A Haar-type theory of best uniform approximation with constraints

A Haar-type theory of best 𝐿₁-approximation with constraints

“…We note that in the uniform norm setting analogous results hold when Aq > 0 [6]. When aci = 0, uniqueness of a best constrained uniform approximation requires that the function / being approximated satisfy the constraint vx < f <ux (see [2,12]).…”

“…It turns out that our theory can be widely applied for different operators and spaces of polynomial and spline functions. Finally, let us mention that a similar study of constrained approximation in the Loo-norm is given in our recent paper [6].…”

“…The L-Haar and L'-Haar properties were found in [6] to completely characterize those subspaces U" of C[a, b] for which U"(v , u) admits unique best uniform approximations to all / e C[a, b] for all continuous or smooth, respectively, boundary functions v and u with \ntUn(v, u) ^ <j>. Among other conditions, the L-Haar property was found to be equivalent to G{y¡y being a Haar space for all {y¡}si=x Q K with dim LUn\{y¡y =s.…”

“…It turns out that L°-^-property is quite restrictive and we shall illustrate this in §3.1 by providing a "piecewise" bound on the dimension of L°-^4-spaces. Never-the-less, in Lx -approximation the C-boundary case is not as restrictive as in uniform approximation (see [6]). We shall present in §3.2 a certain family of spline functions providing a useful example of L°-v4-spaces.…”

“…As was previously mentioned, the L°-y4-property is restrictive but not as restrictive as the L-Haar property for uniform approximation. In [6], it was found that if L is a nontrivial Ac-Rolle operator on U" and Un is L-Haar, then dim Un = n < k + 1. Although the "local dimensions" of L°-^4-spaces are bounded by Ac + 1 when L is Ac-Rolle, the example of this section demonstrates that for L -Dk (k > 1) there are L°-^-spaces of arbitrarily large dimension.…”