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
:
![]() | (1) |
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
![]() | (2) |
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 :
![]() | (3) |
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).
References
[1] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
[2] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
[3] | 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=12593