Difference between revisions of "Semi-continuous summation method"
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− | + | A summation method (cf. [[Summation methods|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|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 | + | 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|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|Abel summation method]], the [[Borel summation method|Borel summation method]], the [[Lindelöf summation method|Lindelöf summation method]], and the [[Mittag-Leffler summation method|Mittag-Leffler summation method]]. The class of semi-continuous methods also includes methods with semi-continuous matrices of the form | 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|Abel summation method]], the [[Borel summation method|Borel summation method]], the [[Lindelöf summation method|Lindelöf summation method]], and the [[Mittag-Leffler summation method|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. | where the denominator is an entire function that does not reduce to a polynomial. | ||
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Conditions for the regularity of semi-continuous summation methods are analogous to regularity conditions for matrix summation methods. For example, the conditions | 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 | + | 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 | + | 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|Regularity criteria]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)</TD></TR></table> |
Revision as of 08:13, 6 June 2020
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=12593