On the divergence of Lagrange interpolation processes on a set of second category
The paper contains the full proofs of the next assertions: Let \(\omega\) be the nondecreasing semiadditive and continuous function on [0,1] such that \(\omega (0)=0\) and let \({\mathfrak M}\subset [0,1]\) be the matrix of the interpolation knots. If \(\limsup_{n\to \infty}\omega (1/n)\log n>0,\) then there exists a point \(x_ 0\in [0,1]\) and a function \(f\in C[0,1]\), such that \(\omega (f,\delta)=O(\omega (\delta))\) and the Lagrange interpolation process of f interpolating in \({\mathfrak M}\) does not converge to f in the point \(x_ 0\). The author [Mat. Sb., Nov. Ser. 27(169), No.4(8), 519-537 (1985; Zbl 0581.41001)] showed that it is possible to substitute the words ``point \(x_ 0''\) for ``a set \({\mathcal E}\) of second category''. If \(\limsup_{n\to \infty}\omega (1/n)\log n=\infty\) then there exists a set \({\mathcal E}\) of second category and a function \(f\in C[0,1]\) such that \(\omega (f,\delta)=o(\omega (\delta))\) and the Lagrange interpolation process of f interpolating in \({\mathfrak M}\) diverge unlimitedly everywhere on \({\mathcal E}\). The technical error admitted by the author in ibid. 66(108), 272-286 (1965; Zbl 0147.051) and noted by \textit{P. Vertesi} [Acta Math. Hungar. 13, 137-151 (1984; Zbl 0556.41004)] is corrected.
scientific article; zbMATH DE number 4046311
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