Piecewise monotone pointwise approximation

Constructive Approximation

Volumn 14, Issue 3, 1998, Pages 311-348

  Dzyubenko, G A ^a,c   Gilewicz, J ^a   Shevchuk, I A ^b,d  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

^c INSTITUTE OF HYDROBIOLOGY   (Ukraine)

^d CNRS   (France)

Author keywords

Piecewise monotone approximation; Polynomial approximation

Indexed keywords

EID: 0032399245     PISSN: 01764276     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003659900077     Document Type: Article

References (47)

1. R. K. BEATSON (1978): The degree of monotone approximation. Pacific J. Math., 74:5-14.
2. R. K. BEATSON, D. LEVIATAN (1983): On comonotone approximation. Canad. Math. Bull., 26:220-224.
3. YU, A. BRUDNYI (1963): Summarizing of a theorem by Timan. Dokl. Akad. Nauk SSSR, 188:1237-1240 (Russian).
4. R. A. DEVORE (1976): Degree of approximation. In: Approximation Theory II (G. G. Lorentz, C. K. Chui, L. Schumaker, eds.). New York: Academic Press, pp. 117-162.
5. R. A. DEVORE (1977): Monotone approximation by polynomials. SIAM J. Math. Anal., 8:906-921.
6. R. A. DEVORE, X. M. YU (1985): Pointwise estimates for monotone polynomial approximation. Constr. Approx., 1:323-331.
7. R. A. DEVORE, D. LEVIATAN, I. A. SHEVCHUK (1997): Approximation of monotone functions: A counter-example. In: Proceedings Chamonix 1996 (A. Le Méhauté, C. Rabut, L. L. Schumaker, eds.). Nashville, TN: Vanderbilt University Press, pp. 95-102.
8. Z. DITZIAN, V. TOTIK (1987): Moduli of Smoothness. Berlin: Springer-Verlag.
9. Z. DITZIAN, D. JIANG, D. LEVIATAN (1992): Shape-preserving polynomial approximation in C[-1, 1]. Math. Proc. Cambridge Philos. Soc., 112:309-316.
10. V. K. DZYADYK (1958): Further improvement of Jackson's theorem on approximation of continuous functions by ordinary polynomials. Dokl. Akad. Nauk SSSR, 121:403-406 (Russian).
11. V. K. DZYADYK (1977): Introduction to the Theory of Uniform Approximation of Functions by Polynomials. Moscow: Nauka, 512 pp. (Russian).
12. G. A. DZYUBENKO (1994): Pointwise estimate of comonotone approximation. Ukrain. Mat. Zh., 46:1467-1472; English translation (1994): Ukrainian Math. J., 46.
13. G. A. DZYUBENKO (1994): Pointwise estimate of comonotone approximation. Ukrain. Mat. Zh., 46:1467-1472; English translation (1994): Ukrainian Math. J., 46.
14. G. A. DZYUBENKO, V. V. LISTOPAD, I. A. SHEVCHUK (1993): Uniform estimates for monotonic polynomial approximation. Ukrain. Mat. Zh., 45:38-43; English translation (1993): Ukrainian Math. J., 45:40-47.
15. G. A. DZYUBENKO, V. V. LISTOPAD, I. A. SHEVCHUK (1993): Uniform estimates for monotonic polynomial approximation. Ukrain. Mat. Zh., 45:38-43; English translation (1993): Ukrainian Math. J., 45:40-47.
16. G. A. DZYUBENKO, V. V. LISTOPAD, I. A. SHEVCHUK (1994): Comonotone approximation pointwise estimates for twice differentiable functions. Preprint. September 1994. CPT-94/P.3067. Marseille, France: CNRS Luminy.
17. G. FREUD (1959): Über die Approximation Reelen Stetiger Functionen Durch Gewohnliche Polynome. Math. Ann., 137:17-25.
18. J. GILEWICZ, I. A. SHEVCHUK (1996): Comonotone approximation. Fundament. i Prik. Mat., 2:319-363 (Russian).
19. G. L. ILIEV (1978): Exact estimates for partially monotone approximation. Anal. Math., 4:181-197.
20. K. A. KOPOTUN, V. V. LISTOPAD (1994): Remarks on monotone and convex approximation by algebraic polynomials. Ukrain. Mat. Zh., 46:1266-1270.
21. D. LEVIATAN (1986): Pointwise estimates for convex polynomial approximation. Proc. Amer. Math. Soc., 98:471-174.
22. D. LEVIATAN (1999): Monotone and comonotone polynomial approximation revisited. J. Approx. Theory, 53:1-16.
23. p. Rocky Mountain J. Math., 1:231-241.
24. G. G. LORENTZ (1972): Monotone approximation. In: Inequalities, III (O. Shisha, ed.). New York: Academic Press, pp. 201-215.
25. G. G. LORENTZ, K. L. ZELLER (1968): Degree of approximation by monotone polynomials I. J. Approx. Theory, 1:501-504.
26. G. G. LORENTZ, K. L. ZELLER (1969): Degree of approximation by monotone polynomials II. J. Approx. Theory, 2:265-269.
27. D. J. NEWMAN (1979): Efficient co-monotone approximation. J. Approx. Theory, 25:189-192.
28. D. J. NEWMAN, E. PASSOW, L. RAYMON (1972): Piecewise monotone polynomial approximation. Trans. Amer. Math. Soc., 172:465-472.
29. S. M. NIKOLSKY (1946): About best approximation of functions by polynomials satisfying Lipschitz condition. Izv. Akad. Nauk SSSR Ser. Math., 10:295-317 (Russian).
30. E. PASSOW, L. RAYMON (1974): Monotone and comonotone approximation. Proc. Amer. Math. Soc., 42:390-394.
31. E. PASSOW, L. RAYMON, J. A. ROULIER (1974): Comonotone polynomial approximation. J. Approx. Theory, 11:221-224.
32. I. A. SHEVCHUK (1989): On co-approximation of monotone functions. Dokl. Akad. Nauk SSSR, 308:537-541; English translation (1990): Soviet Math. Dokl., 40:349-354.
33. I. A. SHEVCHUK (1989): On co-approximation of monotone functions. Dokl. Akad. Nauk SSSR, 308:537-541; English translation (1990): Soviet Math. Dokl., 40:349-354.
34. I. A. SHEVCHUK (1992): Polynomial Approximation and Traces of Functions Continuous on a Segment. Kyiv: Naukova Dumka, 225 pp. (Russian).
35. I. A. SHEVCHUK (1992): Approximation of monotone functions by monotone polynomials. Mat. Sb., 183:63-78; English translation (1993): Russian Acad. Sci. Sb. Math., 76:51-64.
36. I. A. SHEVCHUK (1992): Approximation of monotone functions by monotone polynomials. Mat. Sb., 183:63-78; English translation (1993): Russian Acad. Sci. Sb. Math., 76:51-64.
37. p, 0 < p < 1, for algebraic polynomials and orders of comonotone approximation. Mat. Zametki, 25:107-117; English translation (1979): Math. Notes, 25.
38. p, 0 < p < 1, for algebraic polynomials and orders of comonotone approximation. Mat. Zametki, 25:107-117; English translation (1979): Math. Notes, 25.
39. A. S. SHVEDOV (1980): Comonotone approximation by polynomials. Dokl. Akad. Nauk SSSR, 250:39-42; English translation (1980): Soviet Math. Dokl., 21.
40. A. S. SHVEDOV (1980): Comonotone approximation by polynomials. Dokl. Akad. Nauk SSSR, 250:39-42; English translation (1980): Soviet Math. Dokl., 21.
41. A. S. SHVEDOV (1981): Orders of coapproximation of functions by algebraic polynomials. Mat. Zametki, 6:839-846; English translation (1981): Math. Notes, 30.
42. A. S. SHVEDOV (1981): Orders of coapproximation of functions by algebraic polynomials. Mat. Zametki, 6:839-846; English translation (1981): Math. Notes, 30.
43. A. F. TIMAN (1951): Improvement of Jackson's theorem on the best approximation of continuous functions on a finite segment of the real axis. Dokl. Akad. Nauk SSSR, 78:17-20 (Russian).
44. X. WU, S. P. ZHOU (1992): A conterexample in comonotone approximation. J. Approx. Theory, 69:205-211.
45. X. M. YU (1988): Degree of comonotone polynomial approximation. Approx. Theory Appl., 4:73-78.
46. X. M. YU, Y. P. MA (1993): Private communication.
47. p space. Analysis, 13:363-376.