Kernel of a summation method

A function $ K _ {n} ( t) $( depending on a parameter) the values of which are the averages of the given method of summation applied to the series

$$ \tag{1 } { \frac{1}{2} } + \sum _ {\nu = 1 } ^ \infty \cos \nu t. $$

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 $ \| a _ {nk} \| _ {n,k= 0 } ^ \infty $, then the kernel of this method is the function

$$ K _ {n} ( t) = \sum _ {k = 0 } ^ \infty a _ {nk} D _ {k} ( t), $$

where $ D _ {k} ( t) $ are the partial sums of the series (1):

$$ \tag{2 } D _ {k} ( t) = \ { \frac{1}{2} } + \sum _ {\nu = 1 } ^ { k } \cos \nu t = \ \frac{\sin \{ ( k + {1 / 2 } ) t \} }{2 \sin ( t / 2) } . $$

In this case the averages of the Fourier series for a $ 2 \pi $- periodic function $ f $ can be expressed in terms of $ f $ and the kernel by the formula

$$ \sigma _ {n} ( f, x) = \ { \frac{1} \pi } \int\limits _ {- \pi } ^ \pi f ( t) K _ {n} ( t - x) dt. $$

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

$$ K _ {n} ( t) = \ \frac{2}{n + 1 } \left [ \frac{\sin \{ ( n + 1) t /2 \} }{2 \sin ( t/2) } \right ] ^ {2} , $$

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

$$ K ( r, t) = \ { \frac{1}{2} } \frac{1 - r ^ {2} }{1 - 2r \cos t + r ^ {2} } ,\ \ 0 \leq r < 1, $$

and is called the Poisson kernel. The function $ D _ {k} ( t) $ in (2) is called the Dirichlet kernel.

The function $ \overline{K}\; _ {n} ( t) $ whose values are the averages of a summation method applied to the series

$$ \sum _ {\nu = 1 } ^ \infty \sin \nu t $$

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)