Difference between revisions of "Weierstrass criterion (for uniform convergence)"
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A theorem which gives sufficient conditions for the [[Uniform convergence|uniform convergence]] of a [[Series|series]] or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series | A theorem which gives sufficient conditions for the [[Uniform convergence|uniform convergence]] of a [[Series|series]] or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series | ||
− | + | $$ | |
+ | \sum _ { n= } 1 ^ \infty | ||
+ | u _ {n} ( x) | ||
+ | $$ | ||
− | of real- or complex-valued functions defined on some set | + | of real- or complex-valued functions defined on some set $ E $ |
+ | there exists a convergent series of non-negative numbers | ||
− | + | $$ | |
+ | \sum _ { n= } 1 ^ \infty a _ {n} $$ | ||
such that | such that | ||
− | + | $$ | |
+ | | u _ {n} ( x) | \leq a _ {n} ,\ \ | ||
+ | n = 1, 2 \dots | ||
+ | $$ | ||
+ | |||
+ | then the initial series converges uniformly and absolutely on the set $ E $( | ||
+ | cf. [[Absolutely convergent series|Absolutely convergent series]]). For instance, the series | ||
− | + | $$ | |
+ | \sum _ { n= } 1 ^ \infty | ||
− | + | \frac{\sin nx }{n ^ {2} } | |
+ | |||
+ | $$ | ||
converges uniformly and absolutely on the entire real axis, since | converges uniformly and absolutely on the entire real axis, since | ||
− | + | $$ | |
+ | \left | | ||
+ | \frac{\sin nx }{n ^ {2} } | ||
+ | \right | | ||
+ | \leq | ||
+ | \frac{1}{n ^ {2} } | ||
+ | |||
+ | $$ | ||
and the series | and the series | ||
− | + | $$ | |
+ | \sum _ { n= } 1 ^ \infty | ||
+ | \frac{1}{n ^ {2} } | ||
+ | |||
+ | $$ | ||
is convergent. | is convergent. | ||
− | If, for a sequence of real- or complex-valued functions | + | If, for a sequence of real- or complex-valued functions $ {f _ {n} } $, |
+ | $ n = 1, 2 \dots $ | ||
+ | which converges to a function $ f $ | ||
+ | on a set $ E $ | ||
+ | there exists a sequence of numbers $ \alpha _ {n} $, | ||
+ | $ \alpha _ {n} > 0 $ | ||
+ | and $ \alpha _ {n} \rightarrow 0 $ | ||
+ | as $ n \rightarrow \infty $, | ||
+ | such that $ | f( x) - {f _ {n} } ( x) | \leq \alpha _ {n} $, | ||
+ | $ x \in E $, | ||
+ | $ n = 1, 2 \dots $ | ||
+ | then the sequence converges uniformly on $ E $. | ||
+ | For instance, the sequence | ||
− | + | $$ | |
+ | f _ {n} ( x) = 1 - | ||
− | + | \frac{(- 1) ^ {n} }{x ^ {2} + n } | |
− | + | $$ | |
+ | |||
+ | converges uniformly to the function $ f( x) = 1 $ | ||
+ | on the entire real axis, since | ||
+ | |||
+ | $$ | ||
+ | | 1- f _ {n} ( x) | < | ||
+ | \frac{1}{n} | ||
+ | |||
+ | \ \textrm{ and } \ \lim\limits _ {n \rightarrow \infty } \ | ||
+ | |||
+ | \frac{1}{n} | ||
+ | = 0 . | ||
+ | $$ | ||
The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces. | The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces. | ||
Line 37: | Line 100: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> K. Weierstrass, "Math. Werke" , '''1–7''' , G. Olms & Johnson, reprint (1927)</TD></TR></table> | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> K. Weierstrass, "Math. Werke" , '''1–7''' , G. Olms & Johnson, reprint (1927)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24</TD></TR></table> |
Revision as of 08:28, 6 June 2020
A theorem which gives sufficient conditions for the uniform convergence of a series or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series
$$ \sum _ { n= } 1 ^ \infty u _ {n} ( x) $$
of real- or complex-valued functions defined on some set $ E $ there exists a convergent series of non-negative numbers
$$ \sum _ { n= } 1 ^ \infty a _ {n} $$
such that
$$ | u _ {n} ( x) | \leq a _ {n} ,\ \ n = 1, 2 \dots $$
then the initial series converges uniformly and absolutely on the set $ E $( cf. Absolutely convergent series). For instance, the series
$$ \sum _ { n= } 1 ^ \infty \frac{\sin nx }{n ^ {2} } $$
converges uniformly and absolutely on the entire real axis, since
$$ \left | \frac{\sin nx }{n ^ {2} } \right | \leq \frac{1}{n ^ {2} } $$
and the series
$$ \sum _ { n= } 1 ^ \infty \frac{1}{n ^ {2} } $$
is convergent.
If, for a sequence of real- or complex-valued functions $ {f _ {n} } $, $ n = 1, 2 \dots $ which converges to a function $ f $ on a set $ E $ there exists a sequence of numbers $ \alpha _ {n} $, $ \alpha _ {n} > 0 $ and $ \alpha _ {n} \rightarrow 0 $ as $ n \rightarrow \infty $, such that $ | f( x) - {f _ {n} } ( x) | \leq \alpha _ {n} $, $ x \in E $, $ n = 1, 2 \dots $ then the sequence converges uniformly on $ E $. For instance, the sequence
$$ f _ {n} ( x) = 1 - \frac{(- 1) ^ {n} }{x ^ {2} + n } $$
converges uniformly to the function $ f( x) = 1 $ on the entire real axis, since
$$ | 1- f _ {n} ( x) | < \frac{1}{n} \ \textrm{ and } \ \lim\limits _ {n \rightarrow \infty } \ \frac{1}{n} = 0 . $$
The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces.
References
[1a] | K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866) |
[1b] | K. Weierstrass, "Math. Werke" , 1–7 , G. Olms & Johnson, reprint (1927) |
Comments
References
[a1] | T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974) |
[a2] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
[a3] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 |
Weierstrass criterion (for uniform convergence). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_criterion_(for_uniform_convergence)&oldid=19294