Yang, Chengmin A counterexample to a conjecture of Hasson. (English) Zbl 0696.41005 J. Approximation Theory 56, No. 3, 330-332 (1989). 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 Cited in 1 Document MSC: 41A10 Approximation by polynomials 42A55 Lacunary series of trigonometric and other functions; Riesz products Keywords:lacunary algebraic polynomials; Lipschitz condition Citations:Zbl 0456.41016 PDFBibTeX XMLCite \textit{C. Yang}, J. Approx. Theory 56, No. 3, 330--332 (1989; Zbl 0696.41005) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.