Uapproximation

Uapproximation is a term used in approximation theory to denote uniform approximation, i.e., approximating a function as closely as desired uniformly over a domain. The letter U is used as shorthand for uniform in some texts. Given a compact set K and a function f: K → R (or C), a sequence {f_n} provides a U-approximation of f if sup_{x∈K} |f_n(x) - f(x)| → 0 as n → ∞.

One of the foundational results is the Weierstrass approximation theorem: every continuous function on a closed interval [a,b] can be uniformly approximated by polynomials. Constructive proofs yield sequences such as Bernstein polynomials, which converge uniformly to f. For periodic functions on [0, 2π], trigonometric polynomials (Fourier partial sums or Fejér means) provide uniform approximations under suitable regularity. In multiple dimensions, polynomial or spline systems are used for uniform approximation on compact sets.

Rates of uniform approximation are studied through Jackson-type theorems, linking the uniform error to smoothness via the modulus of continuity. Uniform approximation differs from L^p or pointwise approximation and is especially important where worst-case error matters, such as numerical analysis, computer-aided design, and stability analyses. Extensions cover approximation on manifolds, multivariate settings, and adaptive schemes that seek uniform control of the error across a domain.

While closely related to the broader idea of approximation, U-approximation here refers to uniform convergence concepts. Related notions include the uniform norm, Chebyshev (minimax) approximation, and the classical Weierstrass theorem.