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Small-normed projections onto polynomial and spline spaces

Summary

Small-normed projections onto a $n$-dimensional subspace $V$ of $\mathcal{C}(K)$ are investigated by way of small-conditioned bases of $V$. The $\ell_\infty$-condition number $\kappa_\infty(V)$ of the space $V$, or its Banach-Mazur distance to $\ell_\infty^n$, is shown to equal the minimal norm of generalized interpolating projections onto $V$ when $K$ is the interval $[-1,1]$ or the unit circle. The subsequent inequalities involving the projection constant and the interpolating projection constant, namely

\begin{displaymath}p(V,\mathcal{C}(K)) \; \le \; \kappa_\infty(V) \; \le \; p_{\rm int}(V,\mathcal{C}(K)),\end{displaymath}

are studied in details for some polynomial spaces of low dimension.


For spline spaces, it is relevant to estimate these projection constants only in terms of the order of the splines. For instance, if $\kappa_\infty^{WD}(V)$ represents the least condition number of a weak Descartes basis of $V$, the inequality

\begin{displaymath}p_{\rm int}(V,\mathcal{C}[-1,1]) \; \le \; \kappa_\infty^{WD}(V)\end{displaymath}

gives the opportunity to bound the interpolating projection constant of a spline space by the condition number of the B-spline basis. In an attempt to evaluate the latter accurately, we are led to establish a Markov-type interlacing property for Chebyshevian B-splines. More precisely, we prove that if the knots of two Chebyshevian B-splines of identical support interlace, then the zeros of their appropriate derivatives also interlace.


As for the projection constant of a spline space, it may be estimated independently of the underlying knot sequence via the orthogonal projection onto the space. Indeed, the supremum over all knot sequences of the max-norm of this projection is known to be finite, although the exact value of the supremum, or merely its order, remains to be found. We make a step in this direction by determining precise bounds of order $\sqrt{k}$ in the cases of continuous and of differentiable splines.
If you wish to obtain the whole dissertation, do contact me at simon.foucart@centraliens.net and I will e-mail you the pdf file.