PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 82, Number 4, August 1981 ELEMENTARY PROOF OF THE RUDIN-CARLESON AND THE F. AND M. RIESZ THEOREMS RAOUF DOSS Abstract. A very elementary proof is given of the theorem that on a set of measure zero on T, any continuous function is equal to a continuous function of analytic type. The same elementary method proves that a measure of analytic type is absolutely continuous. A complex Borel measure 11on T, in particular an/ analytic type if an = i»-1 [ e-* dMO = 0, G L\T), is said to be of n = -1, -2, . . T The theorems mentioned in the title are: Rudin-Carleson Theorem. Let F be a closed subset of T of Lebesgue measure zero. If q> is a continuous function on F, then there is a continuous function f, of analytic type, such that At) = <p(t), t e F, sup|/(/)|<A/sup|<p(i)| (*) where M is a constant. (Rudin proves that M = 1. See [8] and [1].) The First F. and M. Riesz Theorem. If the function f in Ll(T) is of analytic type and if f vanishes on a set S* of positive measure, then f = 0. The Second F. and M. Riesz Theorem. If a complex Borel measure ¡i on T is of analytic type, then /x is absolutely continuous {with respect to Lebesgue measure). See [7]. The proofs of these theorems most often use boundary values of functions analytic in the unit disc and the theory of //^-spaces. For the Second F. and M. Riesz Theorem, for example, see three variants in [3], [5] and [9]; other proofs of that theorem use Hilbert-space theory: see e.g. [2] and [4]; a direct short proof is given in [6]. The aim of the present paper is to present a method which gives an elementary proof of all the above theorems. Received by the editors August 7, 1980; presented to the "Séminaire d'Analyse Harmonique d'Orsay", June 13, 1980. 1980 Mathematics Subject Classification. Primary 42A68, 42A72; Secondary 26A15. © 1981 American Mathematical 0002-9939/81/0000-0367/$02.00 599 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Society 600 RAOUF DOSS Lemma. Let F be a closed subset of T of measure zero and <pa continuous function on F. Given e > 0 and an open set G D F there is a continuous function g of analytic type such that sup|g(r) - <p(f)| < esup|<p(i)|, teF teF \g(t)\<e, t£G, sup|g(r)| < 3sup|<p(i)|. ter (♦*) teF Proof. Without loss of generality we may assume that sup,ef|<p(f)| = 1 and also that (p is a trigonometric polynomial <p(t)= 2 oLkeikt \k\<m such that \<p(t)\< e/3, t 0: G. Let e~A = e and let A be a continuous function on T, lying between -2A and 2e, such that \h(t) + 2A\<e, teF. Since m(F) = 0 we may take ||A||, arbitrarily small and hence we may suppose h(k) = 0, |fc| < m. Take a Fejér sump of h such that \p(t) + 2A\ < e, t S F. We write />(')= 2 ßkeik,+ 2 ß^-p-W+p+O) k<.-m k>m where P+(t)= S ßkeikt. k>m We have Re(/>+) = p/2 < s. Put now g(i) = <p(/)[l - ep+(,)]. The expansion of [1 — ep*W] is of the form "Zk>m y^"". The function g is therefore continuous of analytic type. We have \g(t) - <p(')l= MOI K+W| < e»'1 < eA+< <2e ((G F). Moreover ls(')l < IqpCOl |i - e'^\ < i + e' < 3 |g(r)|<(e/3)3 = e ((er). (i £ G). The Lemma is now proved. Proof of the Rudin-Carleson Theorem, e < 5 being fixed, denote by v(<p)any continuous function of analytic type associated to <pby the Lemma. Starting with <p0= <pwe put <Pm+i= <P«- r(<PJ- We have sup|«pm+1| < e sup|<pj < • • • < em+1sup|<p0|, FF sup|v(<pj| T F < 3sup|<pJ F < 3emsup|<p0|. F License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use RUDIN-CARLESON AND F. AND M. RIESZ THEOREMS 601 The series 2^_0 y(<pm)is therefore uniformally convergent on T; its sum / is of analytic type and satisfies the relation/(i) = <p(t) (i e F). Moreover sup |/(/)| < 3(1 — e)~ sup|<p0| < 4sup|<p|. T F The theorem is now proved. Remark. The factor M in the estimate (*) can easily be reduced to 1 + e. In fact, given an open set G D F and using (**) we can manage to have |/(0I<* By the continuity (t$G). of /, there is an open set G' D F such that G' d G and |/(0| < 1 + e (t S G'). Thus we can have |/(0| > 1 + e only if t S G \ G'. Starting with G' we get/' coinciding with <pon F, bounded by 4 where |/'(0| > 1 + e only if t e G' \ G" for an appropriate G" ^ F, with G"cC. Observing that the sets G \ G', G' \ G", G" \ G'", . . . are disjoint and taking an arithmetic mean we get a function bounded everywhere by 1 + 2e. Proof of the First F. and M. Riesz Theorem. It is sufficient to prove that a0 = (27T)-1f f(t) dt = 0 jt for, applying the same process to the function e~"f(t), we deduce a, = 0, and next a2 = 0, . . . and finally / = 0. We shall follow the same pattern of proof as for the Rudin-Carleson Theorem. Denote by 5 the set {(G T: /(/) ¥= 0}. Given e > 0 let e~A = e and let A be a bounded real function equal to -2A on S and such that h(0) = 0. There are such functions since m(S*) > 0. Let p„ be the sequence of Fejér polynomials of h. We write as before PÂ0 = 2 ßkeik' + S ßkeik'= p-(t) + P;(t) k<0 k>0 where p:o) = 2 ßkeik'. k>0 Then, boundedly, Re(/C(0) = ift(0 ^5A(') = ~A a-e-on SPut now &,(0=/(0[l-e'"+(O]The expansion of gn is of the form 2fc>0 yke'kt and therefore /g„ í/í = 0. Hence |2™0|=|//j = |/(/-g„)|<|j/^"+| <Jj/|e^/2^e->.J|/| = e||/||i. Since e is arbitrary we have a0 = 0 and the theorem is proved. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 602 Proof F be a G„ D F analytic RAOUF DOSS of the Second F. and M. Riesz Theorem. We may assume a0 = 0. Let closed set of measure zero. Choose a decreasing sequence of open sets such that D G„ = F, and by the Lemma a sequence of functions g„ of type, such that |1 - g„(0| < \/n, |S„I<3; t S F, |g„(0l<l/n fori^G„. Then, boundedly, g„ -» Xf (characteristic function of F). Hence 0 = jg„ dp -* JXf d¡i = n(F). This proves that /i is absolutely continuous. The author would like to thank Y. Katznelson for a useful conversation. References 1. L. Carleson, Representations of continuous 2. R. G. Douglas, Banach algebra techniques 3. P. Duren, Theory of Hp-spaces, Academic 4. K. Hoffman, Banach spaces of analytic functions, Math. Z. 66 (1957), 447-451. in operator theory, Academic Press, New York, 1972. Press, New York, 1970. functions, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 5. Y. Katznelson, An introduction to harmonic analysis, Dover, New York, 1976. 6. B. 0ksendal, A short proof of the F. and M. Riesz Theorem, Proc. Amer. Math. Soc. 30 (1971), 204. 7. F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion, 4e Congrès des Mathématiciens Scandinaves (Stockholm, 1916), pp. 27-44. 8. W. Rudin, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc. 7 (1956), 808-811. 9. _, Real and complex analysis, McGraw-Hill, New York, 1974. Department of Mathematics, State University of New York, Stony Brook, New York 11794 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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