Weighted trigonometric approximation and inner—Outer functions on higher dimensional Euclidean spaces

Abstract

In this paper we consider a class of weighted trigonometric approximation problems in ip,(~%‘~). Many of these questions go back to the classic papers of Beurling [ 1,2]. One may describe the general problem as that of finding the best approximation to an arbitrary function from the closed subspace generated by functions of the form er@‘.‘+‘)f(w), where f is a fixed function and where the arguments u are to be taken from a given set S. For 9z(9”) and S a half-line, these problems, and their solution, are intimately related to the theory of inner and outer functions. The latter theory does not exist for 9”, n > 2. However, by use of functional analytic and group theoretic methods we are able to treat the trigonometric approximation problem for 4p2(9”) with S either a half-space or a quadrant. In so doing, we also arrive at a beginning of a new theory of inner and outer functions for 9”. The theory of inner and outer functions also plays a central role in the prediction problem for one-parameter processes; see Dym and McKean [4]. In Gustafson and Misra [5] regular one-parameter processes were characterized in terms of the canonical commutation relations of quantum mechanics. Particular emphasis was placed on the use of cyclic vectors. This approach enables us to investigate the weighted trigonometric approximation problems via several-parameter regular processes without appeal to the theory of several complex variables. As a byproduct, we obtain an abstract characterization of the regular representation of 9”. This answers a question raised by Chatterji [3, p. 241. We also observe a new proof for the existence and uniqueness of outer functions on 9’. This comes out of our (new) formulation and results for inner and outer functions on 9”. The theory of inner and outer functions for compact abelian groups whose dual is linearly ordered has been developed. See, for example, Helson and