Abstract
The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).
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1 Introduction
Denote by \(\Delta \) 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 continuous functions on the closed unit disk \(\overline{\Delta }\) that are analytic on \(\Delta \). The following theorem is fundamental; in particular it implies the F. and M. Riesz theorem on analytic measures (cf. [11], pp. 28-31).
Theorem A
(P. Fatou, 1906). Let E be a closed set of Lebesgue measure zero on T. Then there exists a function \(\lambda _E(z)\) in the disk algebra A such that \(\lambda _E(z)=1\) on E and \(|\lambda _E(z)|<1\) on \(T {\setminus } E\).
In its original form Fatou’s theorem states the existence of an element of A which vanishes precisely on E, but it is equivalent to the above version (cf. [11], p. 30, or [9], pp. 80–81).
The following famous theorem, due to W. Rudin [15] and L. Carleson [4], has been the starting point of many investigations in complex and functional analysis (including several complex variables).
Theorem B
(Rudin - Carleson). Let E be a closed set of Lebesgue measure zero on T and let f be a continuous (complex valued) function on E. Then there exists a function g in the disk algebra A agreeing with f on E.
It is obvious from Theorem A and Theorem B that for any \(\epsilon >0\) one can choose the extension function g in Theorem B such that it is bounded by \(||f||_E+ \epsilon \), where \(||f||_E\) is the sup norm of f on E. Rudin has shown that one can even choose the function g such that it is bounded by \(||f||_E\).
Quite naturally, as mentioned already by Rudin, Theorem B may be regarded as a strengthened form of Theorem A (cf. [15], p. 808).
The present paper shows that Theorem B also is an elementary corollary of Theorem A. To be more specific, we present a brief proof of Theorem B merely using Theorem A and the Heine - Cantor theorem (from Calculus I course); this approach may find further applications. We use a simple argument based on uniform continuity, which has been known (at least since 1930s) in particular to M.A. Lavrentiev [12], M.V. Keldysh, and S.N. Mergelyan, but has not been used for the proof of Theorem B before.
We close the introduction by mentioning some references related to Theorem B. An abstract theorem of E. Bishop [3] generalizes Theorem B to any situation where the F. and M. Riesz theorem is valid. A version of this result is the known peak-interpolation theorem of Bishop (see [16], p. 135); a new approach to this theorem is given in [6]. Further generalizations of Theorem B have been proved by S.Ya. Khavinson [10], and A. Pelczyński [14]. Some other developments and extensions of Theorem B have been given by D. Oberlin [13], and by S. Berhanu and J. Hounie [1, 2]. The paper by R. Doss [7] provides elementary proofs for Theorem B and the F. and M. Riesz theorems; note that the argument of [7] (see p. 600) is based on the Weierstrass approximation theorem (cf. the remark below).
2 Proof of Theorem B
Let \(\epsilon >0\) be given. By uniform continuity we cover E by disjoint open intervals \(I_k \subset T\) of a finite number n such that \(|f(z_1)-f(z_2)| < \epsilon \) for any \(z_1, z_2 \in E \cap I_k\) (\(k=1,2,..., n\)). Since the intervals \(I_k\) are disjoint, the endpoints of \(I_k\) are not in E. Thus each \(E\cap I_k\) is closed. Denote \( E_k= E \cap I_k\) and let \(\lambda _{E_k}(z) \) be the function provided by Theorem A. Fix a natural number N so large that \(|\lambda _{E_k}(z)|^N<\frac{\epsilon }{n}\) on \(T{\setminus } I_k\) for all k. Fix a point \(t_k \in E_k\) for each k and denote \(h(z)=\sum _{k=1}^nf(t_k) [\lambda _{E_k}(z)]^{N}\). Obviously the function \(h \in A\) is bounded on T by the number \( (1+\epsilon ) ||f||_E\) and \(|f(z)-h(z)| < \epsilon (1+ ||f||_E) \) if \(z\in E\). Replacing h by \(\frac{1}{1+\epsilon }h\) allows to assume that h is bounded on T simply by \(||f||_E\) and \(|f(z)-h(z)| < \epsilon (1+ 2 ||f||_E) \) if \(z\in E\). Letting \(\epsilon = \frac{1}{m}\) provides a sequence \(\{h_m\}\), \(h_m \in A\), which is uniformly bounded on T by \( ||f||_E\) and uniformly converges to f on E.Footnote 1
To complete the proof, we use the following known steps (cf. e.g. [5]). Let \(\eta >0\) be given and let \(\eta _p>0\) be such \(\sum \eta _p< \eta \). We can find \(H_1=h_{m_1} \in A\) such that \(|H_1(z)| \le ||f||_E\) on T and \(|f(z)-H_1(z)| < \eta _1\) on E. Letting \(f_1=f - H_1\) on E, the same reasoning yields \(H_2 \in A\) with \(|H_2(z)| \le ||f_1||_E < \eta _1\) on T and \(|f_1(z)-H_2(z)| < \eta _2\) on E. Similarly we find \(H_p \in A\) for \(p=3, 4, ...,\) with appropriate properties. The convergence of the series \(||f||_E + \eta _1 + \eta _2 + ... \) implies that the series \(\sum H_p(z) \) converges uniformly on \(\overline{\Delta }\) to a function \(g \in A\), which is bounded by \(||f||_E+\eta \). On E holds \(|f-g|=|(f-H_1)-H_2-...-H_p-...|=|(f_1-H_2)-H_3-...-H_p-...|=...=|(f_{p-1} -H_p)-...| \le \eta _p + \sum _{k=p}^{\infty } \eta _k\). Since \(\lim _{p \rightarrow \infty } (\eta _p + \sum _{k=p}^{\infty } \eta _k)=0\), it follows that \(g=f\) on E, which completes the proof.
Remark
The known proofs of Theorem B use Theorem A and a polynomial approximation theorem (cf. [8], p. 125; or [9], pp. 81–82). The latter is needed to approximate f on E by the elements of the disc algebra A. The above proof uses just Theorem A to provide such approximation of f on E by the elements of A, which in addition are bounded by \(||f||_E\) on T.
Notes
Note that using the constructed sequence \(\{h_m\}\) one can easily derive the lemma on which the proofs of Doss [7] are based. Indeed, the functions \(h_m(z)[\lambda _E(z)]^m \in A\), being uniformly bounded on T by \(||f||_E\), obviously uniformly converge to f on E and converge to zero uniformly inside the complementary intervals of E (which implies the lemma used in [7]).
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The author wishes to thank the referees for the careful reading of the manuscript and useful suggestions.
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Communicated by Hans G. Feichtinger.
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Danielyan, A.A. Fatou’s Interpolation Theorem Implies the Rudin–Carleson Theorem. J Fourier Anal Appl 23, 656–659 (2017). https://doi.org/10.1007/s00041-016-9481-y
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DOI: https://doi.org/10.1007/s00041-016-9481-y