Univariate splines: Equivalence of moduli of smoothness and applications
Author:
Kirill A. Kopotun
Journal:
Math. Comp. 76 (2007), 931945
MSC (2000):
Primary 65D07, 41A15, 26A15; Secondary 41A10, 41A25, 41A29
Published electronically:
November 27, 2006
MathSciNet review:
2291843
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any , , and , the inequality , , is satisfied, where is a piecewise polynomial of degree on a quasiuniform (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 DitzianTotik 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.
 [1]
R.
K. Beatson, Convex approximation by splines, SIAM J. Math.
Anal. 12 (1981), no. 4, 549–559. MR 617714
(82h:41012), http://dx.doi.org/10.1137/0512048
 [2]
Peter
Borwein and Tamás
Erdélyi, Polynomials and polynomial inequalities,
Graduate Texts in Mathematics, vol. 161, SpringerVerlag, New York,
1995. MR
1367960 (97e:41001)
 [3]
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 (81h:41019), http://dx.doi.org/10.1137/0511041
 [4]
Ronald
A. DeVore, Monotone approximation by polynomials, SIAM J.
Math. Anal. 8 (1977), no. 5, 906–921. MR 0510582
(58 #23252)
 [5]
Ronald
A. DeVore, Monotone approximation by splines, SIAM J. Math.
Anal. 8 (1977), no. 5, 891–905. MR 0510725
(58 #23259)
 [6]
R.
A. DeVore, Y.
K. Hu, and D.
Leviatan, Convex polynomial and spline approximation in
𝐿_{𝑝},\0<𝑝<∞, Constr. Approx.
12 (1996), no. 3, 409–422. MR 1405006
(97j:41008), http://dx.doi.org/10.1007/s003659900021
 [7]
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
(93f:41011), http://dx.doi.org/10.1007/BF01238268
 [8]
Ronald
A. DeVore and George
G. Lorentz, Constructive approximation, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 303, SpringerVerlag, Berlin, 1993. MR 1261635
(95f:41001)
 [9]
Z.
Ditzian and V.
Totik, Moduli of smoothness, Springer Series in Computational
Mathematics, vol. 9, SpringerVerlag, New York, 1987. MR 914149
(89h:41002)
 [10]
Yingkang
Hu, On equivalence of moduli of smoothness, J. Approx. Theory
97 (1999), no. 2, 282–293. MR 1682954
(2000a:41044), http://dx.doi.org/10.1006/jath.1997.3264
 [11]
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
(95a:41018), http://dx.doi.org/10.1007/BF01205165
 [12]
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
(96h:41010), http://dx.doi.org/10.1137/0732065
 [13]
Kamen
G. Ivanov and Boyan
Popov, On convex approximation by quadratic splines, J.
Approx. Theory 85 (1996), no. 1, 110–114. MR 1382054
(97c:41024), http://dx.doi.org/10.1006/jath.1996.0032
 [14]
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
(95k:41014), http://dx.doi.org/10.1007/BF01263061
 [15]
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
(83j:41014), http://dx.doi.org/10.1137/0513060
 [16]
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
(89i:41022)
 [17]
I. A. Shevchuk, Approximation by Polynomials and Traces of the Functions Continuous on an Interval, Naukova Dumka, 1992.
 [18]
I.
A. Shevchuk, On coapproximation 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
(91c:41050)
 [19]
A.
S. Švedov, Orders of coapproximations of functions by
algebraic polynomials, Mat. Zametki 29 (1981),
no. 1, 117–130, 156 (Russian). MR 604156
(82c:41009)
 [1]
 R. K. Beatson, Convex approximation by splines, SIAM J. Math. Anal. 12 (1981), no. 4, 549559. MR 617714 (82h:41012)
 [2]
 P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, SpringerVerlag, 1995. MR 1367960 (97e:41001)
 [3]
 C. K. Chui, P. W. Smith, and J. D. Ward, Degree of approximation by monotone splines, SIAM J. Math. Anal. 11 (1980), no. 3, 436447. MR 572194 (81h:41019)
 [4]
 R. A. DeVore, Monotone approximation by polynomials, SIAM J. Math. Anal. 8 (1977), no. 5, 906921. MR 0510582 (58:23252)
 [5]
 R. A. DeVore, Monotone approximation by splines, SIAM J. Math. Anal. 8 (1977), no. 5, 891905. MR 0510725 (58:23259)
 [6]
 R. A. DeVore, Y. K. Hu, and D. Leviatan, Convex polynomial and spline approximation in , Constr. Approx. 12 (1996), no. 3, 409422. MR 1405006 (97j:41008)
 [7]
 R. A. DeVore, D. Leviatan, and X. M. Yu, Polynomial approximation in , Constr. Approx. 8 (1992), no. 2, 187201. MR 1152876 (93f:41011)
 [8]
 R. A. DeVore and G. G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, SpringerVerlag, 1993. MR 95f:41001
 [9]
 Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, vol. 9, SpringerVerlag, 1987. MR 0914149 (89h:41002)
 [10]
 Y. Hu, On equivalence of moduli of smoothness, J. Approx. Theory 97 (1999), no. 2, 282293. MR 1682954 (2000a:41044)
 [11]
 Y. Hu, D. Leviatan, and X.M. Yu, Convex polynomial and spline approximation in , Constr. Approx. 10 (1994), no. 1, 3164. MR 1260358 (95a:41018)
 [12]
 Y. Hu and X. M. Yu, Discrete modulus of smoothness of splines with equally spaced knots, SIAM J. Numer. Anal. 32 (1995), no. 5, 14281435. MR 1352197 (96h:41010)
 [13]
 K. G. Ivanov and B. Popov, On convex approximation by quadratic splines, J. Approx. Theory 85 (1996), no. 1, 110114. MR 1382054 (97c:41024)
 [14]
 K. A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10 (1994), no. 2, 153178. MR 1305916 (95k:41014)
 [15]
 D. Leviatan and H. N. Mhaskar, The rate of monotone spline approximation in the norm, SIAM J. Math. Anal. 13 (1982), no. 5, 866874. MR 668327 (83j:41014)
 [16]
 P. P. Petrushev and V. A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge University Press, 1987. MR 940242 (89i:41022)
 [17]
 I. A. Shevchuk, Approximation by Polynomials and Traces of the Functions Continuous on an Interval, Naukova Dumka, 1992.
 [18]
 I. A. Shevchuk, On coapproximation of monotone functions, Dokl. Akad. Nauk SSSR 308 (1989), no. 3, 537541 (Russian). MR 1021110 (91c:41050)
 [19]
 A. S. Švedov, Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki 29 (1981), no. 1, 117130, 156 (Russian). MR 0604156 (82c:41009)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
65D07,
41A15,
26A15,
41A10,
41A25,
41A29
Retrieve articles in all journals
with MSC (2000):
65D07,
41A15,
26A15,
41A10,
41A25,
41A29
Additional Information
Kirill A. Kopotun
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
Email:
kopotunk@cc.umanitoba.ca
DOI:
http://dx.doi.org/10.1090/S002557180601920X
PII:
S 00255718(06)01920X
Keywords:
Univariate splines,
moduli of smoothness,
degree of approximation,
Jackson type estimates,
polynomial and spline approximation
Received by editor(s):
June 1, 2005
Received by editor(s) in revised form:
August 25, 2005
Published electronically:
November 27, 2006
Additional Notes:
The author was supported in part by NSERC of Canada.
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
