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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a partition of the unit interval, , , denotes the space of piecewise-polynomials of order *k* and of smoothness ; 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 ) in the -norm of the orthogonal projections onto , 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 if . The explicit calculations and estimates of zeros of the "characteristic" polynomial show that in the case , me geometric mesh with the parameter *x*, , the orthogonal projectors are uniformly bounded.

**[1]**Z. Ciesielski,*Properties of the orthonormal Franklin system*, Studia Math.**23**(1963), 141–157. MR**0157182****[2]**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****[3]**Carl de Boor, "A bound on the -norm of the -approximation by splines in terms of a global mesh ratio,"*Math. Comp.*, v. 30, 1976, pp. 767-771.**[4]**Carl de Boor,*On a Max-Norm Bound for the Least-Squares Spline Approximant*, Conf. on Approximation Theory, Gdansk, Poland, August, 1979. (Preprint.)**[5]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, https://doi.org/10.1090/S0025-5718-1975-0371077-0**[6]**Stephen Demko,*Inverses of band matrices and local convergence of spline projections*, SIAM J. Numer. Anal.**14**(1977), no. 4, 616–619. MR**0455281**, https://doi.org/10.1137/0714041**[7]**Stephen Demko,*On bounding 𝐴⁻¹_{∞} for banded 𝐴*, Math. Comp.**33**(1979), no. 148, 1283–1288. MR**537972**, https://doi.org/10.1090/S0025-5718-1979-0537972-8**[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****[9]**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****[10]**B. Mityagin, "Factorization of quasiselfadjoint quadratic pencil,"*Integral Equations Operator Theory*. (To appear.)**[11]**K. Höllig,*𝐿_{∞}-boundedness of 𝐿₂-projections on splines for a geometric mesh*, J. Approx. Theory**33**(1981), no. 4, 318–333. MR**646153**, https://doi.org/10.1016/0021-9045(81)90063-0

Retrieve articles in *Mathematics of Computation*
with MSC:
41A15,
47A68

Retrieve articles in all journals with MSC: 41A15, 47A68

Additional Information

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

Article copyright:
© Copyright 1983
American Mathematical Society