Pointwise error estimates for interpolation
Let f have an uniformly and absolutely convergent expansion of the form \(f(x)=d_ 0T_ 0(x)+\sum^{\infty}_{k=1}d_ kT_ k(x),\)-1\(\leq x\leq 1\), where \(T_ k(x)\) denotes the kth Chebyshev polynomial of the first kind. Let \(L_{n-1}(x;f)\) denote the unique Lagrange polynomial interpolant of f in the roots of \(T_ n(x)\). For the error \(R_{n- 1}(x;f):=f(x)-L_{n-1}(x;f)\) the author proves that \[ R_{n- 1}(x;f)=2T_ n(x)\sum^{\infty}_{m=1}(-1)^{m-1}[D_{(2m-1)n}T_ 0(x)+\sum^{\infty}_{k=1}d_{(2m-1)n+k}T_ k(x)], \] \[ | R_{n-1}(x;f)| \leq 2| T_ n(x)| (D_ 0+\sum^{\infty}_{k=1}D_ k), \] where \(D_ k:=| \sum^{\infty}_{m=1}(-1)^{m-1}d_{(2m-1)n+k}|\), \(k=0,1,... \). The author obtains a similar error representation and bound for polynomial interpolants in the extrema of \(T_ k(x)\).
scientific article; zbMATH DE number 4065577