Abstract
V. M. Tikhomirov expressed Kolmogorov widths of the class Wr := W∞r[−1, 1] in the space C := C[−1, 1] as a norm of special splines: dN(Wr, C) = ‖xN − r, r‖C, N ≥ r; these splines were named Chebyshev splines. The function xn,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 xn,r and prove that xn,r/xn,r(1) = Tn+r+o(1). As a corollary, we obtain that dn+r(Wr, C)/dr(Wr, C) ~ Anr−n/2 as r→∞.
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