# §1.15 Summability Methods

1.15.1

1.15.2

if

1.15.3

1.15.4

if

1.15.5

For ,

1.15.6

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

1.15.10

then

1.15.11

## §1.15(iii) Summability of Fourier Series

### ¶ Poisson Kernel

1.15.13

As

1.15.14

uniformly for . (Here and elsewhere in this subsection is a constant such that .)

For ,

1.15.15

As

1.15.17

uniformly for .

### ¶ Abel Means

1.15.18

where

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

Let

1.15.21

, where

Then

### ¶ Convergence

If is periodic and integrable on , then as the Abel means and the (C,1) means converge to

1.15.24

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

1.15.29

when

### ¶ Cesàro Summability

is (C,1) summable to , or

1.15.31

when

1.15.32

If converges and equals , then the integral is Abel and Cesàro summable to .

## §1.15(v) Summability of Fourier Integrals

### ¶ Poisson Kernel

1.15.33, .

For each ,

1.15.35as .

Let

where is the Fourier transform of 1.14(i)). Then

is the Poisson integral of .

If is integrable on , then

1.15.38

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

1.15.41

For each ,

1.15.43as .

If is integrable on , then

1.15.46

## §1.15(vi) Fractional Integrals

For and , the fractional integral operator of order is defined by

1.15.47

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

1.15.48, .

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

If

1.15.49

then

## §1.15(vii) Fractional Derivatives

For , an integer, and ,

1.15.51

When none of , , and is an integer

1.15.53

## §1.15(viii) Tauberian Theorems

If

1.15.54
,, ,

then

1.15.55

If

1.15.56

and either or , then

1.15.57