Martin Bartelt

On Poreda's problem on the strong unicity constants

Let c‘(I) denote the space of continuous. real~valued functions on the inter\,al I ICI. 6) endowed with the uniform norm /I 11. Let U,, denote the set of all algebraic polynomials of degree II or less. and let II denote the set of all algebraic polynomials. For a given SF C(I). with best uniform approximation r,,(.f) from n,,. Newman and Shapiro 1 IO] showed that there is a constant ;’ > 0 such that

Lipschitz conditions, strong uniqueness, and almost Chebyshev subspaces of C(X)

Abstract For a finite dimensional subspace M of C ( X ), X a compact metric space, it is well known that the (set valued) metric projection P M is (Hausdorff) continuous at any f ϵ C ( X ) having a unique best approximation from M and is point Lipschitz continuous at any f ϵ C ( X ) having a strongly unique best approximation from M . The converses of these classical results are studied. It is shown that if f has a unique best approximation and P M is point Lipschitzian at f , then f has a strongly unique best approximation. If M is an almost Chebyshev subspace of C ( X ), then the converses of both statements above are shown to hold. Using a theorem of Garkavi, the validity of these converses actually characterizes the almost Chebyshev subspaces of C(X) .

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

Abstract.In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities.

Uniform Lipschitz constants in Chebyshev polynomial approximation

Let M denote an n-dimensional Haar subspace of C[a,b]. In Chebyshev approximation, any f∈C[a,b] has a strongly unique best approximation B(f) from M, i.e. there exists a number γ>0 such that ∥f−m∥≥∥f−B(f)∥+γ∥B(f)−m∥ for all m∈M, where γ is taken to be the largest such number. Here γ=γ(f, M,n). There have been many results on the existence of uniform strong unicity constants which are independent of n, f, or M; we are here concerned with uniformity with respect to f

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of CX,R^k, the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in CX,R^k the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Holder) condition of order 12. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in CX,R^k.

An Elementary Extension of Tietze's Theorem

Here C(X) and C(A) are the sets of continuous real-valued functions on X and A, respectively. F is an extension of f means F(x) =f(x) if x E A. While Theorem T is included in almost any text on point-set topology, none of the many books we surveyed mentions anything more general than Theorem T, either as a corollary or as an exercise. Of course, some texts observe that by translation one can extend a function f satisfying M1 <f(x) < M2, x E A, to a ftinction F satisfying M1 < F(x) < M2, x E X when M1 and M2 are any two constants, not just M2 = c = M as given in Theorem T. It should be observed that the original Tietze Theorem was stated for metric spaces and later generalized by Urysohn to normal Hausdorff spaces. Also, some extensions of Theorem T different from what is presented here appear in the literature (cf. [2]). This note considers the following question: Let A be a closed subset of X and assume g, h E C(X) such that g(x) < h(x) for every x E X. If f is a function in C(A) lying between g and h, i.e. g(x) <f(x) < h(x) for every x in A, can f be extended to a function F in C(X) lying between the same two functions? The following easy exercises lead directly from Theorem T to an affirmative answer to the question. However, we show later that there is a more general version that can be verified immediately from Theorem T.

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Holder) condition of order 12. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.