Constrained Approximation in Sobolev Spaces

Hu, Y. K. ; Kopotun, K. A. ; Yu, X. M.

Canadian journal of mathematics, Tome 49 (1997) no. 1, pp. 74-99

Résumé

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.

Détail

DOI : 10.4153/CJM-1997-004-2

Mots-clés : 41A10, 41A15, 41A25, 41A29, Constrained approximation, polynomials, splines, degree of approximation, Lp space, Sobolev space

Bibliographie

[1] 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] 2. Beatson, R.K., Restricted range approximation by splines and variational inequalities, SIAMJ.Numer. Anal. 19(1982), 372–380. Google Scholar

[3] 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] 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] 5. DeVore, R.A. and Lorentz, G.G., Constructive Approximation, Berlin, Springer-Verlag, 1993. Google Scholar

[6] 6. DeVore, R.A. and Popov, V.A., Interpolation of Besov spaces, Trans. Amer. Math. Soc. (1. 305(1988), 397–414. Google Scholar

[7] 7. Ditzian, Z. and Totik, V., Moduli of Smoothness, Berlin, Springer-Verlag, 1987. Google Scholar

[8] 8. Dzyubenko, G.A., Copositive and positive pointwise approximation, preprint. Google Scholar

[9] 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] 10. Hristov, V.H., Operators for onesided approximation by algebraic polynomials inp([-1, 1]d),Math. Balkanica (new series) (4) 2(1988), 374–390. Google Scholar

[11] 11. Hristov, V.H., Realization of K-functionals on subsets and constrained approximation, Math. Balkanica (3. 4(1990), 236–257. Google Scholar

[12] 12. Hu, Y.K., Positive and copositive spline approximation inp[0, 1], Comput. Math. Appl. 30(1995), 137–146. Google Scholar

[13] 13. Hu, Y.K., Kopotun, K. and Yu, X.M., On positive and copositive polynomial and spline approximation inp[-1, 1],0 < p< ∞, J. Approx. Theory 86(1996), 320–334. Google Scholar

[14] 14. Hu, Y.K., Leviatan, D. and Yu, X.M., Copositive polynomial approximation in[0, 1], J. Analysis 1(1993), 85–90. Google Scholar

[15] 15. Hu, Y.K., Copositive polynomial and spline approximation, J. Approx. Theor. 80(1995), 204–218. Google Scholar

[16] 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] 17. Ivanov, K.G., On a newcharacteristic of functions. II. Direct and converse theorems for the best algebraic approximation in[-1, 1] andp[-1, 1], PLISKA Stud. Math. Bulgar. 5(1983), 151–163. Google Scholar

[18] 18. Kopotun, K.A., Coconvex polynomial approximation of twice differentiable functions, J.Approx. Theor. 83(1995), 141–156. Google Scholar

[19] 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] 20. Kopotun, K.A. , On copositive approximation by algebraic polynomials, Anal. Math. 21(1995), 269–283. Google Scholar

[21] 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] 22. Kopotun, K.A., Unconstrained and convex polynomial approximation in[-1, 1], Approx. Theor. Appl. (2) 11(1995), 41–66. Google Scholar

[23] 23. Kornejchuk, N.P., Exact Constants in Approximation Theory, Moscow: Izdat. Nauka, 1987. English translation, Cambridge, Cambridge Univ. Press, 1991. Google Scholar

[24] 24. Leviatan, D., Monotone and comonotone polynomial approximation revisited, J. Approx. Theor. 53(1988), 1–16. Google Scholar

[25] 25. Leviatan, D., The degree of copositive approximation by polynomials, Proc. Amer. Math. Soc. 88(1983), 101–105. Google Scholar

[26] 26. Passow, E. and Raymon, L., Copositive polynomial approximation, J.Approx. Theory 12(1974), 299–304. Google Scholar

[27] 27. Popov, V.A., The one-sided K-functional and its interpolation space, Proc. Steklov Inst. Math. 4(1985). Google Scholar

[28] 28. Roulier, J.A., The degree of copositive approximation, J. Approx. Theor. 19(1977), 253–258. Google Scholar

[29] 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] 30. Shevchuk, I.A., Approximation by Polynomials and Traces of the Functions Continuous on an Interval, Kiev, Naukova dumka, 1992. Google Scholar

[31] 31. Milena Stojanova, The best onesided algebraic approximation inp[-1, 1] (1 ≤ p≤ ∞), Math. Balkanica 2(1988), 101–113. Google Scholar

[32] 32. Yu, X.M., Degree of copositive polynomial approximation, Chinese Ann. Math. 10(1989), 409–415. Google Scholar

[33] 33. Zhou, S.P., A counterexample in copositive approximation, Israel J. Math. 78(1992), 75–83. Google Scholar

[34] 34. Zhou, S.P. , On copositive approximation, Approx. Theor. Appl. (2. 9(1993), 104–110. Google Scholar