Weierstrass criterion (for uniform convergence)
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
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of real- or complex-valued functions defined on some set there exists a convergent series of non-negative numbers
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such that
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then the initial series converges uniformly and absolutely on the set (cf. Absolutely convergent series). For instance, the series
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converges uniformly and absolutely on the entire real axis, since
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and the series
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is convergent.
If, for a sequence of real- or complex-valued functions ,
which converges to a function
on a set
there exists a sequence of numbers
,
and
as
, such that
,
,
then the sequence converges uniformly on
. For instance, the sequence
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converges uniformly to the function on the entire real axis, since
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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=49190