Asymptotic properties of Chebyshev splines with fixed number of knots

Yu. V. Malykhin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 5, pp. 143-166

Résumé

V. M. Tikhomirov expressed Kolmogorov widths of the class $W^r:=W^r_\infty[-1,1]$ in the space $C:=C[-1,1]$ as a norm of special splines: $d_N(W^r,C)=\|x_{N-r,r}\|_C$, $N\ge r$; these splines were named Chebyshev splines. The function $x_{n,r}$ is a perfect spline of order $r$ with $n$ knots. We study the asymptotic behaviour of Chebyshev splines for $r\to\infty$ and fixed $n$. We calculate the asymptotics of knots and the $C$-norm of $x_{n,r}$ and prove that $x_{n,r}/x_{n,r}(1)=T_{n+r}+o(1)$. As a corollary, we obtain that $d_{n+r}(W^r,C)/d_r(W^r,C)\sim A_nr^{-n/2}$ as $r\to\infty$.