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Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities. (English) Zbl 1446.41010

Suppose that \(L_rf\) is the Birkhoff interpolation of a function \(f\in W^r_q[a,b]\) based on \(r\) pairs of numbers \((x_1, k_1)\dots, (x_r, k_r)\) with its Pólya interpolation matrix to be regular. For \(1\leq p,q\leq\infty\), \(0\leq s \leq r-1\), the problem of estimating the best constant \(C(r, s, p, q)\) in the inequality \[ \|(f-L_rf)^{(s)}\|_p \leq C(r, s, p, q)(b-a)^{r-s+1/p-1/q}\|f^{(r)}\|_q \quad\mbox{for all}\quad f\in W^r_q[a,b] \] is studied.

MSC:

41A44 Best constants in approximation theory
41A80 Remainders in approximation formulas
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