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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Quadratic pencils and least-squares piecewise-polynomial approximation
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by Boris Mityagin PDF
Math. Comp. 40 (1983), 283-300 Request permission

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.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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