Spans in Lebesgue and Uniform Spaces of Translations of Step Functions

For each p^ 1, Lp is a linear metric complete separable space. Let E be a set in Lp. The linear manifold M(E) determined by E is the set of all linear combinations (finite) of elements of £ . The span SP(E) of E in Lp is the closure in Lp of M (E) ; an element cj> of Lp belongs to SP(E) if and only if to each e > 0 corresponds an element f€ of M(E) such that | | 0 /« | | <e . L e t / G L p . For each real A, the translation f (x+h) of f(x) is also in Lp. Let 7/ denote the set of translations of/. Wiener [2, pp. 7-9] showed that i f /GL 2 , then ^ ( T / ) is the whole space L2 if and only if the real zeros of the Fourier transform of ƒ form a set of measure 0. He [2, pp. 9-19] showed also (and this was much more difficult) that if / G i i i then Si(T/) is the whole space L\ if and only if the Fourier transform of ƒ has no real zeros. He [2, p. 93] raised the question whether similar propositions hold for other values of p and expressed a "suspicion" that they do, at least when 1^PS2. In view of the similar suspicions held by Wiener and others, a result recently announced by Segal [l ] is surprising. Segal has shown that if Kp < 2 , then there is an element/of Lp such that (i) the zeros of the Fourier transform of ƒ form a set of measure 0 and (ii) the span Sp(Tf) of the translations of ƒ does not include the whole space Lp. This development will doubtless create interest in the search for criteria for Sp(Tf) = Lp. With the hope that both the result and the