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Erschienen in:
Introduction of q-Calculus Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

5.  q-Summation–Integral Operators

verfasst von : Ali Aral, Vijay Gupta, Ravi P. Agarwal

Erschienen in: Applications of q-Calculus in Operator Theory

Verlag: Springer New York

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Abstract

Aral and Gupta [32], proposed a q-analogue of the Baskakov operators and investigated its approximation properties. In continuation of their work they introduced Durrmeyer-type modification of q-Baskakov operators. These operators, opposed to Bernstein–Durrmeyer operators, are defined to approximate a function f on \(\left [0,\ \infty \right )\). The Durrmeyer-type modification of the q-Bernstein operators was first introduced in [48].

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Vorheriges Kapitel q-Bernstein-Type Integral Operators
Nächstes Kapitel Statistical Convergence of q-Operators
Literatur
1.
U. Abel, V. Gupta, An estimate of the rate of convergence of a Bezier variant of the Baskakov–Kantorovich operators for bounded variation functions. Demons. Math. 36 (1), 123–136 (2003)MathSciNetMATH
2.
U. Abel, M. Ivan, Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences. Calcolo 36, 143–160 (1999)MathSciNetMATH
3.
U. Abel, V. Gupta, R.N. Mohapatra, Local approximation by a variant of Bernstein Durrmeyer operators. Nonlinear Anal. Ser. A: Theor. Meth. Appl. 68 (11), 3372–3381 (2008)MathSciNetMATH
4.
M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards Applied Mathematics, Series 55, Issued June (Dover, New York, 1964)
5.
R.P. Agarwal, V. Gupta, On q-analogue of a complex summation-integral type operators in compact disks. J. Inequal. Appl. 2012, 111 (2012). doi:10.1186/1029-242X-2012-111MathSciNet
6.
O. Agratini, Approximation properties of a generalization of Bleimann, Butzer and Hahn operators. Math. Pannon. 9, 165–171 (1988)MathSciNet
7.
O. Agratini, A class of Bleimann, Butzer and Hahn type operators. An. Univ. Timişoara Ser. Math. Inform. 34, 173–180 (1996)MathSciNet
8.
O. Agratini, Note on a class of operators on infinite interval. Demons. Math. 32, 789–794 (1999)MathSciNetMATH
9.
O. Agratini, Linear operators that preserve some test functions. Int. J. Math. Math. Sci. 8, 1–11 (2006) [Art ID 94136]
10.
O. Agratini, G. Nowak, On a generalization of Bleimann, Butzer and Hahn operators based on q-integers. Math. Comput. Model. 53 (5–6), 699–706 (2011)MathSciNetMATH
11.
J.W. Alexander, Functions which map the interior of the unit circle upon simple region. Ann. Math. Sec. Ser. 17, 12–22 (1915)MATH
12.
F. Altomare, R. Amiar, Asymptotic formula for positive linear operators. Math. Balkanica (N.S.) 16 (1–4), 283–304 (2002)
13.
F. Altomare, M. Campiti, Korovkin Type Approximation Theory and Its Application (Walter de Gruyter Publications, Berlin, 1994)
14.
R. Álvarez-Nodarse, M.K. Atakishiyeva, N.M. Atakishiyev, On q-extension of the Hermite polynomials H n x with the continuous orthogonality property on R. Bol. Soc. Mat. Mexicana (3) 8, 127–139 (2002)
15.
G.A. Anastassiou, Global smoothness preservation by singular integrals. Proyecciones 14 (2), 83–88 (1995)MathSciNetMATH
16.
G.A. Anastassiou, A. Aral, Generalized Picard singular integrals. Comput. Math. Appl. 57 (5), 821–830 (2009)MathSciNetMATH
17.
G.A. Anasstasiou, A. Aral, On Gauss–Weierstrass type integral operators. Demons. Math. XLIII (4), 853–861 (2010)
18.
G.A. Anastassiou, S.G. Gal, Approximation Theory: Moduli of Continuity and Global Smoothness Preservation (Birkhäuser, Boston, 2000)
19.
G.A. Anastassiou, S.G. Gal, Convergence of generalized singular integrals to the unit, univariate case. Math. Inequal. Appl. 3 (4), 511–518 (2000)MathSciNetMATH
20.
G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)MATH
21.
T.M. Apostal, Mathematical Analysis (Addison-Wesley, Reading, 1971)
22.
A. Aral, On convergence of singular integrals with non-isotropic kernels. Comm. Fac. Sci. Univ. Ank. Ser. A1 50, 88–98 (2001)
23.
A. Aral, On a generalized λ-Gauss–Weierstrass singular integrals. Fasc. Math. 35, 23–33 (2005)MathSciNetMATH
24.
A. Aral, On the generalized Picard and Gauss Weierstrass singular integrals. J. Comput. Anal. Appl. 8 (3), 246–261 (2006)MathSciNet
25.
A. Aral, A generalization of Szász–Mirakyan operators based on q-integers. Math. Comput. Model. 47 (9–10), 1052–1062 (2008)MathSciNetMATH
26.
A. Aral, Pointwise approximation by the generalization of Picard and Gauss–Weierstrass singular integrals. J. Concr. Appl. Math. 6 (4), 327–339 (2008)MathSciNetMATH
27.
A. Aral, O. Doğru, Bleimann Butzer and Hahn operators based on q-integers. J. Inequal. Appl. 2007, 12 pp. (2007) [Article ID 79410]
28.
A. Aral, S.G. Gal, q-Generalizations of the Picard and Gauss–Weierstrass singular integrals. Taiwan. J. Math. 12 (9), 2501–2515 (2008)
29.
A. Aral, V. Gupta, q-Derivatives and applications to the q-Szász Mirakyan operators. Calcalo 43 (3), 151–170 (2006)
30.
31.
A. Aral, V. Gupta, On the Durrmeyer type modification of the q Baskakov type operators. Nonlinear Anal.: Theor. Meth. Appl. 72 (3–4), 1171–1180 (2010)
32.
33.
A. Aral, V. Gupta, Generalized Szász Durrmeyer operators. Lobachevskii J. Math. 32 (1), 23–31 (2011)MathSciNetMATH
34.
N.M. Atakishiyev, M.K. Atakishiyeva, A q-analog of the Euler gamma integral. Theor. Math. Phys. 129 (1), 1325–1334 (2001)MATH
35.
A. Attalienti, M. Campiti, Bernstein-type operators on the half line. Czech. Math. J. 52 (4), 851–860 (2002)MathSciNetMATH
36.
D. Aydin, A. Aral, Some approximation properties of complex q-Gauss–Weierstrass type integral operators in the unit disk. Oradea Univ. Math. J. Tom XX(1), 155–168 (2013)
37.
V.A. Baskakov, An example of sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk. SSSR 113, 249–251 (1957)MathSciNetMATH
38.
C. Berg, From discrete to absolutely continuous solution of indeterminate moment problems. Arap. J. Math. Sci. 4 (2), 67–75 (1988)
39.
G. Bleimann, P.L. Butzer, L. Hahn, A Bernstein type operator approximating continuous function on the semi-axis. Math. Proc. A 83, 255–262 (1980)MathSciNet
40.
K. Bogalska, E. Gojtka, M. Gurdek, L. Rempulska, The Picard and the Gauss–Weierstrass singular integrals of functions of two variable. Le Mathematiche LII (Fasc 1), 71–85 (1997)
41.
F. Cao, C. Ding, Z. Xu, On multivariate Baskakov operator. J. Math. Anal. Appl. 307, 274–291 (2005)MathSciNetMATH
42.
E.W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966)MATH
43.
I. Chlodovsky, Sur le développement des fonctions d éfinies dans un interval infini en séries de polynômes de M. S. Bernstein. Compos. Math. 4, 380–393 (1937)MathSciNet
44.
W. Congxin, G. Zengtai, On Henstock integral of fuzzy-number-valued functions, I. Fuzzy Set Syst. 115 (3), 377–391 (2000)MATH
45.
O. Dalmanoglu, Approximation by Kantorovich type q-Bernstein operators, in Proceedings of the 12th WSEAS International Conference on Applied Mathematics, Cairo, Egypt (2007), pp. 113–117
46.
P.J. Davis, Interpolation and Approximation (Dover, New York, 1976)
47.
M.-M. Derriennic, Sur l’approximation de functions integrable sur [0, 1] par des polynomes de Bernstein modifies. J. Approx. Theor. 31, 323–343 (1981)MathSciNet
48.
M.-M. Derriennic, Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rend. Circ. Mat. Palermo, Ser. II 76 (Suppl.), 269–290 (2005)
49.
A. De Sole, V.G. Kac, On integral representation of q-gamma and q-beta functions. AttiAccad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (1), 11–29 (2005)
50.
R.A. DeVore, G.G. Lorentz, Constructive Approximation (Springer, Berlin, 1993)MATH
51.
Z. Ditzian, V. Totik, Moduli of Smoothness (Springer, New York, 1987)MATH
52.
O. Doğru, On Bleimann, Butzer and Hahn type generalization of Balázs operators, Dedicated to Professor D.D. Stancu on his 75th birthday. Studia Univ. Babeş-Bolyai Math. 47, 37–45 (2002)
53.
O. Doğru, O. Duman, Statistical approximation of Meyer–König and Zeller operators based on the q-integers. Publ. Math. Debrecen 68, 190–214 (2006)
54.
O. Dogru, V. Gupta, Monotonocity and the asymptotic estimate of Bleimann Butzer and Hahn operators on q integers. Georgian Math. J. 12, 415–422 (2005)MathSciNetMATH
55.
O. Dogru, V. Gupta, Korovkin type approximation properties of bivariate q-Meyer König and Zeller operators. Calcolo 43, 51–63 (2006)MathSciNetMATH
56.
D. Dubois, H. Prade, Fuzzy numbers: an overview, in Analysis of Fuzzy Information, vol.1: Mathematics and Logic (CRC Press, Boca Raton, 1987), pp. 3–39
57.
O. Duman, C. Orhan, Statistical approximation by positive linear operators. Stud. Math. 161, 187–197 (2006)MathSciNet
58.
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
59.
T. Ernst, The history of q-calculus and a new method, U.U.D.M Report 2000, 16, ISSN 1101-3591, Department of Mathematics, Upsala University, 2000
60.
S. Ersan, O. Doğru, Statistical approximation properties of q-Bleimann, Butzer and Hahn operators. Math. Comput. Model. 49 (7–8), 1595–1606 (2009)MATH
61.
Z. Finta, N.K. Govil, V. Gupta, Some results on modified Szász–Mirakyan operators. J. Math. Anal. Appl. 327 (2), 1284–1296 (2007)MathSciNetMATH
62.
Z. Finta, V. Gupta, Approximation by q Durrmeyer operators. J. Appl. Math. Comput. 29 (1–2), 401–415 (2009)MathSciNetMATH
63.
Z. Finta, V. Gupta, Approximation properties of q-Baskakov operators. Cent. Eur. J. Math. 8 (1), 199–211 (2009)MathSciNet
64.
J.A. Friday, H.I. Miller, A matrix characterization of statistical convergence. Analysis 11, 59–66 (1991)MathSciNet
65.
A.D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theoems analogous to that of P.P. Korovkin. Dokl. Akad. Nauk. SSSR 218 (5), 1001–1004 (1974) [in Russian]; Sov. Math. Dokl. 15 (5), 1433–1436 (1974) [in English]
66.
A.D. Gadjiev, On Korovkin type theorems. Math. Zametki 20, 781–786 (1976) [in Russian]
67.
A.D. Gadjiev, A. Aral, The weighted L p -approximation with positive operators on unbounded sets, Appl. Math. Letter 20 (10), 1046–1051 (2007)MathSciNetMATH
68.
A.D. Gadjiev, Ö. Çakar, On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semi-axis. Trans. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 19, 21–26 (1999)MATH
69.
A.D. Gadjiev, Orhan, Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32 (1), 129–138 (2002)MathSciNetMATH
70.
A.D. Gadjiev, R.O. Efendiev, E. Ibikli, Generalized Bernstein Chlodowsky polynomials. Rocky Mt. J. Math. 28 (4), 1267–1277 (1998)MathSciNetMATH
71.
A.D. Gadjiev, R.O. Efendiyev, E. İbikli, On Korovkin type theorem in the space of locally integrable functions. Czech. Math. J. 53 (128)(1), 45–53 (2003)
72.
S.G. Gal, Degree of approximation of continuous function by some singular integrals. Rev. Anal. Numér. Théor. Approx. XXVII (2), 251–261 (1998)
73.
S.G. Gal, Remark on degree of approximation of continuous function by some singular integrals. Math. Nachr. 164, 197–199 (1998)MathSciNet
74.
S.G. Gal, Remarks on the approximation of normed spaces valued functions by some linear operators, in Mathematical Analysis and Approximation Theory (Mediamira Science Publisher, Cluj-Napoca, 2005), pp. 99–109
75.
S.G. Gal, Shape-Preserving Approximation by Real and Complex Polynomials (Birkhäuser, Boston, 2008)MATH
76.
S.G. Gal, Approximation by Complex Bernstein and Convolution-Type Operators (World Scientific, Singapore, 2009)MATH
77.
S.G. Gal, V. Gupta, Approximation of vector-valued functions by q-Durrmeyer operators with applications to random and fuzzy approximation. Oradea Univ. Math. J. 16, 233–242 (2009)MathSciNetMATH
78.
S.G. Gal, V. Gupta, Approximation by a Durrmeyer-type operator in compact disk. Ann. Univ. Ferrara 57, 261–274 (2011)MathSciNetMATH
79.
S.G. Gal, V. Gupta, Quantative estimates for a new complex Durrmeyer operator in compact disks. Appl. Math. Comput. 218 (6), 2944–2951 (2011)MathSciNetMATH
80.
S. Gal, V. Gupta, N.I. Mahmudov, Approximation by a complex q Durrmeyer type operators. Ann. Univ. Ferrara 58 (1), 65–87 (2012)MathSciNet
81.
G. Gasper, M. Rahman, Basic Hypergeometrik Series. Encyclopedia of Mathematics and Its Applications, vol. 35 (Cambridge University Press, Cambridge, 1990)
82.
T.N.T. Goodman, A. Sharma, A Bernstein type operators on the simplex. Math. Balkanica 5, 129–145 (1991)MathSciNetMATH
83.
N.K. Govil, V. Gupta, Convergence rate for generalized Baskakov type operators. Nonlinear Anal. 69 (11), 3795–3801 (2008)MathSciNetMATH
84.
N.K. Govil, V. Gupta, Convergence of q-Meyer–König-Zeller–Durrmeyer operators. Adv. Stud. Contemp. Math. 19, 97–108 (2009)MathSciNetMATH
85.
V. Gupta, Rate of convergence by the Bézier variant of Phillips operators for bounded variation functions. Taiwan. J. Math. 8 (2), 183–190 (2004)MATH
86.
V. Gupta, Some approximation properties on q-Durrmeyer operators. Appl. Math. Comput. 197 (1), 172–178 (2008)MathSciNetMATH
87.
V. Gupta, A. Aral, Convergence of the q analogue of Szász-Beta operators. Appl. Math. Comput. 216, 374–380 (2010)MathSciNetMATH
88.
V. Gupta, A. Aral, Some approximation properties of q Baskakov Durrmeyer operators. Appl. Math. Comput. 218 (3), 783–788 (2011)MathSciNetMATH
89.
V. Gupta, Z. Finta, On certain q Durrmeyer operators. Appl. Math. Comput. 209, 415–420 (2009)MathSciNetMATH
90.
V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szász-Beta operators. J. Math. Anal. Appl. 321, 1–9 (2006)MathSciNetMATH
91.
V. Gupta, C. Radu, Statistical approximation properties of q Baskakov Kantorovich operators. Cent. Eur. J. Math. 7 (4), 809–818 (2009)MathSciNetMATH
92.
V. Gupta, H. Sharma, Recurrence formula and better approximation for q Durrmeyer operators. Lobachevskii J. Math. 32 (2), 140–145 (2011)MathSciNetMATH
93.
V. Gupta, G.S. Srivastava, On the rate of convergence of Phillips operators for functions of bounded variation. Comment. Math. XXXVI, 123–130 (1996)
94.
V. Gupta, H. Wang, The rate of convergence of q-Durrmeyer operators for 0 < q < 1. Math. Meth. Appl. Sci. 31 (16), 1946–1955 (2008)MathSciNetMATH
95.
V. Gupta, T. Kim, J. Choi, Y.-H. Kim, Generating functions for q-Bernstein, q-Meyer–König–Zeller and q-Beta basis. Autom. Comput. Math. 19 (1), 7–11 (2010)
96.
97.
W. Heping, Properties of convergence for the q-Meyer–König and Zeller operators. J. Math. Anal. Appl. 335 (2), 1360–1373 (2007)MathSciNetMATH
98.
W. Heping, Properties of convergence for ω, q Bernstein polynomials. J. Math. Anal. Appl. 340 (2), 1096–1108 (2008)MathSciNetMATH
99.
W. Heping, F. Meng, The rate of convergence of q-Bernstein polynomials for 0 < q < 1. J. Approx. Theor. 136, 151–158 (2005)MATH
100.
W. Heping, X. Wu, Saturation of convergence of q-Bernstein polynomials in the case q ≥ 1. J. Math. Anal. Appl. 337 (1), 744–750 (2008)MathSciNetMATH
101.
V.P. II’in, O.V. Besov, S.M. Nikolsky, The Integral Representation of Functions and Embedding Theorems (Nauka, Moscow, 1975) [in Russian]
102.
A. II’inski, S. Ostrovska, Convergence of generalized Bernstein polynomials. J. Approx. Theor. 116, 100–112 (2002)
103.
F.H. Jackson, On a q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)MATH
104.
V. Kac, P. Cheung, Quantum Calculus (Springer, New York, 2002)MATH
105.
T. Kim, q-Generalized Euler numbers and polynomials. Russ. J. Math. Phys. 13 (3), 293–298 (2006)
106.
T. Kim, Some identities on the q-integral representation of the product of several q-Bernstein-type polynomials. Abstr. Appl. Anal. 2011, 11 pp. (2011). doi:10.1155/2011/634675 [Article ID 634675]
107.
T.H. Koornwinder, q-Special functions, a tutorial, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics, ed. by M. Gerstenhaber, J. Stasheff. Contemporary Mathematics, vol. 134 (American Mathematical Society, Providence, 1992)
108.
S.L. Lee, G.M. Phillips, Polynomial interpolation at points of a geometric mesh on a triangle. Proc. R. Soc. Edinb. 108A, 75–87 (1988)MathSciNet
109.
B. Lenze, Bernstein–Baskakov–Kantorovich operators and Lipscitz type maximal functions, in Approximation Theory (Kecskemét, Hungary). Colloq. Math. Soc. János Bolyai, vol. 58 (1990), pp. 469–496MathSciNet
110.
A. Lesniewicz, L. Rempulska, J. Wasiak, Approximation properties of the Picard singular integral in exponential weighted spaces. Publ. Mat. 40, 233–242 (1996)MathSciNetMATH
111.
Y.-C. Li, S.-Y. Shaw, A proof of Hölder’s inequality using the Cauchy–Schwarz inequality. J. Inequal. Pure Appl. Math. 7 (2), 1–3 (2006) [Article 62]
112.
A.-J. López-Moreno, Weighted silmultaneous approximation with Baskakov type operators. Acta Math. Hung. 104, 143–151 (2004)MATH
113.
G.G. Lorentz, Bernstein Polynomials. Math. Expo., vol. 8 (University of Toronto Press, Toronto, 1953)
114.
G.G. Lorentz, Approximation of Functions (Holt, Rinehart and Wilson, New York, 1966)MATH
115.
L. Lupas, A property of S. N. Bernstein operators. Mathematica (Cluj) 9 (32), 299–301 (1967)
116.
L. Lupas, On star shapedness preserving properties of a class of linear positive operators. Mathematica (Cluj) 12 (35), 105–109 (1970)
117.
A. Lupas, A q-analogue of the Bernstein operator, in Seminar on Numerical and Statistical Calculus (Cluj-Napoca, 1987), pp. 85–92. Preprint, 87-9 Univ. Babes-Bolyai, Cluj. MR0956939 (90b:41026)
118.
N. Mahmoodov, V. Gupta, H. Kaffaoglu, On certain q-Phillips operators. Rocky Mt. J. Math. 42 (4), 1291–1312 (2012)
119.
N.I. Mahmudov, On q-parametric Szász–Mirakjan operators. Mediterr. J. Math. 7 (3), 297–311 (2010)MathSciNetMATH
120.
N.I. Mahmudov, P. Sabancıgil, q-Parametric Bleimann Butzer and Hahn operators. J. Inequal. Appl. 2008, 15 pp. (2008) [Article ID 816367]
121.
N.I. Mahmudov, P. Sabancigil, On genuine q-Bernstein–Durrmeyer operators. Publ. Math. Debrecen 76 (4) (2010)
122.
G. Mastroianni, A class of positive linear operators. Rend. Accad. Sci. Fis. Mat. Napoli 48, 217–235 (1980)MathSciNet
123.
C.P. May, On Phillips operator. J. Approx. Theor. 20 (4), 315–332 (1977)MATH
124.
I. Muntean, Course of Functional Analysis. Spaces of Linear and Continuous Mappings, vol. II (Faculty of Mathematics, Babes-Bolyai’ University Press, Cluj-Napoca, 1988) [in Romanian]
125.
S. Ostrovska, q-Bernstein polynomials and their iterates. J. Approx. Theor. 123, 232–255 (2003)MathSciNetMATH
126.
S. Ostrovska, On the improvement of analytic properties under the limit q-Bernstein operator. J. Approx. Theor. 138 (1), 37–53 (2006)MathSciNetMATH
127.
S. Ostrovska, On the Lupas q-analogue of the Bernstein operator. Rocky Mt. J. Math. 36 (5), 1615–1629 (2006)MathSciNetMATH
128.
S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives. J. Math. Anal. Approx. Theor. 2, 35–51 (2007)MathSciNetMATH
129.
S. Ostrovska, The sharpness of convergence results for q Bernstein polynomials in the case q > 1. Czech. Math. J. 58 (133), 1195–1206 (2008)MathSciNetMATH
130.
S. Ostrovska, On the image of the limit q Bernstein operator. Math. Meth. Appl. Sci. 32 (15), 1964–1970 (2009)MathSciNetMATH
131.
132.
G.M. Phillips, On generalized Bernstein polynomials, in Numerical Analysis, ed. by D.F. Griffiths, G.A. Watson (World Scientific, Singapore, 1996), pp. 263–269
133.
G.M. Phillips, Bernstein polynomials based on the q- integers, The heritage of P.L. Chebyshev: A Festschrift in honor of the 70th-birthday of Professor T. J. Rivlin. Ann. Numer. Math. 4, 511–518 (1997)
134.
G.M. Phillips, Interpolation and Approximation by Polynomials (Springer, Berlin, 2003)MATH
135.
R.S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators. Ann. Math. Sec. Ser. 59, 325–356 (1954)MATH
136.
C. Radu, On statistical approximation of a general class of positive linear operators extended in q-calculus. Appl. Math. Comput. 215 (6), 2317–2325 (2009)MathSciNetMATH
137.
P.M. Rajković, M.S. Stanković, S.D. Marinkovic, Mean value theorems in q-calculus. Math. Vesnic. 54, 171–178 (2002)MATH
138.
T.J. Rivlin, An Introduction to the Approximation of Functions (Dover, New York, 1981)MATH
139.
A. Sahai, G. Prasad, On simultaneous approximation by modified Lupaş operators. J. Approx. Theor. 45, 122–128 (1985)MathSciNetMATH
140.
I.J. Schoenberg, On polynomial interpolation at the points of a geometric progression. Proc. R. Soc. Edinb. 90A, 195–207 (1981)MathSciNet
141.
K. Seip, A note on sampling of bandlinited stochastic processes. IEEE Trans. Inform. Theor. 36 (5), 1186 (1990)
142.
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)MathSciNetMATH
143.
Z. Song, P. Wang, W. Xie, Approximation of second-order moment processes from local averages. J. Inequal. Appl. 2009, 8 pp. (2009) [Article Id 154632]
144.
H.M. Srivastava, V. Gupta, A certain family of summation-integral type operators. Math. Comput. Model. 37, 1307–1315 (2003)MathSciNetMATH
145.
D.D. Stancu, Approximation of functions by a new class of linear polynomial operators. Rev. Roumanine Math. Pures Appl. 13, 1173–1194 (1968)MathSciNetMATH
146.
E.M. Stein, G. Weiss, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)MATH
147.
E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971)MATH
148.
O. Szász, Generalization of S. Bernstein’s polynomials to infinite interval. J. Research Nat. Bur. Stand. 45, 239–245 (1959)
149.
J. Thomae, Beitrage zur Theorie der durch die Heinsche Reihe. J. Reine. Angew. Math. 70, 258–281 (1869)MATH
150.
T. Trif, Meyer, König and Zeller operators based on the q-integers. Rev. Anal. Numér. Théor. Approx. 29, 221–229 (2002)MathSciNet
151.
V.S. Videnskii, On some class of q-parametric positive operators, Operator Theory: Advances and Applications 158, 213–222 (2005)MathSciNet
152.
V.S. Videnskii, On q-Bernstein polynomials and related positive linear operators, in Problems of Modern Mathematics and Mathematical Education Hertzen readings (St.-Petersburg, 2004), pp. 118–126 [in Russian]
153.
I. Yuksel, N. Ispir, Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52 (10–11), 1463–1470 (2006)MathSciNetMATH
154.
Metadaten
Titel
q-Summation–Integral Operators
verfasst von
Ali Aral
Vijay Gupta
Ravi P. Agarwal
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Applications of q-Calculus in Operator Theory

Print ISBN: 978-1-4614-6945-2

Electronic ISBN: 978-1-4614-6946-9

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-6946-9_5