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However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere. It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by [9]
The series can be compared to an integral to establish convergence or divergence. Let : [,) + be a non-negative and monotonically decreasing function such that () =.If = <, then the series converges.
The continuous mapping theorem states that for a continuous function g, if the sequence {X n} converges in distribution to X, then {g(X n)} converges in distribution to g(X). Note however that convergence in distribution of {X n} to X and {Y n} to Y does in general not imply convergence in distribution of {X n + Y n} to X + Y or of {X n Y n} to XY.
A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.
A sequence of discretized approximations () of some continuous-domain function that converges to this target, together with a corresponding sequence of discretization scale parameters () that converge to 0, is said to have asymptotic order of convergence and asymptotic rate of convergence if
If diverges and converges, then necessarily =, that is, =. The essential content here is that in some sense the numbers a n {\displaystyle a_{n}} are larger than the numbers b n {\displaystyle b_{n}} .
First we want to show that (X n, c) converges in distribution to (X, c). By the portmanteau lemma this will be true if we can show that E[f(X n, c)] → E[f(X, c)] for any bounded continuous function f(x, y). So let f be such arbitrary bounded continuous function. Now consider the function of a single variable g(x) := f(x, c).
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.