7 - Closure

Summary

Let K be a bounded set in Rⁿ. For a given d × n matrix A, set

M(A;K) := {f(Ax) : x ∊ K, all possible f : R^d → R},

and for each p ∊ [1, ∞], let

L^p(A; K) := M(A; K) ∩ L^p(K).

That is, L^p(A; K) is the set of functions in L^p(K) of the form f(Ax). In this chapter we consider the question of the closure of

in L^p(K). This question has been studied with an eye to computerized tomography. It is relevant to the problem of the existence of best approximations, see Chapter 8, and also to the rate of convergence of some best approximation algorithms, see Chapter 9. It is also an interesting question in and of itself. In Section 7.1 we present various theorems providing conditions for when closure holds, and in Section 7.2 some examples of when closure is lacking. Some additional results are given in Section 7.3. In Section 7.4 we consider the case of C(K). For two directions closure in C(K) is equivalent to certain geometric properties of the set K and the directions. However, nothing seems to be known in the case of more than two directions. This chapter is more of a survey as the theorems are presented without proofs. The inclusion of their proofs would lead us too far afield.

The space L^p(K), p ∊ [1, ∞), is the standard linear space of functions G defined on K ⊆ Rⁿ for which |G|^p is Lebesgue integrable, and with norm given by

L^∞(K) is the space of Lebesgue measurable, essentially bounded functions G, with norm

The first general results on closure seem to be due to Hamaker and Solmon [1978] who considered the case where p = 2, K is a disk with center at the origin in R², and is a set of pairwise linearly independent vectors in R².