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Erschienen in:
Preliminaries Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

2. Approximation by Certain Operators

verfasst von : Vijay Gupta, Ravi P. Agarwal

Erschienen in: Convergence Estimates in Approximation Theory

Verlag: Springer International Publishing

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Abstract

In the theory of approximation following the well-known Weierstrass theorem, the study on direct results was initiated by Jackson’s classical work [160] on algebraic and trigonometric polynomials of best approximation.

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Vorheriges Kapitel Preliminaries
Nächstes Kapitel Complete Asymptotic Expansion
Literatur
4.
U. Abel, V. Gupta, An estimate of the rate of convergence of a Bézier variant of the Baskakov–Kantorovich operators for bounded variation functions. Demonstr. Math. 36(1), 123–136 (2003)MATHMathSciNet
6.
U. Abel, M. Ivan, On a generalization of an approximation operator defined by A. Lupaş. Gen. Math. 15(1), 21–34 (2007)MATHMathSciNet
7.
U. Abel, V. Gupta, M. Ivan, On the rate of convergence of a Durrmeyer variant of the Meyer–König and Zeller operators. Arch. Inequal. Appl. 1, 1–10 (2003)MATHMathSciNet
14.
T. Acar, A. Aral, V. Gupta, On approximation properties of a new type Bernstein–Durrmeyer operators. Math. Slovaca (to appear)
18.
O. Agratini, B.D. Vecchia, Mastroianni operators revisited. Facta Univ. Ser. Math. Inform. 19, 53–63 (2004)MATH
20.
P.N. Agrawal, V. Gupta, L p -approximation by iterative combination of Phillips operators. Publ. Inst. Math. Nouv. Sér. 52, 101–109 (1992)MathSciNet
21.
P.N. Agrawal, V. Gupta, On convergence of derivatives of Phillips operators. Demonstr. Math. 27, 501–510 (1994)MATHMathSciNet
23.
P.N. Agrawal, H.S. Kasana, On simultaneous approximation by modified Bernstein polynomials. Boll. Unione Mat. Ital. VI. Ser. A 6(3-A), 267–273 (1984)
24.
P.N. Agrawal, A.J. Mohammad, On L p approximation by a linear combination of a new sequence of linear positive operators. Turk. J. Math. 27, 389–405 (2003)MATHMathSciNet
25.
P.N. Agrawal, G. Prasad, Degree of approximation to integrable functions by Kantoorovic polynomials. Boll. Unione Mat. Ital. VI. Ser. A 6(4-A), 323–326 (1985)
31.
A. Aral, V. Gupta, R.P. Agrawal, Applications of q Calculus in Operator Theory, xii (Springer, New York, 2013), p. 262. ISBN:978-1-4614-6945-2/hbk; 978-1-4614-6946-9/ebook
34.
V.A. Baskakov, Primer posledovatel’nosti lineinyh polozitel’nyh operatorov v prostranstve neprerivnyh funkeil (An example of a sequence of linear positive operators in the space of continuous functons). Dokl. Akad. Nauk SSSR 113, 249–251 (1957)MATHMathSciNet
36.
M. Becker, Global approximation theoorems for Szász–Mirakjan and Baskakov operators in polynomial weight spaces. Indiana Univ. Math. J. 27, 127–142 (1978)CrossRefMATHMathSciNet
37.
S. Bernstein, Demonstration du theoreme de Weierstrass, fonde sur le probabilities. Commun. Soc. Math. Kharkow 13, 1–2 (1912–13)
38.
S. Bernstein, Complement a l’ article de E. Voronovskaya “Determination de la forme asymptotique de l’approximation des fonctions par les polynomes de M. Bernstein”. C. R. Acad. Sci. URSS 1932, 86–92 (1932)
46.
R. Bojanic, O. Shisha, Degree of L 1 approximaton to integrable functions by modified Bernstein polynomials. J. Approx. Theory 13, 66–72 (1975)CrossRefMATHMathSciNet
58.
M.-M. Derriennic, Sur l’approximation de functions integrable sur [0, 1] par des polynomes de Bernstein modifies. J. Approx. Theory 31, 323–343 (1981)CrossRefMathSciNet
59.
R. A. DeVore, The Approximation of Continuous Functions by Positive Linear Operators. Lecture Notes in Mathematics, vol. 293 (Springer, New York, 1972)
60.
61.
62.
65.
J.L. Durrmeyer, Une formule d’ inversion de la Transformee de Laplace: Applications a la Theorie des Moments. These de 3e Cycle, Faculte des Sciences de l’ Universite de Paris (1967)
69.
M. Felten, direct and inverse estimates for Bernstein polynomials. Constr. Approx. 14, 459–468 (1998)
70.
G. Feng, Direct and inverse approximation theorems for Baskakov operators with the Jacobi-type weight. Abstr. Appl. Anal. 2011, 13 (2011). Article ID 101852
71.
72.
74.
75.
A.D. Gadzjiv, Theorems of the type of P.P. Korovkin type theorems. Math. Zametki 20(5), 781–786 (1976); English Translation, Math. Notes 20(5–6), 996–998 (1976)
76.
S.G. Gal, On the lower pointwise estimate by Bernstein polynomials. Math. Anal. Approx. Theory 103–108 (2002). ISBN:973-85647-4-3
84.
92.
V. Gupta, A note on modified Baskakov type operators. Approx. Theory Appl. 10(3), 74–78 (1994)MATHMathSciNet
95.
V. Gupta, Simultaneous approximation by Szász–Durrmeyer operators. Math. Stud. 64(1–4), 27–36 (1995)MATH
99.
103.
V. Gupta, Approximation for modified Baskakov Durrmeyer operators. Rocky Mt. J. Math. 29(3), 825–841 (2009)CrossRef
105.
V. Gupta, A new class of Durrmeyer operators. Adv. Stud. Contemp. Math. 23(2), 219–224 (2013)MathSciNet
107.
V. Gupta, U. Abel, The rate of convergence by a new type of Meyer–König and Zeller operators. Fasc. Math. 34, 15–23 (2004)MATHMathSciNet
112.
V. Gupta, P. Gupta, Direct theorem in simultaneous approximation for Szász–Mirakyan Baskakov type operators. Kyungpook Math. J. 41(2), 243–249 (2001)MATHMathSciNet
113.
V. Gupta, N. Ispir, On simultaneous approximation for some modified Bernstein type operators. Int. J. Math. Math. Sci. 2004(71), 3951–3958 (2004)CrossRefMATHMathSciNet
116.
V. Gupta, A. Lupas, Direct results for mixed Beta–Szász type operators. Gen. Math. 13(2), 83–94 (2005)MATHMathSciNet
118.
V. Gupta, P. Maheshwari, Bézier variant of a new Durrmeyer type operators. Rivista di Matematica della Università di Parma 7(2), 9–21 (2003)MathSciNet
119.
V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szász Beta operators. J. Math. Anal. Appl. 321(1), 1–9 (2006)CrossRefMATHMathSciNet
122.
V. Gupta, C. Radu, Statistical approximation properties of q-Baskakov–Kantorovich operators. Cent. Eur. J. Math. 7(4), 809–818 (2009)CrossRefMATHMathSciNet
123.
V. Gupta, A. Sahai, On linear combination of Phillips operators. Soochow J. Math. 19, 313–323 (1993)MATHMathSciNet
124.
V. Gupta, J. Sinha, Simultaneous approximation for generalized Baskakov–Durrmeyer-type operators. Mediterr. J. Math. 4, 483–495 (2007)CrossRefMATHMathSciNet
125.
V. Gupta, G.S. Srivastava, Simultaneous approximation by Baskakov–Szász type operators. Bull. Math. Soc. Sci. de Roumanie (N.S.) 37(85)(3–4), 73–85 (1993)
126.
V. Gupta, G.S. Srivastava, On convergence of derivatives by Szász–Mirakyan–Baskakov type operators. Math. Stud. 64(1–4), 195–205 (1995)MATHMathSciNet
133.
V. Gupta, R. Yadav, Direct estimates in simultaneous approximation for BBS operators. Appl. Math. Comput. 218(22), 11290–11296 (2012)CrossRefMATHMathSciNet
137.
V. Gupta, G.S. Srivastava, A. Sahai, On simultaneous approximation by Szász–Beta operators. Soochow J. Math. 11(1), 1–11 (1995)MathSciNet
140.
V. Gupta, R.N. Mohapatra, Z. Finta, On certain family of mixed summation integral type operators. Math. Comput. Model. 42, 181–191 (2005)CrossRefMATHMathSciNet
145.
V. Gupta, D.K. Verma, P.N. Agrawal, Simultaneous approximation by certain Baskakov–Durrmeyer–Stancu operators. J. Egyptian Math. Soc. 20(3), 183–187 (2012)CrossRefMATHMathSciNet
146.
V. Gupta, R.P. Agarwal, D.K. Verma, Approximation for a new sequence of summation-integral type operators. Adv. Math. Sci. Appl. 23(1), 35–42 (2013)
149.
M. Heilmann, On simultaneous approximation by the method of Baskakov–Durrmeyer operators. Numer. Funct. Anal. Optim. 10, 127–138 (1989)CrossRefMATHMathSciNet
150.
H. Heilmann, M.W. Muller, Direct and converse results on weighted simultaneous approximation by the method of operators of Baskakov–Durrmeyer type. Result. Math. 16, 228–242 (1989)CrossRefMATHMathSciNet
151.
M. Heilmann, G. Tachev, Commutativity, direct and strong converse results for Phillips operators. East J. Approx. 17 (3), 299–317 (2011)MATHMathSciNet
155.
157.
M. Ismail, P. Simeonov, On a family of positive linear integral operators. Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics (Springer, Basel, 2011), pp. 259–274
160.
D. Jackson, The Theory of Approximation. American Mathematical Society Colloquium Publications, vol. 11 (American Mathematical Society, New York, 1930)
161.
164.
L.V. Kantorovich, Sur certains developpments suivant les polynomes de la forme de S. Bernstein, I, II. C. R. Acad. Sci. USSR 20, 563–568 (1930)
168.
H.S. Kasana et al., Modified Szász operators. In Proceedings of International Conference on Mathematical Analysis and Applications, Kuwait (Pergamon, Oxford, 1985), pp. 29–41
169.
H.S. Kasana, P.N. Agrawal, Approximation by linear combination of Szász–Mirakian operators. Colloquium Math. 80(1), 123–130 (1999)MATHMathSciNet
173.
180.
G.G. Lorentz, Zur Theoorie der Polynome von S. Bernstein. Mat. Sb. 2, 543–556 (1937)
181.
G.G. Lorentz, Bernstein Polynomials (University of Toronto Press, Toronto, 1953)MATH
188.
G. Mastroianni, Su un operatore lineare e positivo. Rend. Acc. Sc. Fis. Mat. Napoli 46(4), 161–176 (1979)MATHMathSciNet
190.
C.P. May, On Phillips operators. J. Approx. Theory 20, 315–322 (1997)CrossRef
191.
S.M. Mazhar, V. Totik, Approximation by modified Szász operators. Acta Sci. Math. 49, 257–269 (1985)MATHMathSciNet
192.
W. Meyer-König, K. Zeller, Bernsteinsche Potenzreihen. Stud. Math. 19, 89–94 (1960)MATH
194.
G.M. Mirakjan, Approximation des fonctions continues au moyen de polynomes de la forme \({e}^{-nx}\sum {k = 0}^{m_{n}}C_{k,n}{x}^{k}\) [Approximation of continuous functions with the aid of polynomials of the form \({e}^{-nx}\sum {k = 0}^{m_{n}}C_{k,n}{x}^{k}\)]” (in French). Comptes rendus de l’Acad. des Sci. de l’URSS 31, 201–205 (1941)MATH
199.
R.S. Phillips, An inversion formula for semi-groups of linear operators. Ann. Math. (Ser.2) 59, 352–356 (1954)
204.
Q. Qi, Y. Zhang, Pointwise approximation for certain mixed Szász–Beta operators. Further Progress in Analysis, ed. by H.G.W. Begehr, A.O. Celebi, R.P. Gilbert. Proceedings of 6th International ISAAC Congress (World Scientific Co., Singapore, 2009), pp. 152–163
208.
A. Sahai, G. Prasad, On the rate of convergence for modified Szász–Mirakyan operators on functions of bounded variation. Publ. Inst. Math. Nouv. Sér. (N.S.) 53(67), 73–80 (1993)
209.
F. Schurer, On linear positive operators in approximation theory. Ph.D. Thesis, Technical University of Delft, 1965
212.
O. Shisha, B. Mond, The degree of convergence of sequences of linear positive operators. Proc. Natl. Acad. Sci. 60(4), 196–200 (1968)CrossRefMathSciNet
213.
215.
R.P. Sinha, P.N. Agrawal, V. Gupta, On simultaneous approximation by modified Baskakov operators. Bull. Soc. Math. Belg. Ser. B 43(2), 217–231 (1991)MATHMathSciNet
220.
D.D. Stancu, Approximation of functions by a new class of linear polynomial operators. Rev. Roumanie Math. Pure. Appl. 13, 1173–1194 (1968)MATHMathSciNet
221.
D.D. Stancu, Use of probabilistic methods in the theory of uniform approximation of continuous functions. Rev. Roumanie Math. Pures Appl. 14, 673–691 (1969)MATHMathSciNet
222.
225.
X.H. Sun, On the simultaneous approximation of functions and their derivatives by the Szász–Mirakian operators. J. Approx. Theory 55, 279–288 (1988)CrossRefMATHMathSciNet
227.
O. Szász, Generalizations of S. Bernstein’s polynomial to the infinite interval. J. Res. Nat. Bur. Standards 45, 239–245 (1950)CrossRef
228.
S. Tarabie, On Jain-Beta linear operators. Appl. Math. Inform. Sci. 6(2), 213–216 (2012)MathSciNet
229.
230.
V. Totik, Uniform approximation by Baskakov and Meyer–König Zeller operators. Period. Math. Hungar. 14(3–4), 209–228 (1983)CrossRefMATHMathSciNet
231.
232.
235.
S. Umar, Q. Razi, Approximation of function by a generalized Szász operators. Commun. Fac. Sci. L’Univ D’Ankara 34, 45–52 (1985)
238.
D.K. Verma, V. Gupta, P.N. Agrawal, Some approximation properties of Baskakov–Durrmeyer–Stancu operators. Appl. Math. Comput. 218(11), 6549–6556 (2012)CrossRefMATHMathSciNet
239.
E. Voronovskaja, Determination de la forme asyptotique de approximation des fonctions par les polynomes de M. Bernstein. C. R. Acad. Sci. URSS 1932, 79–85 (1932)MATH
240.
241.
Z. Walczak, V. Gupta, Uniform convergence with certain linear operators. Indian J. Pure. Appl. Math. 38(4), 259–269 (2007)MATHMathSciNet
244.
P.C. Xun, D.X. Zhou, Rate of convergence for Baskakov operators with Jacobi-weights. Acta Math. Appl. Sinica 18, 127–139 (1995)
Metadaten
Titel
Approximation by Certain Operators
verfasst von
Vijay Gupta
Ravi P. Agarwal
Copyright-Jahr
2014
Buch
Convergence Estimates in Approximation Theory

Print ISBN: 978-3-319-02764-7

Electronic ISBN: 978-3-319-02765-4

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-02765-4_2