Difference between revisions of "Hardy criterion"
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''for uniform convergence of series of functions'' | ''for uniform convergence of series of functions'' | ||
| − | If a sequence of real-valued functions | + | If a sequence of real-valued functions $ a _ {n} ( x) $, |
| + | $ n = 1, 2 \dots $ | ||
| + | is monotone for every $ x \in E $, | ||
| + | where $ E $ | ||
| + | is a certain set, and converges uniformly to zero on $ E $, | ||
| + | and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $ | ||
| + | is bounded on $ E $( | ||
| + | the functions $ b _ {n} ( x) $ | ||
| + | may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $ | ||
| + | converges uniformly on $ E $. | ||
This criterion was established by G.H. Hardy [[#References|[1]]]. | This criterion was established by G.H. Hardy [[#References|[1]]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Some theorems connected with Abel's theorem on the continuity of power series" ''Proc. London. Math. Soc. (2)'' , '''4''' (1907) pp. 247–265</TD></TR></table> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Some theorems connected with Abel's theorem on the continuity of power series" ''Proc. London. Math. Soc. (2)'' , '''4''' (1907) pp. 247–265 {{ZBL|37.0429.01}}</TD></TR> | |
| − | + | </table> | |
====Comments==== | ====Comments==== | ||
| − | This criterion can be proved using the [[ | + | This criterion can be proved using the [[Euler–MacLaurin formula]] (cf. the proof of Thm. 3.42 in [[#References|[a1]]]). |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD></TR> | ||
| + | </table> | ||
Latest revision as of 07:38, 1 November 2023
for uniform convergence of series of functions
If a sequence of real-valued functions $ a _ {n} ( x) $, $ n = 1, 2 \dots $ is monotone for every $ x \in E $, where $ E $ is a certain set, and converges uniformly to zero on $ E $, and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $ is bounded on $ E $( the functions $ b _ {n} ( x) $ may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $ converges uniformly on $ E $.
This criterion was established by G.H. Hardy [1].
References
| [1] | G.H. Hardy, "Some theorems connected with Abel's theorem on the continuity of power series" Proc. London. Math. Soc. (2) , 4 (1907) pp. 247–265 Zbl 37.0429.01 |
Comments
This criterion can be proved using the Euler–MacLaurin formula (cf. the proof of Thm. 3.42 in [a1]).
References
| [a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108 |
How to Cite This Entry:
Hardy criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_criterion&oldid=11277
Hardy criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_criterion&oldid=11277
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article