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)