Bruce L. Chalmers

The determination of minimal projections and extensions in

Equations are derived which are shown to be necessary and sufficient for finite rank projections in L1 to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of L1 and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the L1 setting. These equations are solved in terms of an L1 best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal L1 projection onto the quadratics.

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L1, so-called regular symmetric subspaces. Using this notion, we show that for any k⩾2, there exist k-dimensional symmetric subspaces of L1 which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L1 is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.

A unified approach to uniform real approximation by polynomials with linear restrictions

Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

Determination of a minimal projection from C[-1, 1] onto the quadratics

In this paper we obtain a solution to the long-standing question of what is a minimal projection from C[—1,1] onto the quadratics. A recently developed general theory characterizing minimal projections (see [2]) is used both for obtaining the form of the minimal projection and for determining the values of the parameters appearing in the form.

Minimal shape-preserving projections onto Π n

In this paper are determined minimal shape–preserving projections onto the n–th degree algebraic polynomials. 1985 Mathematics Subject Classification: Primary 46B20; Secondary 47A30.

Convex Lp approximation

In [7] Karlovitz developed an algorithm for finding best Lp approximations from certain finite-dimensional subspaces for p an even integer. It will be shown that this algorithm converges for 2 < p < co when the approximating subspace is replaced by certain closed convex subsets of a linitedimensional subspace. Furthermore, the restrictions which must be placed on the functions involved to ensure convergence are weakened. We shall consider the problem of approximating 0 by elements of a closed nonempty convex subset K which is contained in a finite-dimensional subspace and which does not contain 0. This is seen to be equivalent to the general problem of approximating a function f by elements of a closed convex subset G, with f 65 G and G contained in a finite-dimensional subspace, by simply translating all functions involved by -f. That is,

Extension constants of unconditional two-dimensional operators

It is shown that the (absolute) extension constant e(T) of an operator T such that Tvk = λkvk, k = 1, 2, for some unconditional basis (v1, v2) of a two-dimensional real normed space is less than or equal to λ1| + |λ2| + 2√λ21 − |λ1λ2| + λ22)3. In fact, it is demonstrated that e(T) is attained by exactly one unconditional two-dimensional space (up to an isometry).

A unified theory of strong uniqueness in uniform approximation with constraints

Abstract The previously developed unified theory of constrained uniform approximation from a finite dimensional subspace is extended to treat strong uniqueness and continuity of the best approximation operator.

Remarks on the Rank of Hermite–Birkhoff Interpolation

We pose the following extension of the real polynomial Hermite–Birkhoff interpolation problem : Given an incidence matrix, what is the rank of the associated system of point and derivative point evaluations? Bounds are obtained for both first order and general incidence matrices. Several examples provide motivation.

A characterization and equations for minimal shape-preserving projections

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B=B(X, V) the space of all bounded linear operators from X into V; let P(X, V) be the set of all projections in B. For a given cones S ⊂ X, we denote by P=PS(X, V) the set of operators P ∈ P such that PS ⊂ S. When PS ≠ O, we characterize those P ∈ PS for which ||P|| is minimal. This characterization is then utilized in several applications and examples.