Fatou's interpolation theorem implies the Rudin-Carleson theorem
aa r X i v : . [ m a t h . C V ] O c t Fatou’s interpolation theorem implies theRudin-Carleson theorem
Arthur A. DanielyanJuly 30, 2018
Abstract
The purpose of this paper is to show that the Rudin-Carleson interpolation theorem isa direct corollary of Fatou’s much older interpolation theorem (of 1906).
Denote by ∆ and T the open unit disk and the unit circle in the complex plane, respectively.Recall that the disk algebra A is the algebra of all functions on the closed unit disk ∆ thatare analytic on ∆.The following theorem is fundamental; in particular it implies the F. and M. Riesz theo-rem on analytic measures (cf. [5], pp. 28-31). Theorem A (P. Fatou, 1906).
Let E be a closed subset of T such that m ( E ) = 0 ( m isthe Lebesgue measure on T ). Then there exists a function λ E ( z ) in the disk algebra A suchthat λ E ( z ) = 1 on F and | λ E ( z ) | < on T \ E . In its original form Fatou’s theorem states the existence of an element of A which vanishesprecisely on F , but it is equivalent to the above version (cf. [5], p. 30, or [4], pp. 80-81).The following famous theorem, due to W. Rudin [7] and L. Carleson [1], has been thestarting point of many investigations in complex and functional analysis (including severalcomplex variables). Theorem B (Rudin - Carleson).
Let E be a closed set of measure zero on T and let f be a continuous (complex valued) function on E . Then there exists a function g in the diskalgebra A agreeing with f on E.
1t is obvious from Theorem A and Theorem B that for any ǫ > g in Theorem B such that it is bounded by || f || E + ǫ , where || f || E is thesup norm of f on E. Rudin has shown that one can even choose the function g such that itis bounded by || f || E .Quite naturally, as mentioned already by Rudin, Theorem B may be regarded as astrengthened form of Theorem A (cf. [7], p. 808).The present paper shows that Theorem B also is an elementary corollary of TheoremA. To be more specific, we present a brief proof of Theorem B merely using Theorem Aand the Heine - Cantor theorem (from Calculus 1 course); this approach may find furtherapplications. We use a simple argument based on uniform continuity, which has been known(at least since 1930s) in particular to M.A. Lavrentiev [6], M.V. Keldysh, and S.N. Mergelyan,but has not been used for the proof of Theorem B before. Let ǫ > E by disjoint open intervals I k ⊂ T of afinite number n such that | f ( z ) − f ( z ) | < ǫ for any z , z ∈ E ∩ I k ( k = 1 , , ..., n ). Denote E k = E ∩ I k and let λ E k ( z ) be the function provided by Theorem A. Fix a natural number N so large that | λ E k ( z ) | N < ǫn on T \ I k for all k. Fix a point t k ∈ E k for each k and denote h ( z ) = P nk =1 f ( t k )[ λ E k ( z )] N . Obviously the function h ∈ A is bounded on T by the number(1 + ǫ ) || f || E and | f ( z ) − h ( z ) | < ǫ (1 + || f || E ) if z ∈ E . Replacing h by ǫ h allows to assumethat h is bounded on T simply by || f || E and | f ( z ) − h ( z ) | < ǫ (1 + 2 || f || E ) if z ∈ E . Letting ǫ = m provides a sequence { h m } , h m ∈ A , which is uniformly bounded on T by || f || E anduniformly converges to f on E .To complete the proof, we use the following known steps (cf. e.g. [2]). Let η > η n > P η n < η . We can find H = h m ∈ A such that | H ( z ) | ≤ || f || E on T and | f ( z ) − H ( z ) | < η on E . Letting f = f − H on E , the same reasoning yields H ∈ A with | H ( z ) | ≤ || f || E < η on T and | f ( z ) − H ( z ) | < η on E . Similarly wefind H n ∈ A for n = 3 , , ..., with appropriate properties. The convergence of the series || f || E + η + η + ... implies that the series P H n ( z ) converges uniformly on ∆ to a function g ∈ A , which is bounded by || f || E + η . On E holds | f − g | = | ( f − H ) − H − ... − H n − .. | = | ( f − H ) − H − ... − H n − ... | = ... = | ( f n − − H n ) − ... | ≤ η n + P ∞ k = n η k . Sincelim n →∞ ( η n + P ∞ k = n η k ) = 0, it follows that g = f on E , which completes the proof. Remark.
The known proofs of Theorem B use Theorem A and a polynomial approxi-mation theorem (cf. [3], p. 125; or [4], pp. 81-82). The latter is needed to approximate f on E by elements of the disc algebra A . The above proof uses just Theorem A to provide suchapproximation of f on E by elements of A , which in addition are bounded by || f || E on T . References [1] L. Carleson, Representations of continuous functions, Math. Z., (1957), 447-451.[2] A.A. Danielyan, On a polynomial approximation problem, Journal of Approx. Theory, (2010), 717-722.[3] J.B. Garnett, Bounded Analytic Functions, Academic Press, 1981.[4] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NewJersey, 1962.[5] P. Koosis, Introduction to H p Spaces, Cambridge University Press, Cambridge, 1980.[6] M. Lavrentieff, Sur les fonctions d’une variable complexe repr´esntables par des s´eries depolynˆomes, Actual. Sci. et Industr., 441, Paris, Hermann 1936.[7] W. Rudin, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc., (1956), 808-811.Arthur A. DanielyanDepartment of Mathematics andStatisticsUniversity of South FloridaTampa, Florida 33620USA e-mail: [email protected]: [email protected]