On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds

Chui, Charles K.; Smith, Philip W.; Ward, Joseph D.

doi:10.1090/S0025-5718-1977-0422955-7

On the smoothness of best $L_{2}$ approximants from nonlinear spline manifolds
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by Charles K. Chui, Philip W. Smith and Joseph D. Ward PDF

Math. Comp. 31 (1977), 17-23 Request permission

Abstract:

Let $S_n^k$ be the nonlinear spline manifold of order k and with n - k interior variable knots. We prove that all best ${L_2}[0,1]$ approximants from $S_n^k$ to a continuous function on [0, 1] are also continuous there. We also prove that there exists a ${C^\infty }[0,1]$ function with no ${C^2}[0,1]$ best ${L_2}[0,1]$ approximants from $S_n^k$.

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