Quadratic pencils and least-squares piecewise-polynomial approximation

Abstract:

For a partition $\xi = (0 = {\xi _0} < {\xi _1} < \cdots < {\xi _n} < {\xi _{n + 1}} = 1)$ of the unit interval, $S_\xi ^{km}$, $k > m$, denotes the space of piecewise-polynomials of order k and of smoothness $m - 1$; this space can be represented as the graph of the appropriate linear operator between two finite-dimensional Hilbert spaces. It gives an approach to the C. de Boor problem, 1972, on uniform boundedness (with respect to $\xi$) in the ${L_\infty }$-norm of the orthogonal projections onto $S_\xi ^{km}$, and we give the detailed analysis of a quadratic pencil (matrix-valued polynomial of the second degree) which appears in the case of a geometric mesh $\xi$ if $2m \leqslant k$. The explicit calculations and estimates of zeros of the "characteristic" polynomial show that in the case $S_{\xi (x)}^{63}$, $\xi (x)$ me geometric mesh with the parameter x, $0 < x < \infty$, the orthogonal projectors are uniformly bounded.