The rate of convergence of bounded linear processes on spaces of continuous functions

Heiner Gonska

doi:10.33993/jnaat522-1326

Authors

Heiner Gonska Faculty of Mathematics, University of Duisburg-Essen, Germany

DOI:

https://doi.org/10.33993/jnaat522-1326

Keywords:

quantitative Korovkin-type theorems, compact metric spaces, bounded linear operators, positive linear operators, K-functional, multivariate modulus of continuity, least concave majorant of a modulus, optimal constants

Abstract views: 65

Abstract

Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on \(C(X,d)\) are given, where \((X,d)\) is a compact metric space. Special emphasis is on positive linear operators.
As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique.
We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.

Downloads

Download data is not yet available.

References

F. Altomare, M. Campiti, Korovkin-type Approximation Theory and its Applications. Berlin - New York: de Gruyter 1994. https://doi.org/10.1515/9783110884586 DOI: https://doi.org/10.1515/9783110884586

D. Andrica, C. Mustata, An abstract Korovkin-type theorem and applications, Studia Univ. Babes-Bolyai Math., 34 (1989), pp. 44–51, MR 91j:41043.

C. Badea, K-functionals and moduli of smoothness of functions defined on compact metric spaces, Comput. Math. Appl., 30 (1995), pp. 23–31. Zbl. 836.46065. https://doi.org/10.1016/0898-1221(95)00083-6 DOI: https://doi.org/10.1016/0898-1221(95)00083-6

C. Badea, C. Cottin, H.H. Gonska, B ̈ogel functions, tensor products and blending approximation, Math. Nachr., 173 (1995), pp. 25–48. Zbl. 830.41027. http://doi.org/10.1002/mana.19951730103 DOI: https://doi.org/10.1002/mana.19951730103

H. Berens, G.G. Lorentz, Theorems of Korovkin type for positive linear operators on Banach lattices, in: Approximation Theory, (Proc. Int. Sympos. Austin 1973; ed. by G.G. Lorentz), pp. 1–30. New York – San Francisco – London: Acad. Press 1973. MR 48#761.

P.L. Butzer, H. Berens, Semi Groups of Operators and Approximation, Berlin – Heidelberg – New York: Springer 1967. MR 37#5588. https://doi.org/10.1007/978-3-642-46066-1 DOI: https://doi.org/10.1007/978-3-642-46066-1

E. Censor, Quantitative results for positive linear approximation operators, J. Approx. Theory, 4 (1971), pp. 442–450. MR 44#4441. https://doi.org/10.1016/0021-9045(71)90009-8 DOI: https://doi.org/10.1016/0021-9045(71)90009-8

Tian-ping Chen and Wen-ge Zhu, On quantitative approximation, Chin. Sci. Bull.,39 (1994), pp. 1585–1587, Zbl. 817.41030

Gh. Coman, Shepard operators of Birkhoff-type, Calcolo, 35 (1998), pp. 197–203. https://doi.org/10.1007/s100920050016 DOI: https://doi.org/10.1007/s100920050016

R.A. DeVore, The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Mathematics, 293, Berlin – Heidelberg – New York: Springer 1972. MR 54#8100. https://doi.org/10.1007/BFb0059493 DOI: https://doi.org/10.1007/BFb0059493

J. Dieudonne, Grundzuge der modernen Analysis, Bd. 2, Berlin: VEB Deutscher Verlag der Wissenschaften 1975. MR 57#1400b.

V.K. Dzjadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Moscow: Izdat. “Nauka” 1977. MR 58#29579, (in Russian).

G. Freud, Uber positive lineare Approximationsfolgen von stetigen reellen Funktionen auf kompakten Mengen, in: On Approximation Theory, (ed. by P.L. Butzer and J. Korevaar), pp. 233–238. Basel: Birkh ̈auser 1964. MR 31#6088. https://doi.org/10.1007/978-3-0348-4131-3_23 DOI: https://doi.org/10.1007/978-3-0348-4131-3_23

H. Gonska, Konvergenzsatze vom Bohman-Korovkin-Typ fur positive lineare Operatoren, Diplomarbeit, Ruhr-Universit ̈at Bochum, 1975.

H. Gonska, On Mamedov estimates for the approximation of finitely defined operators, in: Approximation Theory III (Proc. Int. Sympos. Austin 1980; ed. by E.W. Cheney), pp. 443-448. New York – San Francisco – London: Acad. Press 1980. MR 82e:41033

H. Gonska, On approximation in spaces of continuous functions, Bull. Austral. Math. Soc., 28 (1983), pp. 411-432. MR 85d:41021. https://doi.org/10.1017/S0004972700021134 DOI: https://doi.org/10.1017/S0004972700021134

H. Gonska, On approximation in C(X), in: Constructive Theory of Functions, (Proc. Int. Conf. Varna 1984; ed. by Bl. Sendov et al.), pp. 364-369. Sofia: Publishing House of the Bulgarian Academy of Sciences 1984. Zbl. 603.41011 .

H. Gonska, Quantitative Approximation in C(X), Habilitationsschrift, Universitat Duisburg 1985. (Schriftenreihe des Fachbereichs Mathematik SM-DU-94 (1986), Universitat Duisburg/Germany). Zbl. 688.41024.

H. Gonska, The second order modulus again: some (still?) open problems. Schriftenreihe des Fachbereichs Mathematik, SM-DU-426 (1998), Universitat Duisburg/Germany.

H. Gonska, J. Meier, On approximation by Bernstein type operators: Best constants, Studia Sci. Math. Hungar., 22 (1987), pp. 287-297. MR 89d:41041.

M.A. Jimenez Pozo, Sur les operateurs lineaires positifs et la methode des fonctions tests, C.R. Acad. Sci. Paris Ser. A, 278 (1974), pp. 149-152. MR 48#11861.

M.A. Jimenez Pozo, On the problem of the convergence of a sequence of linear operators, Moscow Univ. Math. Bull., 33 (1978) no. 4, pp. 1–8. MR 80c:41007

M.A. Jimenez Pozo, Deformation de la convexite et theoremes du type Korovkin, C.R. Acad. Sci. Paris Ser. A, 290 (1980), pp. 213-215. MR 81a:41054.

M.A. Jimenez Pozo, Approximacion polinomial de funciones de varias variables con ayuda de teoremas de tipo Korovkin, Cienc. Mat. (Havana), 1 (1980) no. 1, pp. 7–17. MR 82m:41021

M.A. Jimenez Pozo, Convergence of sequences of linear functionals, Z. Angew. Math. Mech., 61 (1981), pp. 495–500. MR 83b:41024. https://doi.org/10.1002/zamm.19810611004 DOI: https://doi.org/10.1002/zamm.19810611004

M.A. Jimenez Pozo, Quantitative theorems of Korovkin type in bounded function spaces, in: Constructive Function Theory ’81, (Proc. Int. Conf. Varna, Bulgaria, 1981; ed. by Bl. Sendov et al.), pp. 488-494. Sofia: Publishing House of the Bulg. Acad. of Sciences 1983. MR 84m:41036.

M.A. Jimenez Pozo, M. Baile Baldet, Estimados del orden de convergencia de una sucesion de operadores lineales en espacios de functiones con peso, Cienc. Mat. (Havana), 2 (1981) no. 1, pp. 16–28. MR 84e:41024

H. Knauf, Studiumuber die Shepard-Methoden, Diplomarbeit, Universitat Duisburg/Germany, 1997.

N.P. Korneicuk, Extremal Problems of Approximation Theory, Moscow: Izdat. “Nauka” 1976. MR 56#6244, (in Russian).

N.P. Korneicuk, On best constants in Jackson’s inequality for continuous periodic functions, Mat. Zametki, 32 (1982) no. 5, pp. 669-674. MR 84c:41015, (in Russian).

P.P. Korovkin, Linear Operators and Approximation Theory, Delhi: Hindustan, 1960. MR 27#561.

O.G. Kukus, Estimate of the rate of convergence of linear positive operators in the case of functions of a finite number of variables, in: Mathematical Analysis and Probability, pp. 95-100. Kiev: “Naukova Dumka” 1978. MR 82m:41022, (in Russian).

A. Lupas ̧, M.W. Muller, Approximation properties of the Meyer-Konig and Zeller operators, Aequationes Math. 5 (1970), pp. 19-37. MR 43#5217. DOI: https://doi.org/10.1007/BF01819267

R.G. Mamedov, On the order of approximation of functions by linear positive operators, Dokl. Akad. Nauk SSSR, 128 (1959), pp. 674-676. MR 22#900, (in Russian).

R.G. Mamedov, Approximation of Functions by Linear Operators (Azerbaijani). Baku: A.D.N., 1967.

G. Mastroianni, Sull’ approssimazione di funzioni continue mediante operatori lineari, Calcolo, 14 (1977), pp. 343-368. MR 80a:41018. https://doi.org/10.1007/BF02575991 DOI: https://doi.org/10.1007/BF02575991

K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann., 100 (1928), pp. 75-163. FdM 1928, 622. https://doi.org/10.1007/BF01448840 DOI: https://doi.org/10.1007/BF01448840

B.S. Mitjagin, E.M. Semenov, Lack of interpolation of linear operators in spaces of smooth functions, Math. USSR-Izv., 11 (1977), pp. 1229-1266. MR 58#2234. https://doi.org/10.1070/IM1977v011n06ABEH001767 DOI: https://doi.org/10.1070/IM1977v011n06ABEH001767

B. Mond, On the degree of approximation by linear positive operators, J. Approx. Theory, 18 (1976), pp. 304–306. MR 54#10949. DOI: https://doi.org/10.1016/0021-9045(76)90022-8

M.W. Muller, H. Walk, Konvergenz- und Guteaussagen f ̈ur die Approximation durch Folgen linearer positiver Operatoren, in: Constructive Theory of Functions, (Proc. Int. Conf. Varna, Bulgaria, 1970; ed. by B. Penkov and D. Vaˇcov), pp. 221-233. Sofia: Publishing House of the Bulgarian Academy of Sciences 1972. MR 51#3761.

D.J. Newman, H.S. Shapiro, Jackson’s theorem in higher dimension, in: On Approximation Theory, (Proc. Conf. Math. Res. Inst. Oberwolfach 1963; ed. by P.L. Butzer and J. Korevaar), pp. 208-219. Basel: Birkhauser 1964. MR 32#310. https://doi.org/10.1007/978-3-0348-4131-3_20

T. Nishishiraho, The rate of convergence of positive linear approximation processes, in: Approximation Theory IV, (Proc. Int. Sympos. College Station; ed. by. C.K. Chui et al.), pp. 635–641. New York – San Francisco – London: Acad. Press 1983. Zbl. 542.41022.

T. Nishishiraho, Convergence of positive linear approximation processes, Tohoku Math. J., 35 (1983), pp. 441-458. Zbl. 525.41019. https://doi.org./10.2748/tmj/1178229002 DOI: https://doi.org/10.2748/tmj/1178229002

T. Nishishiraho, The degree of approximation by positive linear approximation processes, Bull. College of Education, Univ. of the Ryukyus, 28 (1985), pp. 7–36.

J. Peetre, On the connection between the theory of interpolation spaces and approximation theory, in: Approximation Theory, (Proc. Conf. Constructive Theory of Functions, Budapest, 1969; ed. by G. Alexits, S.B. Steˇckin), pp. 351-363. Budapest: Akademia Kiado 1972. MR 53#8760.

T. Popoviciu, Uber die Konvergenz von Folgen positiver Operatoren, An. Sti. Univ. “Al. I. Cuza” Iasi (N.S.), 17 (1971), pp. 123-132. MR 46#9597.

I. Rasa, Test sets in quantitative Korovkin approximation. Studia Univ. Babes-Bolyai Math., 36 (1991), pp. 97–100. Zbl. 887.41024.

Ch. Richter, Entropy, approximation quantities and the asymptotics of the modulus of continuity, To appear in Math. Nachr.

Ch. Richter, A chain of controllable partitions of unity on the cube and the approximation of Holder continuous functions, to appear in Illinois J. Math.

Ch. Richter, I. Stephani, Entropy and the approximation of bounded functions and operators, Arch. Math., 67 (1996), pp. 478–492. Zbl. 865.41024. https://doi.org/10.1007/BF01270612 DOI: https://doi.org/10.1007/BF01270612

W. Rinow, Die innere Geometrie der metrischen Raume Berlin –Gottingen – Heidelberg: Springer 1961. MR 23#A1290. DOI: https://doi.org/10.1007/978-3-662-11499-5

L.L. Schumaker, Spline Functions: Basic Theory, New York: J. Wiley & Sons, 1981. MR 82j:41001.

F. Schurer, F.W. Steutel, On the degree of approximation of functions in C1[0, 1] by Bernstein polynomials, T.H.-Report 75-WSK-07, Eindhoven Univ. of Technology 1975. Zbl. 324.41004.

F. Schurer, F.W. Steutel, On the degree of approximation of functions in C1[0, 1] by the operators of Meyer-K ̈onig and Zeller, J. Math. Anal. Appl., 63 (1978), pp. 719–728. MR 58#12120. https://doi.org/10.1016/0022-247X(78)90067-7 DOI: https://doi.org/10.1016/0022-247X(78)90067-7

H.S. Shapiro, Topics in Approximation Theory, Lecture Notes in Mathematics, 187. Berlin – Heidelberg – New York: Springer 1971. MR 55#10902. DOI: https://doi.org/10.1007/BFb0058976

O. Shisha, B. Mond, The degree of approximation to periodic functions by linear positive operators, J. Approx. Theory, 1 (1968), pp. 335–339. MR 39#1883. DOI: https://doi.org/10.1016/0021-9045(68)90011-7

O. Shisha, B. Mond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60 (1968), pp. 1196–1200. MR 37#5582. https://doi.org./10.1073/pnas.60.4.1196 DOI: https://doi.org/10.1073/pnas.60.4.1196

P.C. Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein Polynomen, Numer. Math., 3 (1961), pp. 107–116. MR 23#A459. https://doi.org/10.1007/BF01386008 DOI: https://doi.org/10.1007/BF01386008

S.B. Steckin, Remarks on the theorem of Jackson (Russian), Tr. Matem. In-ta AN SSSR 88 (1967), 17-19. MR 36#5593.

I. Stephani, Entropy and the approximation of continuous functions, Arch. Math., 58 (1992), pp. 280–287. Zbl. 723.41019. https://doi.org/10.1007/BF01292929 DOI: https://doi.org/10.1007/BF01292929

I. Stephani, Embedding maps of generalized H ̈older spaces-entropy and approximation, Rend. Circ. Mat. Palermo (2) Suppl., 52 (1998), pp. 793–804. Zbl. 980.44823.

M. Wolff, Uber das Spektrum von Verbandshomomorphismen in C(X), Math. Ann., 182 (1969), pp. 161-169. MR 40#776. https://doi.org/10.1007/BF01350319 DOI: https://doi.org/10.1007/BF01350319

Wen-ge Zhu, The Korovkin-type theorem on the compact metric spaces, J. Math. Res. Exp., 14 (1994), pp. 569–573. Zbl. 862.54025, (in Chinese).

V. Zuk and G. Natanson, About the question of approximation of functions by means of positive operators, Tartu Riikl. ̈Ul. Toimetised, (430) (1977), pp. 58-69. MR 58#1888, (in Russian).

V. Zuk and G. Natanson, On the accuracy of approximation of periodic functions by linear methods, Vestnik Leningrad Univ. Math., 8 (1980), pp. 277-284. Zbl. 437.41016.