Difference between revisions of "Equivalent summation methods"
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for the same $ k \geq 0 $), | for the same $ k \geq 0 $), | ||
the Cesàro summation methods $ ( C, k) $ | the Cesàro summation methods $ ( C, k) $ | ||
| − | and the [[Hölder summation methods|Hölder summation methods]] $ ( H, k) $( | + | and the [[Hölder summation methods|Hölder summation methods]] $ ( H, k) $ |
| − | for the same integer $ k \geq 0 $). | + | (for the same integer $ k \geq 0 $). |
There are equivalent summation methods that are not compatible. | There are equivalent summation methods that are not compatible. | ||
| Line 25: | Line 25: | ||
Equivalence of summation methods for special forms of summability (absolute, strong, etc.) is defined similarly. | Equivalence of summation methods for special forms of summability (absolute, strong, etc.) is defined similarly. | ||
| − | [[Matrix summation method|Matrix summation method]], defined by transformations of sequences to sequences using matrices $ \| a _ {nk} \| $ | + | [[Matrix summation method|Matrix summation method]]s, defined by transformations of sequences to sequences using matrices $ \| a _ {nk} \| $ |
and $ \| b _ {nk} \| $, | and $ \| b _ {nk} \| $, | ||
are called absolutely equivalent on a set $ U $ | are called absolutely equivalent on a set $ U $ | ||
of sequences $ \{ s _ {k} \} $ | of sequences $ \{ s _ {k} \} $ | ||
| − | if $ \tau _ {n} ^ {( | + | if $ \tau _ {n} ^ {( A)} - \tau _ {n} ^ {( B)} \rightarrow 0 $, |
$ n \rightarrow \infty $, | $ n \rightarrow \infty $, | ||
for any $ \{ s _ {k} \} \subset U $, | for any $ \{ s _ {k} \} \subset U $, | ||
| Line 35: | Line 35: | ||
$$ | $$ | ||
| − | \tau _ {n} ^ {( | + | \tau _ {n} ^ {( A)} = \ |
\sum _ {k = 1 } ^ \infty | \sum _ {k = 1 } ^ \infty | ||
a _ {nk} s _ {k} ,\ \ | a _ {nk} s _ {k} ,\ \ | ||
| − | \tau _ {n} ^ {( | + | \tau _ {n} ^ {( B)} = \ |
\sum _ {k = 1 } ^ \infty | \sum _ {k = 1 } ^ \infty | ||
b _ {nk} s _ {k} , | b _ {nk} s _ {k} , | ||
$$ | $$ | ||
| − | and the series in the expressions for $ \tau _ {n} ^ {( | + | and the series in the expressions for $ \tau _ {n} ^ {( A)} $ |
| − | and $ \tau _ {n} ^ {( | + | and $ \tau _ {n} ^ {( B)} $ |
converge for all $ n $. | converge for all $ n $. | ||
Latest revision as of 18:05, 16 December 2020
Methods that sum the same sequences (possibly to different limits); in other words, equivalent summation methods are summation methods having the same summability field. Sometimes methods that have the same summability field and that are compatible summation methods (cf. Compatibility of summation methods) are called equivalent. Examples of equivalent and compatible methods are the Cesàro summation methods $ ( C, k) $
and the Riesz summation methods (cf. Riesz summation method) $ ( R, n, k) $(
for the same $ k \geq 0 $),
the Cesàro summation methods $ ( C, k) $
and the Hölder summation methods $ ( H, k) $
(for the same integer $ k \geq 0 $).
There are equivalent summation methods that are not compatible.
Sometimes one considers not the complete summability fields, but subsets of them belonging to some set $ U $. If these subsets coincide for two summation methods, then it is said that the methods are equivalent on $ U $. Summation methods for real sequences are called completely equivalent if the equality of their summability fields remains valid upon the inclusion of sequences summable to $ + \infty $ and $ - \infty $. Equivalence of summation methods for special forms of summability (absolute, strong, etc.) is defined similarly.
Matrix summation methods, defined by transformations of sequences to sequences using matrices $ \| a _ {nk} \| $ and $ \| b _ {nk} \| $, are called absolutely equivalent on a set $ U $ of sequences $ \{ s _ {k} \} $ if $ \tau _ {n} ^ {( A)} - \tau _ {n} ^ {( B)} \rightarrow 0 $, $ n \rightarrow \infty $, for any $ \{ s _ {k} \} \subset U $, where
$$ \tau _ {n} ^ {( A)} = \ \sum _ {k = 1 } ^ \infty a _ {nk} s _ {k} ,\ \ \tau _ {n} ^ {( B)} = \ \sum _ {k = 1 } ^ \infty b _ {nk} s _ {k} , $$
and the series in the expressions for $ \tau _ {n} ^ {( A)} $ and $ \tau _ {n} ^ {( B)} $ converge for all $ n $.
References
| [1] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
| [2] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
| [3] | G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70 |
Comments
References
| [a1] | K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970) |
Equivalent summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalent_summation_methods&oldid=50999