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1 Algebraic and Analytic MethodsTopics of Discussion

§1.15 Summability Methods

  1. §1.15(i) Definitions for Series
  2. §1.15(ii) Regularity
  3. §1.15(iii) Summability of Fourier Series
  4. §1.15(iv) Definitions for Integrals
  5. §1.15(v) Summability of Fourier Integrals
  6. §1.15(vi) Fractional Integrals
  7. §1.15(vii) Fractional Derivatives
  8. §1.15(viii) Tauberian Theorems

§1.15(i) Definitions for Series

1.15.1 sn=k=0nak.

Abel Summability

1.15.2 n=0an=s(A),


1.15.3 limx1n=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!ank=s.

Borel Summability

1.15.8 n=0an=s(B),


1.15.9 limtetn=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)=1r212rcosθ+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,2πδ]. (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,2πδ].

Abel Means

1.15.18 A(r,θ)=n=r|n|F(n)einθ,


1.15.19 F(n)=12π02πf(t)eintdt.

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

1.15.20 A(r,θ)=12π02πP(r,θt)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(θt)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+eϵ|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 limRRR(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 12πP(x,y)dx=1.

For each δ>0,

1.15.35 |x|δP(x,y)dx0,
as y0.


1.15.36 h(x,y)=12πey|t|eixtF(t)dt,

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

1.15.37 h(x,y)=12πf(t)P(xt,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)=iπf(t)1(xt)+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)=1πf(t)xt(xt)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)=1πR1cos(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) =12πRR(1|t|R)eiθtF(t)dt,
1.15.45 σR(θ) =f(t)KR(θt)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 of order α is defined by

1.15.47 𝐼αf(x)=1Γ(α)0x(xt)α1f(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 (xt) in the integrand by (tx) 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)=dndxn𝐼nαf(x),

and satisfies the property

1.15.52 𝐷k𝐼α=𝐷n𝐼α+nk,

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(1x)n=0anxn=s,

and either |an|K or an0, then

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