In [6] Becker and Rosenberg defined the reduced Witt ring WAK) of a formally real field K, with respect to a higher level preorder Tin K. WAK) is a subring of the ring C(X,, @) of continuous c-valued functions on the space X, of signatures of K which are trivial on T, and is built up from higher level T-forms over K. This higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields. Spaces of Signatures were introduced in [18] to provide an abstract framework for the higher level reduced theory over fields, along the lines of Marshall’s Spaces of Orderings [13-H] treatment of the classical reduced theory of quadratic forms. There are many advantages to this axiomatic approach. It allows for a unified study of reduced Witt rings over fields and skew fields (Powers has shown that Becker and Rosenberg’s study of higher level reduced Witt rings can also be carried out over skew fields [22]). The absence of technical considerations such as valuation theory not only yields simpler proofs of known results, but also new results about the theory over fields [ 183. The Space of Orderings version [ 151 of the result which we prove here has found application in real algebraic geometry, and we hope that our abstract higher level result might also prove useful in this context. The question which concerns us here is the so-called representation problem: given FE C(X, @), how can we tell if F=f for some f~ W(X)? This has been solved for the higher level reduced Witt ring of a field by Becker and Rosenberg [6] (see [S] also), and in the case of Spaces of Orderings by Marshall [15], both solutions being generalizations of the original representation theorems of Becker and Brlicker [4, 5.31 and Brown and Marshall [9, 4.21 (in the context of the reduced quadratic form theory over fields). Under a 2-power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting. This entails studying fans and more importantly a generalization called quasifans. We not only obtain a common proof of the representation 105 0021-8693/88 $3.00