Convergence in Variation and Rates of Approximation for Bernstein-Type Polynomials and Singular Convolution Integrals

Analysis (Germany)

Volumn 23, Issue 4, 2003, Pages 299-340

  Butzer, P L ^a   Stens, R L ^a   Bardaro, C ^b   Vinti, G ^b  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b UNIVERSITY OF PERUGIA   (Italy)

Author keywords

absolutely continuous functions; Bernstein type polynomials; convergence in variation; multivariate functions of bounded variation; order of approximation; singular convolution integrals

Indexed keywords

EID: 29944433767     PISSN: 01744747     EISSN: 21966753     Source Type: Journal    
DOI: 10.1524/anly.2003.23.4.299     Document Type: Article

References (61)

1. Bardaro, C., Nuclei, derivata simmetrica e perimetro di una misura [Kemels, the symmetric derivative and the perimeter of a measure], Atti Sem. Mat. Fis. Univ. Modena 28 (1979), no. 2, 373–386.
2. Bardare, C. and Vinti, G., On convergence of moment operators with respect to ip-variation, Appl. Anal. 41 (1991), no. 1–4, 247–256.
3. Becker, M., Global approximation theorems for Szdsz-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J. 27 (1978), no. 1, 127–142.
4. Becker, M., Fink, K.M., and Nessel, R.J., Approximation by Bernstein polynomials in weighted BV-spaces, In: Functions, Series, Operators, vol. II, Proc. Conf., Budapest, Hungary, 1980 (Sz.-Nagy, B. and Szabados, J., eds.), Colloquia Mathematica Societatis Janos Bolyai, vol. 35, North-Holland Publishing Company, Amsterdam, and Janos Bolyai Mathematical Society, Budapest, 1983, pp. 221–231.
5. Becker, M., Lautner, K.J., Nessel, R.J., and Worms, G.J., On global approximation by Kantorovitch polynomials in V, In: Constructive Function Theory 81, Proc. Conf., Varna, Bulgaria, 1981 (Sendov, B., Boyanov, B., Vacov, D., Maleev, R., Markov, S., and Boyanov, T., eds.), Publishing House of the Bulgarian Academy of Sciences, Sofia, 1983, pp. 217–223.
6. Becker, M. and Nessel, R.J., Some global direct estimates for Kantorovitch polynomials, Analysis 1 (1981), no. 2, 117–127.
7. Berens, H. and DeVore, R.A., Quantitative Korovkin theorems for L-p-spaces, In: Lorentz et al. [39], pp. 289–298.
8. Butzer, P.L., Dominated convergence of Kantorovitch polynomials in the space LP, Trans. Roy. Soc. Canada. Sect. III. (3) 46 (1952), 23–27.
9. Butzer, P.L., Linear combinations of Bernstein polynomials, Canadian J. Math. 5 (1953), 559–567.
10. Butzer, P.L., Summability of generalized Bernstein polynomials. I, Duke Math. J. 22 (1955), 617–623.
11. Butzer, P.L., Fischer, A., and Stens, R.L., Generalized sampling approximation of multivariate signals; theory and some applications, Note Mat. 10 Suppl. No. 1 (1990), 173–191.
12. Butzer, P.L., Fischer, A., and Stens, R.L., Generalized sampling approximation of multivariate signals; general theory, Atti Sem. Mat. Fis. Univ. Modena 41 (1993), no. 1, 17–37.
13. Butzer, P.L. and Nessel, R.J., Fourier Analysis and Approximation, Birkhauser Verlag, Basel, and Academic Press, New York, 1971.
14. Butzer, P.L., Schmidt, M., Stark, E.L., and Vogt, L., Central factorial numbers; their main properties and some applications, Numer. Funet. Anal. Optim. 10 (1989), no. 5–6, 419–488.
15. Cesari, L., Sulle funzioni a variazione limitata, Ann. Scuola Norm. Sup. Pisa 5 (1936), 299–313.
16. Chui, C.K., Multivariate Splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.
17. de Boor, C., Splinefunktionen, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1990.
18. de Boor, C. and Hollig, K., Recurrence relations for multivariate B-splines, Proc. Amer. Math. Soc. 85 (1982), no. 3, 397–400.
19. de Boor, C. and Hollig, K., B-splines from parallelepipeds, J. Analyse Math. 42 (1982/1983), 99–115.
20. DeVore, R.A., Degree of approximation, In: Lorentz et al. [39], pp. 117–161.
21. DeVore, R.A. and Lorentz, G.G., Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, SpringerVerlag, Berlin, 1993.
22. Ditzian, Z. and Totik, V., Moduli of Smoothness, Springer Series in Computational Mathematics, vol. 9, Springer-Verlag, New York, 1987.
23. Garg, K.M., Characterizations of absolutely continuous and singular functions, In: Approximation Theory, Proc. Conf, on Constructive Theory of Functions, Budapest, Hungary, 1969 (Alexits, G. and Steckin, S.B., eds.), Akademiai Kiado, Budapest, 1972, pp. 183–188.
24. Giusti, E., Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhauser Verlag, Basel, 1984.
25. Goffman, C., Lower-semi-continuity and area functionals. I. The non-parametric case, Rend. Circ. Mat. Palermo (2) 2 (1953), 203–235 (1954).
26. Gorlich, E. and Stark, E.L., Uber beste Konstanten und asymptotische Entwicklungen positiver Faltungsintegrale und deren Zusammenhang mit dem Saturationsproblem, Jahresber. Deutsch. Math.-Verein. 72 (1970), 18–61.
27. Grundmann, A., Inverse theorems for Kantorovic polynomials, In: Fourier Analysis and Approximation Theory, I, Proc. Coll., Budapest, Hungary, 1976 (Alexits, G. and Turan, P., eds.), Colloquia Mathematica Societatis Janos Bolyai, vol. 19, North-Holland Publishing Co., Amsterdam, 1978, pp. 395–401.
28. Hardy, G.H., Littlewood, J.E., and Polya, G., Inequalities, 2 ed., Cambridge University Press, Cambridge, 1952.
29. Hoeffding, W., The Li norm of the approximation error for Bemstein-type polynomials, J. Approximation Theory 4 (1971), 347–356.
30. Hoffman, K., Analysis in Euclidean Space, Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.
31. Hollig, K., Box splines, In: Approximation Theory V, Proc. Conf., College Station, TX, USA, 1986 (Chui, C.K., Schumaker, L.L., and Ward, J.D., eds.), Academic Press Inc., Boston, MA, 1986, pp. 71–95.
32. Jetter, K., A short survey on cardinal interpolation by box splines, In: Topics in Multivariate Approximation, Proc. Workshop, University of Chile, Santiago, Chile, 1986 (Chui, C.K., Schumaker, L.L., and Utreras, F.I., eds.), Academic Press Inc., Boston, MA, 1987, pp. 125–139.
33. Johnen, H., Inequalities connected with the moduli of smoothness, Mat. Vesnik 9(24) (1972), 289–303.
34. Kantorovich, L.V., Sur certains developpements suivant les polynomes de la forme de S. Bernstein, I, II, C.R. (Doklady) Acad. Sci. URSS 2 (1930), 563–568, 595–600.
35. Krickeberg, K., Distributionen, Funktionen beschrankter Variation und Lebesguescher Inhalt nichtparametrischer Flachen, Ann. Mat. Pura Appl. (4) 44 (1957), 105–133.
36. Lasser, R., Introduction to Fourier series, Monographs and Textbooks in Pure and Applied Mathematics, vol. 199, Marcel Dekker Inc., New York, 1996.
37. Lorentz, G.G., Zur Theorie der Polynome von S. Bernstein, Mat. Sb. 2 (1937), 543–556.
38. Lorentz, G.G., Bernstein Polynomials, University of Toronto Press, Toronto, 1953; 2 ed., Chelsea Publishing Co., New York, 1986.
39. Lorentz, G.G., Chui, C.K., and Schumaker, L.L. (eds.), Approximation Theory, II, Proc. Symp., University of Texas, Austin, TX, USA, 1976, Academic Press Inc., New York, 1976.
40. Maier, V., The L\ saturation class of the Kantorovic operator, J. Approximation Theory 22 (1978), no. 3, 223–232.
41. Mairhuber, J.C., Schoenberg, I.J., and Williamson, R.E., On variation diminishing transformations of the circle, Rend. Circ. Mat. Palermo (2) 8 (1959), 241–270.
42. Micchelli, C.A., On a numerically efficient method for computing multivariate B-splines, In: Multivariate Approximation Theory, Proc. Conf., Mathematical Research Institute, Oberwol-fach, Germany, 1979 (Schempp, W. and Zeller, K., eds.), ISNM, vol. 51, Birkhauser Verlag, Basel, 1979, pp. 211–248.
43. Mirakyan, G.M., Approximation des fonctions continues au moyen de polynomes de la forme C.R. (Doklady) Acad. Sci. URSS (N.S.) 31 (1941), 201–205.
44. Musielak, J. and Orlicz, W., On generalized variations. I, Studia Math. 18 (1959), 11–41.
45. Polya, G. and Schoenberg, I.J., Remarks on de la Vallee Poussin means and convex conformai maps of the circle, Pacific J. Math. 8 (1958), 295–334.
46. Rado, T., Length and Area, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, 1948.
47. Sciamannini, S. and Vinti, G., Convergence and rate of approximation in BV for a class of integral operators, Approx. Theory Appl. (N.S.) 17 (2001), no. 4, 17–35.
48. Serrin, J., On the differentiability of functions of several variables, Arch. Rational Mech. Anal. 7 (1961), 359–372.
49. Stein, E.M., Functions of exponential type, Ann. of Math. (2) 65 (1957), 582–592.
50. Stein, E.M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, N.J., 1971.
51. Szasz, O., Generalization of S. Bernstein's polynomials to the infinite interval, J. Research Nat. Bur. Standards 45 (1950), 239–245.
52. Tonelli, L., Su alcuni concetti dellanalisi moderna, Ann. Scuola Norm. Super. Pisa. (2) 11 (1942), 107–118.
53. Totik, V., Approximation in L1 by Kantorovich polynomials, Acta Sci. Math. (Szeged) 46 (1983), no. 1–4, 211–222.
54. Totik, V., Problems and solutions concerning Kantorovich operators, J. Approx. Theory 37 (1983), no. 1, 51–68.
55. Totik, V., An interpolation theorem and its applications to positive operators, Pacific J. Math. III (1984), no. 2, 447–481.
56. Vinti, C., Perimetro-variazione, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 201–231.
57. Vinti, C., Sullapprossimazione in perimetro e in area, Atti Sem. Mat. Fis. Univ. Modena 13 (1964), 187–197.
58. Vinti, C., Espressioni che danno larea di una superficie, Atti Sem. Mat. Fis. Univ. Modena 17 (1968), no. 2, 289–350.
59. Vinti, G., Generalized ip-variation in the sense of Vitali: estimates for integral operators and applications in fractional calculus, Comment. Math. Prace Mat. 34 (1994), 199–213.
60. Vogt, L., Approximation by linear combinations of positive convolution integrals, J. Approx. Theory 57 (1989), no. 2, 178–201.
61. Wickeren, E. van, Weak-type inequalities for Kantorovitch polynomials and related operators, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 1, 111–120.