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Measurable Darboux functions. (English) Zbl 0651.26005

The authors investigate how certain ten Darboux-like properties of real functions (including connectivity, peripheral continuity, Zahorski’s condition, and almost continuity) are related to each other within the following classes of functions: the class F of all functions f: [0,1]\(\to R\) (R - the real line), the class of Lebesgue measurable functions in F, the class of Borel measurable functions in F, the class of pointwise limits of sequences of functions in F which have only discontinuities of the first kind, the class of pointwise limits of sequences of functions in F which are continuous from the right, and the class of Baire one functions in F. Let P(z) be a polynomial of degree n-1 with coefficients \(a_ k\) satisfying \(| a_ k| =1\). Thus \(\| P\|_ 2=\sqrt{n}\). J.-P. Kahane [Bull. Lond. Math. Soc. 12, 321-342 (1980; Zbl 0443.30005)] showed that, for any \(\epsilon >0\), there are polynomials of this type with \[ (1-\epsilon)\sqrt{n}<| P(z)| <(1+\epsilon)\sqrt{n} \] for all z on \(| z| =1.\)
The authors claim to prove that, in contrast, if \(a_ k=\pm 1\) for all k, then there is a constant \(C>0\) so that \[ (1)\quad \max_{k}| P(e^{2\pi ik/n})| \geq (1+C)\sqrt{n}, \] and hence certainly (2) \(\| P\|_{\infty}\geq (1+C)\sqrt{n}\). They also claim a similar result for the qth power mean of P on the nth roots of unit and on the unit circle for sufficiently large \(q<\infty\). Unfortunately, the result (1) and the analogous result for the qth power mean are false, as was pointed out to the reviewer by the authors. In a “counter note” (entitles, “Sur une Note récente relative aux polynômes à coefficients \(\pm 1\) et à la conjecture d’Erdős”) recently submited to the Comptes Rendus by the authors, they give a counterexample to (1) for certain n, based on Barker codes. They trace the error to the incorrect statement on lines 25 and 26 of p. 697, that \(| P_ 0(e^{2\pi iu})|^ 2\) approaches a rearrangement of \(\phi\). The conjecture of Erdős (2) remains undecided. In their counter note, they sketch an approach to (2) which depends on an unproved conjecture concerning the minimal value of \(\| P\|_ 4/\| P\|_ 2\) for reciprocal polynomials with coefficients \(a_ k=0\), \(\pm 1\) or \(\pm i\). A result of this type, but insufficiently precise, has recently been obtained in a paper by the authors and M. Fredman, to appear in Comptes Rendus.
Reviewer: D.W.Boyd

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
54C05 Continuous maps
30A10 Inequalities in the complex plane
30B20 Random power series in one complex variable
30C10 Polynomials and rational functions of one complex variable
42A10 Trigonometric approximation

Citations:

Zbl 0443.30005
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References:

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