$L_ \infty$-boundedness of $L_ 2$-projections on splines for a multiple geometric mesh

Author:
Rong Qing Jia

Journal:
Math. Comp. **48** (1987), 675-690

MSC:
Primary 41A15; Secondary 15A60

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878699-7

MathSciNet review:
878699

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns the ${L_2}$-projectors from ${L_\infty }$ to the normed linear space of polynomial splines. It is shown that for the multiple geometric meshes the ${L_\infty }$ norms of the corresponding ${L_2}$-projectors are bounded independently of the mesh ratio.

- 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,
*Splines as linear combinations of $B$-splines. A survey*, Approximation theory, II (Proc. Internat. Sympos., Univ.#Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 1–47. MR**0467092** - Carl de Boor,
*Total positivity of the spline collocation matrix*, Indiana Univ. Math. J.**25**(1976), no. 6, 541–551. MR**415138**, DOI https://doi.org/10.1512/iumj.1976.25.25043
C. de Boor, "A bound on the ${L_\infty }$-norm of the ${L_2}$-approximation by splines in terms of a global mesh ratio," - Carl de Boor,
*The inverse of a totally positive bi-infinite band matrix*, Trans. Amer. Math. Soc.**274**(1982), no. 1, 45–58. MR**670917**, DOI https://doi.org/10.1090/S0002-9947-1982-0670917-5 - A. S. Cavaretta Jr., W. Dahmen, C. A. Micchelli, and P. W. Smith,
*On the solvability of certain systems of linear difference equations*, SIAM J. Math. Anal.**12**(1981), no. 6, 833–841. MR**635236**, DOI https://doi.org/10.1137/0512069 - 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 - Y. Y. Feng and J. Kozak,
*On the generalized Euler-Frobenius polynomial*, J. Approx. Theory**32**(1981), no. 4, 327–338. MR**641143**, DOI https://doi.org/10.1016/0021-9045%2881%2990007-1 - A. O. Gel′fond,
*Calculus of finite differences*, Hindustan Publishing Corp., Delhi, 1971. Translated from the Russian; International Monographs on Advanced Mathematics and Physics. MR**0342890** - I. I. Hirschman Jr.,
*Matrix-valued Toeplitz operators*, Duke Math. J.**34**(1967), 403–415. MR**220002** - 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 - Samuel Karlin,
*Total positivity. Vol. I*, Stanford University Press, Stanford, Calif, 1968. MR**0230102** - Charles A. Micchelli,
*Cardinal ${\cal L}$-splines*, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 203–250. MR**0481767** - Boris Mityagin,
*Quadratic pencils and least-squares piecewise-polynomial approximation*, Math. Comp.**40**(1983), no. 161, 283–300. MR**679446**, DOI https://doi.org/10.1090/S0025-5718-1983-0679446-0 - Walter Rudin,
*Real and complex analysis*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR**0344043** - I. J. Schoenberg,
*Cardinal spline interpolation*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. MR**0420078**

*Math. Comp.*, v. 30, 1976, pp. 767-771. C. de Boor, "On a max norm bound for the least-squares spline approximation," in

*Approximation and Function Spaces*(Z. Ciesielski, ed.), North-Holland, New York, 1981, pp. 163-175.

Retrieve articles in *Mathematics of Computation*
with MSC:
41A15,
15A60

Retrieve articles in all journals with MSC: 41A15, 15A60

Additional Information

Article copyright:
© Copyright 1987
American Mathematical Society