A functional analysis proof of titchmarsh's theorem on convolution

We give an elementary proof of the following theorem of Titchmarsh. Suppose f, g are integrable on the interval (0, 2T) and that the convolution f * g(t) = fo f (t x)g(x) dx = 0 on (0, 2T). Then there are nonnegative numbers a, : with a + d > 2T for which f(x) = 0 for almost all x in (0, a) and g(x) = 0 for almost all x in (0,13). Suppose f, g are integrable on the interval (0, 2T). If f = 0 a.e. on (0, a), g = 0 a.e. on (0, A) with ae + = 2T, then the convolution rt f * g(t) =jf(t-x)g(x) dx = 0 for 0 2T for which f(x) = 0 for almost all a in (0, a) and g(x) = 0 for almost all al in (0, A). There are many proofs of this theorem. The first three: Titchmarsh [6], Crum [1] and Dufresnoy [2] were based on the theory of analytic or harmonic functions. An elaborate real variable proof was later given by Mikusinski and Ryll-Nardzewski in [4, (a), (b), (c)]. An entirely different proof, now classical, by the same authors, appears in [5]. Unfortunately it is valid only for T = 00. Recently, Helson [3] gave an elegant proof using the theory of Hardy's HP(R) spaces and invariant subspaces. The present proof, valid for any T, is elementary, using no machinery beyond Fubini's theorem and Parseval's formula for trigonometric series. Put F(x) = ftx f (s) ds. Then, by Fubini's theorem rt rU t t rt 0= j J f(u x)g(x) dx du = J J f(u x)g(x) du dx = J F(t x)g(x) dx. This is F * g(t) = 0, t E (0, 2T). Similarly, putting G(x) = fx g(s) ds we get F * G(t) = 0, t E (0, 2T). Now if Titchmarsh's theorem is true for the functions F, G, it is also true for f, g. Hence, we may assume, in the proof of Titchmarsh 's Theorem, that f, g are several times differentiable and satisfy e.g. relations like f' * g = 0 and g(0) = 0. LEMMA 1. Suppose that h is continuous on (0, 2T) and that (1) j 2te2n(t-x)h(x) dx < Ctn1/2 Received by the editors August 7, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 42A85; Secondary 45E10. (?)1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

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An elementary proof of Titchmarsh’s convolution theorem

In this chapter, we’ll present a proof of Pick’s theorem, Theorem 16.4, due to Sarason [294], that relies on the Hardy spaces H2 and H∞.

Pick Interpolation, IV: Commutant Lifting Proof

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J. G.-Mikusiński

Papers

On generalized exponential functions