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The proof of Carleson’s theorem to be given below was worked up in response to a particular teaching situation. About a year and a half ago, I was assigned our department’s first (semester) course on functional analysis, normally devoted to the three principles of that subject and to some of their applications. My class consisted of quite good students with an adequate preparation in measure theory and integration but only a rather scanty knowlege of analytic functions; I desired nevetheless to show them how Carleson’s result could be made to follow from the rudiments of the subject. The problem with this was that the students could not be expected to know anything about H p spaces or their duality, and class time did not allow for any preliminary discussion of that material.

The theme of this work, the logarithmic integral, lies athwart much of twentieth-century analysis. It is a thread connecting many apparently separate parts of the subject, and so is a natural point at which to begin a serious study of real and complex analysis. Professor Koosis' aim is to show how, from simple ideas, one can build up an investigation which explains and clarifies many different, seemingly unrelated problems; to show, in effect, how mathematics grows. The presentation is straightforward, so that by following the theme, Professor Koosis has produced a work that can be read as a whole. He has brought together here many results, some unpublished, some new, and some available only in inaccessible journals.

A certain minimal extrapolation problem for Fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators. In this paper we estimate the value of the constant in the Fourier-transform problem, by an analytic reformulation.

Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)

1. S. P. Diliberto, On systems of ordinary differential equations, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, no. 20, Princeton, 1950, pp. 1-38. 2. , A note on linear ordinary differential equations, Proc. Amer. Math. Soc. vol. 8 (1957) pp. 462-464. 3. 0. Perron, Ober eine Matrixtransformation, Math. Zeit. vol. 32 (1930) pp. 465473. 4. W. T. Reid, Some results on the growth of solutions of differential systems, Bull. Amer. Math. Soc. Abstract 48-11-319.