Geometry and Topology of Continuous Best and Near Best Approximations

Abstract

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints. Academic Press

Convex geometry and nonlinear approximation

“…In [lo,Theorem 4.21, we found an upper bound on the diameter of the set of best approximants in terms related to the modulus of convexity and the degree to which a near best approximation can be shown to exist, provided that the approximating subset (assumed closed) has compact intersection with every closed ball. This applies to neural manifolds defined by a compact set of parameters.…”

Integral combinations of Heavisides

“…The above expressions are identical, except for the omitted factor a d , with (14), (15), and (21) above since dy ⊥ = d H y, and ∂ ∂y1 and its iterates are directional derivatives in the direction of e, i.e., normal to the hyperplane H e,b . Shifting the partials is permitted if we show that…”

“…Then we define a sequence of functions − y)) dy, the last formula holding provided |α| ≤ d. Since f and all of its derivatives of order ≤ d vanish at infinity, it is straightforward to show that f n converges uniformly to f on R d and ∂ α f n likewise converges uniformly to ∂ α f on R d for |α| ≤ d. If the functions {f n } satisfy the integral formula (4) with w fn as in (15), then f will satisfy this integral formula with w f as in (15).…”

Best approximation by linear combinations of characteristic functions of half-spaces

“…For the proof and extensions to non-strictly convex spaces see Kainen, K urkov a & Vogt (1999), (2000.…”