Exact Markov-Type Inequalities for Oscillating Perfect Splines

Bojanov, Borislav; Naidenov, Nikola
January 2002
Constructive Approximation;2002, Vol. 18 Issue 1, p37
Academic Journal
We prove the inequality |s[SUP(k)]|L[SUBp][-1,1] = |T[SUP(k),[SUBrn]|L[SUBp][-1,1]|S|C[-1,1], 1 = k = r, 1 = p = 8, for each perfect spline s of degree r with n - r knots and n zeros in [-1,1]. Here T[SUBrn] is the Tchebycheff perfect spline of degree r with n - r knots, normalized by the condition |T[SUBrn]|C[-1,1] := max[SUB-1=x=1] |T[SUBrn](x)| = 1. The constant |T[SUP(k),[SUBrn]|L[SUBp][-1,1] in above inequality is the best possible.


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