Journal ArticleDOI

# On generalized exponential functions

01 Jan 1953-Studia Mathematica (Institute of Mathematics Polish Academy of Sciences)-Vol. 13, Iss: 1, pp 48-50

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Journal ArticleDOI
01 Jan 1988
TL;DR: In this article, the authors give an elementary proof of Titchmarsh's theorem for any T, using no machinery beyond Fubini's theorem and Parseval's formula for trigonometric series.
Abstract: 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

17 citations

### Cites background from "On generalized exponential function..."

• ...Jo An integration by parts in (4) then gives...

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• ...rit (4) where h= if = f e<2t-^e-^ht(y)dy JJA' J2t...

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Book ChapterDOI
01 Jan 2019
TL;DR: In this article, a proof of Pick's theorem, Theorem 16.4, due to Sarason [294], that relies on the Hardy spaces H2 and H∞ is presented.
Abstract: 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∞.

1 citations