Uniform Convergence

(redirected from Converges uniformly)

uniform convergence

[′yü·nə‚fȯrm kən′vər·jəns]
A sequence of functions {ƒn (x)} converges uniformly on E to ƒ(x) if given ε > 0 there is an N such that |ƒn (x) - ƒ(x)| < ε="" for="" all="">x in E provided n > N.

Uniform Convergence


an important special case of convergence. A sequence of functions fn (x) (n = 1, 2, 3,…) is said to converge uniformly on a given set to the limit function f(x) if, for every ∊ > 0, there exists a number N = N(∊) such that, when η > N, ǀ f(x) – fn (x)ǀ < ∊ for all points x in the set.

For example, the sequence of functions fn(x) = xn converges uniformly on the closed interval [0, 1 /2] to the limit f(x) = 0. To show that this is true, let us take n > In (1/∊)/In 2. It then follows that ǀf(x)–fn (x)ǀ ≤ (1/2)n < ∊ for all x, 0 ≤ x ≤ 1/2. This sequence of functions, however, does not converge uniformly on the interval [0, 1]. Here, the limit function is f(x) = 0 for 0 ≤ x< 1 and f(1) = 1. The reason for the failure to converge uniformly is that for arbitarily large η there exist points η that satisfy the inequalities Uniform Convergence and for which ǀf (η) - fn(η)ǀ = ηn > 1/2.

The notion of uniform convergence admits of a simple geometric interpretation. The uniform convergence of a sequence of functions fn(x) on some closed interval to the function f(x) means that, for any ∊ > 0, all curves y = fn (x) with large enough n will be located within a strip that is 2∊ in width and is bounded by the curves y = f(x) ± ∊ for any x in the interval (see Figure 1).

Figure 1

Uniformly converging sequences of functions have a number of important properties. For example, the limit of a uniformly converging sequence of continuous functions is also continuous. On the other hand, the example given above shows that the limit of a sequence of functions that does not converge uniformly may be discontinuous. An important role is played in mathematical analysis by Weierstrass’ theorem, which states that every function continuous on a closed interval can be represented as the limit of a uniformly converging sequence of polynomials. (SeeAPPROXIMATION AND INTERPOLATION OF FUNCTIONS.)

References in periodicals archive ?
The Poincare series converges uniformly for any multiply connected domain without any geometrical restriction [14].
Hence, for a = 1 there exists no approximation process that converges uniformly on R to Df for all f [member of] [PW.
phi]]f converges uniformly on R to D f for a > 1 and [phi] [member of] M(a).
We show that under suitable conditions on f, there exists a monotone sequence of solutions of linear problems that converges uniformly and rapidly to unique solution of the original nonlinear problem.
We develop the approximation scheme and show that under suitable conditions on f, there exists a bounded monotone sequence of solutions of linear problems that converges uniformly to a solution of the original problem.
This implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges uniformly on closed balls of X to [d.
converges uniformly on [absolute value of z] < R for all R < [infinity] and [phi] is an entire function of exponential type [pi] [19].
omega) fulfills certain integrability conditions in the vicinity of [omega] = [pi], even the non-symmetric Shannon sampling series converges uniformly on whole of R.
OMEGA] version of f on [- L, L] converges uniformly to f on any strip parallel to R in the complex plane.
Note that this corollary implies the weaker result that the sequence {fN}N[member of]N converges uniformly to the original strictly bandlimited f on any compact subset of C.
2) converges uniformly o,1 any subset over which ||k(t)||H is bounded.