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\(L^p\)-convergence of Lagrange interpolation on an arbitrary system of nodes. (English) Zbl 1018.41004

This paper is inspired by an old problem of Paul Turán on necessary conditions for mean convergence of Lagrange interpolation polynomials in \(L_p\) norms. The authors consider the following problem: Let \(E\) be a compact subset of the real line, and \(d\alpha\) be a measure supported on \(E\). Let \(X\) be an array of interpolation points supported in \(E\), and let \(L_n(X,f)\) denote the Lagrange interpolation polynomial to a function \(f\) at the \(n\)th row of the interpolation array. Let \(p>0\). What are the necessary and sufficient conditions for \[ \lim_{n \to\infty} \int_E\bigl|f-L_n(X,f) \bigr|^p d \alpha=0, \] for all functions continuous on \(E\)? The authors claim some necessary conditions for this convergence to take place, when \(d\alpha\) belongs to an unusual version of the Szegő class on \(E\).

MSC:

41A10 Approximation by polynomials
30E10 Approximation in the complex plane
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