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)
How to Cite This Entry:
Semi-continuous summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_summation_method&oldid=48657
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article