(p,q)-Baskakov Operators

Yıl 2019, Cilt: 8 Sayı: 4, 1223 - 1232, 24.12.2019

Nazlım Deniz Aral

Öz

Bu çalışmada q-Baskakov operatörünün bir genelleşmesi olan (p,q)-Baskakov operatörü olarak adlandırılan Baskakov operatörünün yeni bir türü tanıtılmıştır. Merkezi momentler için formüller elde edilmiştir. Aynı zamanda süreklilik modülü kullanılarak bu operatörlerin yaklaşım özellikleri ve yakınsama oranı çalışılmıştır.

Anahtar Kelimeler

(pq)-analiz, Korovkin teoremi, Baskakov operatörleri, süreklilik mödülü

(p,q)-Baskakov Operators

Öz

In the present paper, we give a new analogue of Baskakov operators and we call them (p,q)-Baskakov operators which are a generalization of q -Baskakov operators. We obtain their respective formulae for central moments. Also, we study the rate of convergence and approximation properties for these operators using the modulus of smoothness.

Anahtar Kelimeler

(pq)-calculus, Korovkin's theorem, Baskakov operators, Modulus of continuity

Kaynakça

• Jackson F.H. 1909. On q-functions and a certain difference operator, Transactions Royal Society Edinburgh, 46(2): 253-281.
• Lupaş A. 1987. A q-analogue of the Bernstein operator, in Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 9: 85-92.
• Phillips G.M. 1997. Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4: 511-518.
• Aral A., Gupta V. 2009. On q-Baskakov type operators, Demons. Math., 42(1): 109-122.
• Aral A., Gupta V. 2010. On the Durrmeyer type modification of the q Baskakov type operators, Nonlinear Anal.:Theor. Meth. Appl., 72(3–4), 1171–1180.
• Aral A., Gupta V. 2011. Generalized q-Baskakov operators, Math. Slovaca, 61(4): 619–634.
• Finta Z., Gupta V. 2010. Approximation properties of q-Baskakov Operators, Cent. Eur. J. Math., 8(1): 199-211.
• Aral A., Gupta V., Agarval R.P. 2013. Applications of q-calculus in Operator Theory, Springer, New York, 262p.
• Ernst T. 1999. A new notation for q-calculus and new q-Taylor formula, Uppsala University, Report Depart. Math., 1-28.
• Heping W. 2007. Properties of convergence for the q-Meyer-König and Zeller operators, J. Math. Anal. Appl., 335(2): 1360-1373.
• Doğru O., Duman O. 2006. Statistical approximation of Meyer-König and Zeller operators based on the q-integers, Publ. Math. Debrecen, 68: 190-214.
• Heping W., Meng F. 2005. The rate of convergence of q-Bernstein polynomials for 0<q<1, J. Approx. Theor., 136(2): 151-158.
• II’inskii A., Ostrovska S. 2002. Convergence of Generalized Bernstein Polynomials, J. Approx. Theor., 116(1): 100-112.
• Ostrovska S. 2003. q-Bernstein polynomials ans their iterates, J. Approx. Theory, 123(2): 232-255.
• Nowak G., Gupta V. 2011. The rate of pointwise approximation of positive linear operators based on q-integer, Ukranian Math. J., 63(3): 350-360.
• Radu C. 2009. On statistical approximation of a general class of positive linear operators extended in q-calculus, Appl. Math. Comput., 215(6): 2317-2325.
• Şimşek E., Tunç T. 2017. On the construction of q-analogues for some positive linear operators, Filomat, 31(13): 4287-4295.
• Şimşek E. 2018. On a New type of q-Baskakov operators, Süleyman Demirel Ünv. Fen Bilim. Der., 22(1): 121-125.
• Şimşek E., Tunç T. 2018. On approximation properties of some class positive linear operators in q-analysis, J. Math. Inq. 12(2): 559-571.
• Sadjang P.N. 2018. On the Fundamental Theorem of (p,q)-calculus and some (p,q)-Taylor formulas, Result. Math., 73: 39.
• Mursaleen M., Ansari K.J., Khan A. 2015. On (p,q)-analogue of Bernstein operators, Appl. Math. Comput., 266: 874-882.
• Aral A., Gupta V. 2016. (p,q)-type Beta functions of second kind, Adv. Oper. Theory 1(1): 134–146.
• Mursaleen M., Alotibi A., Ansari J. 2016. On a Kantrovich variant of (p,q)-Szasz-Mirakjan operators, J. Funct. Spaces, Article ID 1035253, 9 pages.
• Mursaleen M., Ansari K.J., Khan A. 2015. Some Approximation results by (p,q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput., 264: 392-402.
• Acar T. 2016. (p,q)-generalization of Szasz-Mirakyan operators, Math. Methods Appl. Sci., 39(10): 2685-2695.
• Acar T., Aral A., Mohiuddine S.A. 2018. Approximation by bivariate (p,q)-Bernstein-Kantorovich operators, Iran J. Sci. Technol. Trans., 42: 655-662.
• Acar T., Aral A., Mohiuddine S.A. 2016. On Kantorovich modification of (p,q)-Baskakov operators, J. Inequality Appl., 98, doi:10.1186/s13660-016-1045-9.
• Cai Q.B., Zhou G. 2016. On (p,q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput., 276: 12-20.
• Sharma H. 2016. On Durrmeyer-type generalization of (p,q)-Bernstein operators. Arab J. Math., 5: 239-248.
• Gupta V. 2016. (p,q)-Baskakov-Kantorovich operators, Appl. Math. Inf. Sci., 10(4): 1551-1556.
• Şimşek E., Tunç T. 2018. On some sequences of the Positive Linear Operators Based on (p,q)-calculus, ICMSA (International Conference on Mathematical Studies and Applications 4-6 October, Abstract Book, pp 250-257, Kırşehir.