Uniform Convergence (redirected from Cauchy criterion for uniform convergence)

uniform convergence [′yü·nə‚fȯrm kən′vər·jəns]

(mathematics)

A sequence of functions {ƒ_n(x)} converges uniformly onEto ƒ(x) if given ε > 0 there is anNsuch that |ƒ_n(x) - ƒ(x)| < ε for allxinEprovidedn>N.

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Uniform Convergence 

an important special case of convergence. A sequence of functions f_n (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) – f_n (x)ǀ < ∊ for all points x in the set.

For example, the sequence of functions f_n(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)–f_n (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 and for which ǀf (η) - f_n(η)ǀ = ηⁿ > 1/2.

The notion of uniform convergence admits of a simple geometric interpretation. The uniform convergence of a sequence of functions f_n(x) on some closed interval to the function f(x) means that, for any ∊ > 0, all curves y = f_n (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.)