Asymptotic Properties of Chebyshev Splines with Fixed Number of Knots

Abstract

V. M. Tikhomirov expressed Kolmogorov widths of the class W^r := W_∞^r[−1, 1] in the space C := C[−1, 1] as a norm of special splines: d_N(W^r, C) = ‖xN − r, r‖C, N ≥ 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 behavior of Chebyshev splines for r→∞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) ~ A_nr^−n/2 as r→∞.