Semi-continuous summation method
A summation method (cf. Summation methods) for series and sequences, defined by means of sequences of functions. Let , be a sequence of functions defined on some set of variation of the parameter , and let be an accumulation point of (finite or infinite). The functions are used to convert a given sequence into a function :
If the series in (1) is convergent for all sufficiently close to , and if
one says that the sequence is summable to by the semi-continuous summation method defined by the sequence . If is the sequence of partial sums of the series
one says that the series (2) is summable by the semi-continuous method to . A semi-continuous summation method with is an analogue of the matrix summation method defined by the matrix , in which the integer-valued parameter is replaced by the continuous parameter . The sequence of functions 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 :
In this case the series (2) is said to be summable to if
where is an accumulation point of the set of variation of , and the series (3) is assumed to be convergent for all sufficiently close to .
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, the Borel summation method, the Lindelöf summation method, and the Mittag-Leffler summation method. The class of semi-continuous methods also includes methods with semi-continuous matrices of the form
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
for all sufficiently close to ,
are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of into a function to be regular (see Regularity criteria).
|||G.H. Hardy, "Divergent series" , Clarendon Press (1949)|
|||R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)|
|||W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)|
Semi-continuous summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_summation_method&oldid=48657