Kernel of a summation method

A function (depending on a parameter) the values of which are the averages of the given method of summation applied to the series

(1)

The kernel of a summation method gives an integral representation of the averages of the method in the summation of Fourier series. If the summation method is defined by a transformation of sequences into sequences using a matrix , then the kernel of this method is the function

where are the partial sums of the series (1):

(2)

In this case the averages of the Fourier series for a -periodic function can be expressed in terms of and the kernel by the formula

In particular, the kernel of the method of arithmetical averages (cf. Arithmetical averages, summation method of) has the form

and is called the Fejér kernel. The kernel of the Abel summation method is given by

and is called the Poisson kernel. The function in (2) is called the Dirichlet kernel.

The function whose values are the averages of a summation method applied to the series

is called the conjugate kernel of the summation method.

References

[1]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)