$L_ \infty$-boundedness of $L_ 2$-projections on splines for a multiple geometric mesh
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- 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 C. de Boor, "A bound on the ${L_\infty }$-norm of the ${L_2}$-approximation by splines in terms of a global mesh ratio," 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.
- 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
Additional Information
- © Copyright 1987 American Mathematical Society
- 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