View source for Kernel of a summation method

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553101.png" /> (depending on a parameter) the values of which are the averages of the given method of summation applied to the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

The kernel of a summation method gives an integral representation of the averages of the method in the summation of [[Fourier series|Fourier series]]. If the summation method is defined by a transformation of sequences into sequences using a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553103.png" />, then the kernel of this method is the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553104.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553105.png" /> are the partial sums of the series (1):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In this case the averages of the Fourier series for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553107.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553108.png" /> can be expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553109.png" /> and the kernel by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531010.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531011.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531012.png" /></td> </tr></table>

and is called the Poisson kernel. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531013.png" /> in (2) is called the Dirichlet kernel.

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531014.png" /> whose values are the averages of a summation method applied to the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531015.png" /></td> </tr></table>

is called the conjugate kernel of the summation method.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>