§1.15 Summability Methods
Contents
- §1.15(i) Definitions for Series
- §1.15(ii) Regularity
- §1.15(iii) Summability of Fourier Series
- §1.15(iv) Definitions for Integrals
- §1.15(v) Summability of Fourier Integrals
- §1.15(vi) Fractional Integrals
- §1.15(vii) Fractional Derivatives
- §1.15(viii) Tauberian Theorems
§1.15(i) Definitions for Series
¶ Abel Summability
if
¶ Cesàro Summability
if
¶ General Cesàro Summability
For
,
if
¶ Borel Summability
if
§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
then
§1.15(iii) Summability of Fourier Series
¶ Poisson Kernel
As
uniformly for
. (Here and elsewhere in this
subsection
is a constant such that
.)
¶ Fejér Kernel
For
,
As ![]()
uniformly for
.
¶ Abel Means
¶ Cesàro (or (C,1)) Means
Let
, where
Then
¶ Convergence
If
is periodic and integrable on
, then as
the Abel means
and the (C,1) means
converge
to
at every point
where both limits exist. If
is also
continuous, then the convergence is uniform for all
.
For real-valued
, if
is the Fourier series of
, then the series
can be extended to the interior of the unit circle as an analytic function
Here
is the Abel (or Poisson) sum
of
, and
has the series representation
compare §1.15(v).
§1.15(iv) Definitions for Integrals
¶ Abel Summability
is Abel summable to
, or
when
¶ Cesàro Summability
is (C,1) summable to
, or
when
If
converges and equals
, then the
integral is Abel and Cesàro summable to
.
§1.15(v) Summability of Fourier Integrals
¶ Poisson Kernel
For each
,
Let
where
is the Fourier transform of
(§1.14(i)). Then
is the Poisson integral of
.
If
is integrable on
, then
Suppose now
is real-valued and integrable on
. Let
where
and
. Then
is an analytic
function in the upper half-plane and its real part is the Poisson integral
; compare (1.9.34). The imaginary part
is the conjugate Poisson integral
of
. Moreover,
is the Hilbert
transform of
(§1.14(v)).
¶ Fejér Kernel
For each
,
Let
then
If
is integrable on
, then
§1.15(vi) Fractional Integrals
For
and
, the fractional integral operator of order
is defined by
For
see §5.2, and compare
(1.4.31) in the case when
is a positive integer.
If
then
§1.15(vii) Fractional Derivatives
For
,
an integer, and
,
When none of
,
, and
is an integer
Note that
. See also Love (1972b).
§1.15(viii) Tauberian Theorems
If
then
If
and either
or
, then


