Univariate splines: Equivalence of moduli of smoothness and applications

Kopotun, Kirill

doi:10.1090/S0025-5718-06-01920-X

Univariate splines: Equivalence of moduli of smoothness and applications
HTML articles powered by AMS MathViewer

by Kirill A. Kopotun;

Math. Comp. 76 (2007), 931-945

DOI: https://doi.org/10.1090/S0025-5718-06-01920-X

Published electronically: November 27, 2006

PDF | Request permission

Abstract:

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any $1\leq k\leq r+1$, $0\leq m\leq r-1$, and $1\leq p\leq \infty$, the inequality $n^{-\nu } \omega _{k-\nu }(s^{(\nu )}, n^{-1})_p \sim \omega _{k} (s, n^{-1})_p$, $1\leq \nu \leq \min \{ k, m+1\}$, is satisfied, where $s\in \mathbb {C}^m[-1,1]$ is a piecewise polynomial of degree $\leq r$ on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian–Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.

References

R. K. Beatson, Convex approximation by splines, SIAM J. Math. Anal. 12 (1981), no. 4, 549–559. MR 617714, DOI 10.1137/0512048
Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
C. K. Chui, P. W. Smith, and J. D. Ward, Degree of $L_{p}$ approximation by monotone splines, SIAM J. Math. Anal. 11 (1980), no. 3, 436–447. MR 572194, DOI 10.1137/0511041
Ronald A. DeVore, Monotone approximation by polynomials, SIAM J. Math. Anal. 8 (1977), no. 5, 906–921. MR 510582, DOI 10.1137/0508069
Ronald A. DeVore, Monotone approximation by splines, SIAM J. Math. Anal. 8 (1977), no. 5, 891–905. MR 510725, DOI 10.1137/0508068
R. A. DeVore, Y. K. Hu, and D. Leviatan, Convex polynomial and spline approximation in $L_p,\ 0<p<\infty$, Constr. Approx. 12 (1996), no. 3, 409–422. MR 1405006, DOI 10.1007/s003659900021
Ronald A. DeVore, Dany Leviatan, and Xiang Ming Yu, Polynomial approximation in $L_p$ $(0<p<1)$, Constr. Approx. 8 (1992), no. 2, 187–201. MR 1152876, DOI 10.1007/BF01238268
Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, vol. 9, Springer-Verlag, New York, 1987. MR 914149, DOI 10.1007/978-1-4612-4778-4
Yingkang Hu, On equivalence of moduli of smoothness, J. Approx. Theory 97 (1999), no. 2, 282–293. MR 1682954, DOI 10.1006/jath.1997.3264
Yingkang Hu, Dany Leviatan, and Xiang Ming Yu, Convex polynomial and spline approximation in $C[-1,1]$, Constr. Approx. 10 (1994), no. 1, 31–64. MR 1260358, DOI 10.1007/BF01205165
Yingkang Hu and Xiang Ming Yu, Discrete modulus of smoothness of splines with equally spaced knots, SIAM J. Numer. Anal. 32 (1995), no. 5, 1428–1435. MR 1352197, DOI 10.1137/0732065
Kamen G. Ivanov and Boyan Popov, On convex approximation by quadratic splines, J. Approx. Theory 85 (1996), no. 1, 110–114. MR 1382054, DOI 10.1006/jath.1996.0032
Kirill A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10 (1994), no. 2, 153–178. MR 1305916, DOI 10.1007/BF01263061
D. Leviatan and H. N. Mhaskar, The rate of monotone spline approximation in the $L_{p}$-norm, SIAM J. Math. Anal. 13 (1982), no. 5, 866–874. MR 668327, DOI 10.1137/0513060
P. P. Petrushev and V. A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge University Press, Cambridge, 1987. MR 940242
I. A. Shevchuk, Approximation by Polynomials and Traces of the Functions Continuous on an Interval, Naukova Dumka, Kyiv, 1992.
I. A. Shevchuk, On co-approximation of monotone functions, Dokl. Akad. Nauk SSSR 308 (1989), no. 3, 537–541 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 2, 349–354. MR 1021110
A. S. Švedov, Orders of coapproximations of functions by algebraic polynomials, Mat. Zametki 29 (1981), no. 1, 117–130, 156 (Russian). MR 604156