Frank Deutsch

The Angle Between Subspaces of a Hilbert Space

This is a mainly expository paper concerning two different definitions of the angle between a pair of subspaces of a Hilbert space, certain basic results which hold for these angles, and a few of the many applications of these notions. The latter include the rate of convergence of the method of cyclic projections, existence and uniqueness of abstract splines, and the product of operators with closed range.

The Method of Alternating Orthogonal Projections

The method of alternating orthogonal projections is discussed, and some of its many and diverse applications are described.

The rate of convergence of dykstra's cyclic projections algorithm: The polyhedral case

Suppose K is the intersection of a finite number of closed half-spaces in a Hilbert space X. Starting with any point xeX, it is shown that the sequence of iterates {x n } generated by Dykstras cyclic projections algorithm satisfies the inequality for all n, where P K (x) is the nearest point in K to x;, ρ is a constant, and 0 ≤c<1. In the case when K is the intersection of just two closed half-spaces, a stronger result is established: the sequence of iterates is either finite or satisfies for all n, where c is the cosine of the angle between the two functionals which define the half-spaces. Moreover, the constant c is the best possible. Applications are made to isotone and convex regression, and linear and quadratic programming.

Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections

Two new continuity properties for set-valued mappings are defined which are weaker than lower semicontinuity. One of these properties characterizes when approximate selections exist. A few selection theorems characterized by the other property are established. Some applications are made to set-valued metric projections.

Constrained best approximation in Hilbert space

In this paper we study the characterization of the solution to the extremal problem inf{‖x‖x ∈C ∩M}, wherex is in a Hilbert spaceH, C is a convex cone, andM is a translate of a subspace ofH determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated withM is an interior point of the data cone.

Rate of Convergence of the Method of Alternating Projections

A proof is given of a rate of convergence theorem for the method of alternating projections. The theorem had been announced earlier in [8] without proof.

Constrained best approximation in Hilbert space, II

In this paper we study the characterization of the solution to the extremal problem inf{‖x‖x ∈C ∩M}, wherex is in a Hilbert spaceH, C is a convex cone, andM is a translate of a subspace ofH determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated withM is an interior point of the data cone.

On some geometric properties of suns

BRUNO BROSOWSK~ Gesellschaft fiir wissenschaftliche Datenverarbeitung m.b.H. Cattingen, 34 Gtittingen-Nikolausberg, Am Fassberg, West German-v AND FRANK DEUTSCH Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802 Communicated by L. Collatz 1. INTRODUCTION

Strong CHIP, normality, and linear regularity of convex sets

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a. characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets.

Radial continuity of set-valued metric projections

Some new continuity concepts for metric projections are introduced which are simpler and more general than the usual upper and lower semicontinuity. These concepts are strong enough to generalize a number of known results yet weak enough so that now the converses of many of these generalizations are also valid. In particular, in a large class of normed linear spaces, suns and Chebychev sets can be characterized by a certain continuity property of their metric projections.