Asymptotic properties of best 𝐿₂[0,1] approximation by splines with variable knots

Let S N k S_N^k be the set of k k th-order splines on [ 0 , 1 ] \left [ {0, 1} \right ] having at most N − 1 N - 1 interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best L 2 [ 0 , 1 ] {L_2}\left [ {0, 1} \right ] approximation of a sufficiently smooth function f f by the set S N k S_N^k : \[ lim N → ∞ N k d i s t ( f , S N k ) = ( | B 2 k | / ( 2 K ) ! ) 1 / 2 ( ∫ 0 1 | f ( k ) ( τ ) | σ d τ ) 1 / σ \lim \limits _{N \to \infty } {N^k}dist\left ( {f, S_N^k} \right ) = {\left ( {\left | {{B_{2k}}} \right |/\left ( {2K} \right )!} \right )^{1/2}}{\left ( {{{\int _0^1 {\left | {{f^{\left ( k \right )}}\left ( \tau \right )} \right |} }^\sigma }d\tau } \right )^{1/\sigma }} \] , where σ = ( k + 1 / 2 ) − 1 \sigma = {\left ( {k + 1/2} \right )^{ - 1}} and B 2 k {B_{2k}} is the 2 k 2k th 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( f f , S N k ( t ) S_N^k\left ( t \right ) ), where S N k ( t ) S_N^k\left ( t \right ) is the linear space of k k th-order splines having simple knots determined from the fixed function t t by the rule t i = t ( i / N ) , i = 0 , . . . , N {t_i} \\ = t\left ( {i/N} \right ),i = 0,...,N .