Demonstrates improved approximation orders for C-k functions using Lagrange interpolation, suggesting broader applications in function analysis.
We study the optimal order of approximation for C-k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, if x is away from the singular points, the local order of approximation at x can be improved by O(n(-1)). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for \(3)) in a much more general form.
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