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Constrained Approximation in Sobolev Spaces

Published online by Cambridge University Press:  20 November 2018

Y. K. Hu ,
K. A. Kopotun  and
X. M. Yu
Y. K. Hu
Affiliation:
Georgia Southern University, Statesboro, Georgia 30460, U.S.A.
K. A. Kopotun
Affiliation:
University of Alberta, Edmonton, Alberta, T6G 2G1
X. M. Yu
Affiliation:
Southwest Missouri State University, Springfield, Missouri 65804, U.S.A.
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Abstract

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Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

Keywords

41A1041A1541A2541A29Constrained approximationpolynomialssplinesdegree of approximationLp spaceSobolev space
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 74 - 99
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Andreev, A., Popov, V.A. and Sendov, B., Jackson-type theorems for one-sided approximation by trigonometric polynomials and splines, Math. Note. 26(1979), 889–896.Google Scholar
2. Beatson, R.K., Restricted range approximation by splines and variational inequalities, SIAMJ.Numer. Anal. 19(1982), 372–380.Google Scholar
3. Vore, R.A.De , Leviatan, D.,Convex polynomial and spline approximation in L p ,0 < p < ∞, Constr. Approx. 12(1996), 409–422.Google Scholar
4. DeVore, R.A., Leviatan, D. and Yu, X.M., Polynomial approximation in Lp(0 < p < 1) space, Constr. Approx. 8(1992), 187–201.Google Scholar
5. DeVore, R.A. and Lorentz, G.G., Constructive Approximation, Berlin, Springer-Verlag, 1993.Google Scholar
6. DeVore, R.A. and Popov, V.A., Interpolation of Besov spaces, Trans. Amer. Math. Soc. (1. 305(1988), 397–414.Google Scholar
7. Ditzian, Z. and Totik, V., Moduli of Smoothness, Berlin, Springer-Verlag, 1987.Google Scholar
8. Dzyubenko, G.A., Copositive and positive pointwise approximation, preprint.Google Scholar
9. Hristov, V.H. and Ivanov, K.G., Characterization of best approximations from below and from above, Proc. Conf. Approx. Theory, Kecskemet, (1990), 377–403.Google Scholar
10. Hristov, V.H., Operators for onesided approximation by algebraic polynomials in L p([-1, 1]d),Math. Balkanica (new series) (4) 2(1988), 374–390.Google Scholar
11. Hristov, V.H., Realization of K-functionals on subsets and constrained approximation, Math. Balkanica (3. 4(1990), 236–257.Google Scholar
12. Hu, Y.K., Positive and copositive spline approximation in L p[0, 1], Comput. Math. Appl. 30(1995), 137–146.Google Scholar
13. Hu, Y.K., Kopotun, K. and Yu, X.M., On positive and copositive polynomial and spline approximation in L p[-1, 1],0 < p< ∞, J. Approx. Theory 86(1996), 320–334.Google Scholar
14. Hu, Y.K., Leviatan, D. and Yu, X.M., Copositive polynomial approximation in C [0, 1], J. Analysis 1(1993), 85–90.Google Scholar
15. Hu, Y.K., Copositive polynomial and spline approximation, J. Approx. Theor. 80(1995), 204–218.Google Scholar
16. Hu, Y.K. and Yu, X.M., The degree of copositive approximation and a computer algorithm, SIAM J. Numer. Anal. 33(1996), 388–398.Google Scholar
17. Ivanov, K.G., On a newcharacteristic of functions. II. Direct and converse theorems for the best algebraic approximation in C [-1, 1] and L p[-1, 1], PLISKA Stud. Math. Bulgar. 5(1983), 151–163.Google Scholar
18. Kopotun, K.A., Coconvex polynomial approximation of twice differentiable functions, J.Approx. Theor. 83(1995), 141–156.Google Scholar
19. Kopotun, K.A. , Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. (2. 10(1994), 153–178.Google Scholar
20. Kopotun, K.A. , On copositive approximation by algebraic polynomials, Anal. Math. 21(1995), 269–283.Google Scholar
21. Kopotun, K.A., On k-monotone polynomial and spline approximation in Lp,0 < p < ∞ (quasi)norm, Approximation Theory VIII, World Scientific Publishing Co., (eds. Chui, C. and Schumaker, L.), 1995. 295–302.Google Scholar
22. Kopotun, K.A., Unconstrained and convex polynomial approximation in C [-1, 1], Approx. Theor. Appl. (2) 11(1995), 41–66.Google Scholar
23. Kornejchuk, N.P., Exact Constants in Approximation Theory, Moscow: Izdat. Nauka, 1987. English translation, Cambridge, Cambridge Univ. Press, 1991.Google Scholar
24. Leviatan, D., Monotone and comonotone polynomial approximation revisited, J. Approx. Theor. 53(1988), 1–16.Google Scholar
25. Leviatan, D., The degree of copositive approximation by polynomials, Proc. Amer. Math. Soc. 88(1983), 101–105.Google Scholar
26. Passow, E. and Raymon, L., Copositive polynomial approximation, J.Approx. Theory 12(1974), 299–304.Google Scholar
27. Popov, V.A., The one-sided K-functional and its interpolation space, Proc. Steklov Inst. Math. 4(1985).Google Scholar
28. Roulier, J.A., The degree of copositive approximation, J. Approx. Theor. 19(1977), 253–258.Google Scholar
29. Sendov, B. and Popov, V.A., The averaged moduli of smoothness with applications in numerical methods and approximation, John Wiley & Sons, 1988.Google Scholar
30. Shevchuk, I.A., Approximation by Polynomials and Traces of the Functions Continuous on an Interval, Kiev, Naukova dumka, 1992.Google Scholar
31. Milena Stojanova, The best onesided algebraic approximation in L p[-1, 1] (1 ≤ p≤ ∞), Math. Balkanica 2(1988), 101–113.Google Scholar
32. Yu, X.M., Degree of copositive polynomial approximation, Chinese Ann. Math. 10(1989), 409–415.Google Scholar
33. Zhou, S.P., A counterexample in copositive approximation, Israel J. Math. 78(1992), 75–83.Google Scholar
34. Zhou, S.P. , On copositive approximation, Approx. Theor. Appl. (2. 9(1993), 104–110.Google Scholar