View source for Semi-continuous summation method

A summation method (cf. [[Summation methods|Summation methods]]) for series and sequences, defined by means of sequences of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840302.png" /> be a sequence of functions defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840303.png" /> of variation of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840304.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840305.png" /> be an [[Accumulation point|accumulation point]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840306.png" /> (finite or infinite). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840307.png" /> are used to convert a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840308.png" /> into a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840309.png" />:

<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/s/s084/s084030/s08403010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

If the series in (1) is convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403011.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403012.png" />, and if

<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/s/s084/s084030/s08403013.png" /></td> </tr></table>

one says that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403014.png" /> is summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403015.png" /> by the semi-continuous summation method defined by the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403017.png" /> is the sequence of partial sums of 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/s/s084/s084030/s08403018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

one says that the series (2) is summable by the semi-continuous method to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403019.png" />. A semi-continuous summation method with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403020.png" /> is an analogue of the [[Matrix summation method|matrix summation method]] defined by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403021.png" />, in which the integer-valued parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403022.png" /> is replaced by the continuous parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403023.png" />. The sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403024.png" /> is therefore known as a semi-continuous matrix.

A semi-continuous summation method can be defined by direct transformation of a series into a function, using a given sequence of functions, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403025.png" />:

<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/s/s084/s084030/s08403026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

In this case the series (2) is said to be summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403027.png" /> if

<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/s/s084/s084030/s08403028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403029.png" /> is an accumulation point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403030.png" /> of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403031.png" />, and the series (3) is assumed to be convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403032.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403033.png" />.

In some cases a semi-continuous summation method is more convenient than a summation method based on ordinary matrices, since it enables one to utilize tools of function theory. Examples of semi-continuous summation methods are: the [[Abel summation method|Abel summation method]], the [[Borel summation method|Borel summation method]], the [[Lindelöf summation method|Lindelöf summation method]], and the [[Mittag-Leffler summation method|Mittag-Leffler summation method]]. The class of semi-continuous methods also includes methods with semi-continuous matrices of 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/s/s084/s084030/s08403034.png" /></td> </tr></table>

where the denominator is an entire function that does not reduce to a polynomial.

Conditions for the regularity of semi-continuous summation methods are analogous to regularity conditions for matrix summation methods. For example, the conditions

<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/s/s084/s084030/s08403035.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403036.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403037.png" />,

<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/s/s084/s084030/s08403038.png" /></td> </tr></table>

<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/s/s084/s084030/s08403039.png" /></td> </tr></table>

are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403040.png" /> into a function to be regular (see [[Regularity criteria|Regularity criteria]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Beekmann,   K. Zeller,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>