Asymptotic properties of best ₂[0,1] approximation by splines with variable knots
Let SN" be the set of &th-order splines on [0,1] having at most N 1 interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best L2[0, 1] approximation of a sufficiently smooth function / by the lim N* dist(/, SNk) = (|fl2*|/(2£)!)1/2( f \ fKr)]" dr AT—»oo \ J o \/(T where a = (k + 1/2)"1 and B2k is the 2/cth Bernoulli number. Similar results have previously been obtained for piecewise polynomial (i.e., with no continuity constraints) approximation, but with different constant before the integral term. The approach we use is first to study the asymptotic behavior of dist(/, SNk(t)), where SN"(t) is the linear space of /cth-order splines having simple knots determined from the fixed function / by the rule tt = t(i/N), i « 0, • • • , N.