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Semi-continuous summation method

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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)
How to Cite This Entry:
Semi-continuous summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_summation_method&oldid=54832
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article