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Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Authors:	Wilhelm Forst

Affiliation:	(1) Abteilung mathematik der Universität Dortmund, Postfach 500500, D-4600 Dortmund 50, Germany (Fed. Rep.)

Abstract:	Summary Letx₀<x₁<...<x_n–1<x₀+2 be nodes having multiplicitiesv₀,...,v_n–1, 1v_kr (0k<n). We approximate the evaluation functional,x fixed, and the integral respectively by linear functionals of the form $sumlimits_{k = 0}^{n - 1} {sumlimits_{j = 0}^{v_k - 1} {alpha _{k,j} hat x_k D^j } }$ and determine optimal weights $alpha _{k,j} (0 leqq k< n,0 leqq j< v_k )$ for the Favard classesW_rC₂. In the even case $sumlimits_{k = 0}^{n - 1} {v_k = 2N}$ of optimal interpolation these weights are unique except forr=1,x(x_k+x_k–1)/2 mod 2. Moreover we get periodic polynomial splinesw_{k, j} (0k<n, 0j<v_k) of orderr such that $alpha _{k,j} : = w_{k,j} (x)$ are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case $sumlimits_{k = 0}^{n - 1} {v_k = 2N - 1}$ of optimal interpolation we merely have obtained a partial solution. Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert

Keywords:	AMS(MOS): 65D30 CR: 5.16
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