Let R be the real line and
A = A(R) the space of continuous functions on R which are the Fourier transforms of
functions in L1(R).A(R) is a Banach Algebra when it is given the L1(R) norm. For a
closed F ⊆ R one defines A(F) as the restrictions of f ∈ A to F with the norm of
g ∈ A(F) the infimum of the norms of elements of A whose restrictions are g. Let
Fr⊆ R be of the form Fγ={ Σ1∞𝜖jγ(j) : 𝜖j either 0 or 1}. This paper shows that if
Σ(7⋅(j + 1)∕r(j))2< ∞ and ∑(s(j + 1)∕s(j))2< ∞ then A(Fr) is isomorphic to
A(Fs). We also show that, in some sense square summability is the best possible
criterion. In the course of the proof we show that Fr is a set of synthesis and
uniqueness if ∑(r(j + 1)∕r(j))2< ∞. This is almost a converse to a theorem of
Salem.
