𝐿_{∞}-boundedness of 𝐿₂-projections on splines for a multiple geometric mesh

Jia, Rong Qing

doi:10.1090/S0025-5718-1987-0878699-7

$L_ \infty$-boundedness of $L_ 2$-projections on splines for a multiple geometric mesh
HTML articles powered by AMS MathViewer

by Rong Qing Jia PDF

Math. Comp. 48 (1987), 675-690 Request permission

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.

References

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 10.1512/iumj.1976.25.25043

Math. Comp.

Approximation and Function Spaces

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 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 10.1137/0512069
Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141–157. MR 157182, DOI 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 10.1016/0021-9045(81)90007-1
A. O. Gel′fond, Calculus of finite differences, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, 1971. Translated from the Russian. 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 10.1016/0021-9045(81)90063-0
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 10.1090/S0025-5718-1983-0679446-0
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078