Quadratic pencils and least-squares piecewise-polynomial approximation

Author:
Boris Mityagin

Journal:
Math. Comp. **40** (1983), 283-300

MSC:
Primary 41A15; Secondary 47A68

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679446-0

MathSciNet review:
679446

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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.

- Z. Ciesielski,
*Properties of the orthonormal Franklin system*, Studia Math.**23**(1963), 141–157. MR**157182**, DOI https://doi.org/10.4064/sm-23-2-141-157 - Carl de Boor,
*The quasi-interpolant as a tool in elementary polynomial spline theory*, Approximation theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973) Academic Press, New York, 1973, pp. 269–276. MR**0336159**
Carl de Boor, "A bound on the ${L_\infty }$-norm of the ${L_2}$-approximation by splines in terms of a global mesh ratio," - Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,
*Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**371077**, DOI https://doi.org/10.1090/S0025-5718-1975-0371077-0 - Stephen Demko,
*Inverses of band matrices and local convergence of spline projections*, SIAM J. Numer. Anal.**14**(1977), no. 4, 616–619. MR**455281**, DOI https://doi.org/10.1137/0714041 - Stephen Demko,
*On bounding $A^{-1}_{\infty }$ for banded $A$*, Math. Comp.**33**(1979), no. 148, 1283–1288. MR**537972**, DOI https://doi.org/10.1090/S0025-5718-1979-0537972-8 - I. C. Gohberg and I. A. Fel′dman,
*Convolution equations and projection methods for their solution*, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware; Translations of Mathematical Monographs, Vol. 41. MR**0355675** - Harm Bart, Israel Gohberg, and Marinus A. Kaashoek,
*Minimal factorization of matrix and operator functions*, Operator Theory: Advances and Applications, vol. 1, Birkhäuser Verlag, Basel-Boston, Mass., 1979. MR**560504**
B. Mityagin, "Factorization of quasiselfadjoint quadratic pencil," - K. Höllig,
*$L_{\infty }$-boundedness of $L_{2}$-projections on splines for a geometric mesh*, J. Approx. Theory**33**(1981), no. 4, 318–333. MR**646153**, DOI https://doi.org/10.1016/0021-9045%2881%2990063-0

*Math. Comp.*, v. 30, 1976, pp. 767-771. Carl de Boor,

*On a Max-Norm Bound for the Least-Squares Spline Approximant*, Conf. on Approximation Theory, Gdansk, Poland, August, 1979. (Preprint.)

*Integral Equations Operator Theory*. (To appear.)

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Article copyright:
© Copyright 1983
American Mathematical Society