Title:

Weight functions on the torus and the approximation property in Banach spaces

This thesis is divided into two distinct and independent parts. Part 1 concerns the Approximation Property (a.p.) and Radon Nikodym Property (RNP) in Banach Spaces. In Chapter 1 we outline the importance of the a.p. and produce examples of Banach Spaces without the a.p. by modifying a construction due to Szankowski. These spaces are closed subspaces of Rp direct sums of finite dimensional Rq spaces (1 < q < p < ), so with p < 2 we obtain Banach spaces of cotype 2 without the a.p.  this was unknown. In Chapter 2 we discuss the IMP proving in Theorem 2.9 the characterisation in terms of dentable subsets due to Rieffel and Huff (among others), of Banach spaces with the RNP. In theorem 2.18 we prove that dual spaces with the a.p. and RNP have the metric approximation property, obtaining as corollaries results of Grothendieck. We introduce p nuclear and p integral maps between Banach spaces E and F and prove in theorem 2.26 that, if E* has the RNP, all p integral maps are p nuclear, and in theorem 2.29 that, if F has the RNP all integral maps are nuclear. This extends work of Grothendieck, Perrson and Pietsch. Part 2 concerns the prediction theory of doubly stationary processes. In Chapter 3 we outline the basic prediction theory, and state, for the absolutely continuous case, Helson and Lowdenslager's characterisation, for a weight function w and an irrational a, of a process as type 1, 2 or 3. We give an example of a process of type 2, for all irrational a. In Chapter 4 we obtain in Theorem 4.10 an exact analogue of Helson and Szego's result, viz. that the past and future of a process are at positive angle if and only if dµ = wda , w = exp(u + v), where u, v are real L~ functions with null < ~ . We introduce a class of functions  BMO(a) functions, analogous to BMO functions, and prove BMO(a) is the dual of H1 (a) and {u + v u, v c L (a)} = BMO(a) in Theorems 4.19 and 4.20.
