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A counterexample to a conjecture of Hasson. (English) Zbl 0696.41005

Let \(f\in C[-1,1]\) and \(\pi_ n\) be the class of algebraic polynomials of degree \(\leq n\). Define \(E_ n(f)=\min \{\| f-p\|:\) \(p\in \pi_ n\}\) and \(E_ n^{(1)}(f)=\min \{\| f-p\|:\) \(f\in \pi_ n\) and \(p_ n^{(1)}(0)=0\}\). In this paper the author constructs a counterexample to show that a conjecture of M. Hasson [J. Approximation Theory 29, 103-115 (1980; Zbl 0456.41016)] that if \(f\in C[-1,1]\) and \(f^{(1)}\) does not exist at some point in \((-1,1)\) then (*) \(\limsup_{n}E_ n^{(1)}(f)/E_ n(f)<\infty,\) is false. Another problem that is posed here and now left open, is whether the result at (*) is true if ‘lim sup’ is replaced by ‘lim inf’. The author gives another theorem as well in the paper that extends an earlier work of M. Hasson and O. Shisha [“Approximation Theory and Applications” (Proc. Workshop, Technian-Israel Int. Haifor, 1981) pp. 311-317, Academic Press. N.Y. (1981)].
Reviewer: G.D.Dikshit

MSC:

41A10 Approximation by polynomials
42A55 Lacunary series of trigonometric and other functions; Riesz products

Citations:

Zbl 0456.41016
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References:

[1] Hasson, M., Comparison between the degree of approximation by lacunary and ordinary algebraic polynomials, J. Approx. Theory, 29, 103-115 (1980) · Zbl 0456.41016
[2] Hasson, M.; Shisha, O., Approximation by lacunary polynomials: A converse theorem, (Approximation Theory and Applications. Approximation Theory and Applications, Proc. Workshop, Technion-Israel Inst., Haifa, 1981 (1981), Academic Press: Academic Press New York), 311-317
[3] Lorentz, G. G., Approximation by incomplete polynomial (problem and results), (Padé and Rational Approximation: Theory and Applications (1977), Academic Press: Academic Press New York), 259-302
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