Semi-continuous summation method
A summation method (cf. Summation methods) for series and sequences, defined by means of sequences of functions. Let \{ a _ {k} ( \omega ) \} ,
k = 0 , 1 \dots
be a sequence of functions defined on some set E
of variation of the parameter \omega ,
and let \omega _ {0}
be an accumulation point of E (
finite or infinite). The functions a _ {k} ( \omega )
are used to convert a given sequence \{ s _ {n} \}
into a function \sigma ( \omega ) :
\tag{1 } \sigma ( \omega ) = \sum _ {k=0} ^ \infty a _ {k} ( \omega ) s _ {k} .
If the series in (1) is convergent for all \omega sufficiently close to \omega _ {0} , and if
\lim\limits _ {\omega \rightarrow \omega _ {0} } \sigma ( \omega ) = s ,
one says that the sequence \{ s _ {n} \} is summable to s by the semi-continuous summation method defined by the sequence \{ a _ {k} ( \omega ) \} . If \{ s _ {n} \} is the sequence of partial sums of the series
\tag{2 } \sum _ {k=0} ^ \infty u _ {k} ,
one says that the series (2) is summable by the semi-continuous method to s . A semi-continuous summation method with \omega _ {0} = \infty is an analogue of the matrix summation method defined by the matrix \| a _ {nk} \| , in which the integer-valued parameter n is replaced by the continuous parameter \omega . The sequence of functions a _ {k} ( \omega ) 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 \{ g _ {k} ( \omega ) \} :
\tag{3 } \gamma ( \omega ) = \sum _ {k=0} ^ \infty g _ {k} ( \omega ) u _ {k} .
In this case the series (2) is said to be summable to s if
\lim\limits _ {\omega \rightarrow \omega _ {0} } \gamma ( \omega ) = s ,
where \omega _ {0} is an accumulation point of the set E of variation of \omega , and the series (3) is assumed to be convergent for all \omega sufficiently close to \omega _ {0} .
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
a _ {k} ( \omega ) = \frac{p _ {k} \omega ^ {k} }{\sum _ {l=0}^ \infty p _ {l} \omega ^ {l} } ,
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
\sum _ {k=0} ^ \infty | a _ {k} ( \omega ) | \leq M
for all \omega sufficiently close to \omega _ {0} ,
\lim\limits _ {\omega \rightarrow \omega _ {0} } a _ {k} ( \omega ) = 0 ,\ \ k = 0 , 1 \dots
\lim\limits _ {\omega \rightarrow \omega _ {0} } \sum _ {k=0} ^ \infty a _ {k} ( \omega ) = 1
are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of \{ s _ {k} \} 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=54832