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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.


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DOI: https://doi.org/10.1090/S0025-5718-1983-0679446-0
Article copyright: © Copyright 1983 American Mathematical Society

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