Unicity of best mean approximation by second order splines with variable knots

Abstract

Let S N 2 S_N^2 denote the nonlinear manifold of second order splines defined on [0, 1] having at most N N interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function f f by elements of S N 2 S_N^2 . Approximation relative to the L 2 [ 0 , 1 ] {L_2}[0,1] norm is treated first, with the results then extended to the best L 1 {L_1} and best one-sided L 1 {L_1} approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function f f satisfying f > 0 f > 0 has a unique best approximant from S N 2 S_N^2 provided either log ⁡ f \log f is concave, or N N is sufficiently large, N ⩾ N 0 ( f ) N \geqslant {N_0}(f) ; for any N N , there is a smooth function f f , with f > 0 f > 0 , having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

Analysis of an algorithm for generating locally optimal meshes for 𝐿₂ approximation by discontinuous piecewise polynomials

“…The unusual conditions in the second theorem ensure that there is only one stationary point [3]. Those conditions also yield favourable bounds on the eigenvalues of the error hessian, and the inequality in the theorem follows easily.…”

“…It is known [3] that, save for the trivial case where f is a polynomial of degree less than r, the optimal partition ∆ opt satisfies…”

“…With these lemmata at our disposal, the proof of Theorem 3.1 is now straightforward. Using the definition of u n and integration by parts, we may deduce [3] …”

“…If there is only one stationary point of the error functional E, then the entire sequence {∆ n } n∈N must converge to it. Barrow et al [3] showed that the conditions in the hypothesis are sufficient for the uniqueness of the stationary point.…”

“…However, it has been noted by some authors that the best L 2 approximation by discontinuous piecewise linear polynomials on the optimal mesh is often continuous (for instance in the one-dimensional case of a convex function [3]) or "nearly continuous" [20]. On the basis of this observation, and with a view to making the optimization problem as local as possible, Baines [2] recently promoted the use of discontinuous piecewise polynomials as the basic approximation tool.…”