Solarity and Proximinality in Generalized Rational Approximation in Spaces $$C(Q)$$ and $$L^p$$

Existence, uniqueness, and stability of best and near-best approximations

“…The stability of near-best approximation operators was studied by Berdyshev [12], [13], Liskovets, Marinov, Morozov, Wegmann, and others (see, for example, [40], [50]) with applications, in particular, to ill-posed problems. Various results on sets admitting stable ε-selections and on the stability of minimization problems can be found in [3], [32], [8]- [11], [36], [52], [53], [60], [69], and [70]. The book [48] by Repovš and Semenov is also worth mentioning, which surveys results on continuous selections of set-valued mappings.…”

“…Remark 3.2. The set of generalized rational functions R U (with convex U ; see (5.3) below) in C(Q) is stably monotone path-connected and therefore admits a continuous additive ε-selection for any ε > 0 (see Theorem 2.3 in [8]).…”

“…This shows that the set P U is B-and B-contractible in the asymmetric space C ψ+,ψ− (Q) and therefore, is B-infinitely connected. Hence P U admits a continuous additive ε-selection for all ε > 0 (for further details, see [8] and § 5 below).…”

“…where P n is the subspace of algebraic polynomials of degree at most n. It is well known that R n,m is a Chebyshev sun in C[a, b] (see, for example, § 2 in [8]). However in L p [a, b], 1 ⩽ p < ∞, Efimov and Stechkin showed, with the help of general theorems of geometric approximation theory, that R n,m , m ⩾ 1, is an existence set but not a uniqueness set.…”

“…here Q is a metrizable compact set, V, W ⊂ C(Q) are convex sets, and W consists of positive functions. It is well known (see, for example, [8]) that R V W is a strict protosun in C(Q).…”