σR(θ) =12π-RR(1-|t|R)e-iθtF(t)dt,
1.15.45 σR(θ) =-f(t)KR(θ-t)dt.

If f(θ) is integrable on (-,), then

1.15.46 limR-|σR(θ)-f(θ)|dθ=0.

§1.15(vi) Fractional Integrals

For α>0 and x0, the Riemann-Liouville fractional integral operator of order α is defined by

1.15.47 Iαf(x)=1Γ(α)0x(x-t)α-1f(t)dt.

For Γ(α) see §5.2, and compare (1.4.31) in the case when α is a positive integer.

1.15.48 IαIβ=Iα+β,
α>0, β>0.

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


1.15.49 f(x)=k=0akxk,


1.15.50 Iαf(x)=k=0k!Γ(k+α+1)akxk+α.

The lower limit 0 of the integral in (1.15.47) can be replaced by any constant ax . 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=. See (18.17.9), (18.17.11) and (18.17.13) as examples.

§1.15(vii) Fractional Derivatives

For 0<α<n, n an integer, and x0, the fractional derivative of order α is defined by

1.15.51 Dαf(x)=dndxnIn-αf(x),

and satisfies the property

1.15.52 DkIα=DnIα+n-k,

When none of α, β, and α+β is an integer

1.15.53 DαDβ=Dα+β.

Note that D1/2DD3/2. See also Love (1972b).

§1.15(viii) Tauberian Theorems


1.15.54 n=0an =s(A),
an >-Kn,
n>0, K>0,


1.15.55 n=0an=s.


1.15.56 limx1-(1-x)n=0anxn=s,

and either |an|K or an0, then

1.15.57 limna0+a1++ann+1=s.