Vol. 31, No. 1, 1969

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Some theorems in Fourier analysis on symmetric sets

Robert Bruce Schneider

Vol. 31 (1969), No. 1, 175–195

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 ={ Σ1𝜖jγ(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.

Mathematical Subject Classification
Primary: 42.58
Received: 22 April 1968
Published: 1 October 1969
Robert Bruce Schneider