Some approximation properties of a kind of (p, q)-Phillips operators
-
Wentao Cheng
,
Chunyan Gui
Abstract
In this paper, a kind of new analogue of Phillips operators based on (p, q)-integers is introduced. The moments of the operators are established. Then some local approximation for the above operators is discussed. Also, the rate of convergence and weighted approximation by these operators by means of modulus of continuity are studied. Furthermore, the Voronovskaja type asymptotic formula is investigated.
This research is supported by the National Natural Science Foundation of China (Grant No. 11626031), the Key Natural Science Research Project in Universities of Anhui Province (Grant No. KJ2019A0572), the Philosophy and Social Sciences General Planning Project of Anhui Province of China (Grant No. AHSKYG2017D153) and the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29)
Communicated by Marcus Waurick
Acknowledgement
The authors are thankful to the editor and anonymous referees for their helpful comments and suggestion.
References
[1] Acar, T.: (p, q)-Generalization of Szász-Mirakjan operators, Math. Methods Appl. Sci. 39(10) (2016), 2685–2695.10.1002/mma.3721Suche in Google Scholar
[2] Acar, T.—Aral, A.—Mohiuddine, S. A.: Approximation by bivariate (p, q)-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. Sci. 42(2) (2018), 655–662.10.1007/s40995-016-0045-4Suche in Google Scholar
[3] Acar, T.—Aral, A.—Mohiuddine, S. A.: On Kantorovich modifications of (p, q)-Baskakov operators, J. Inequal. Appl. 98 (2016).10.1186/s13660-016-1045-9Suche in Google Scholar
[4] Acar, T.—Aral, A.—Mohiuddine, S. A.: On Kantorovich modifications of (p, q)-Bernstein operators, Iran. J. Sci. Technol. Trans. Sci. 42(3) (2018), 1459–1464.10.1007/s40995-017-0154-8Suche in Google Scholar
[5] Acar, T.—Aral, A.—Mursaleen, M.: Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers, Math. Slovaca 68(4) (2018), 897–906.10.1515/ms-2017-0153Suche in Google Scholar
[6] Acar, T.—Agrawal, P. N.—Kumar, S.: On a modification of (p, q)-Szász-Mirakyan operators, Complex Anal. Oper. Theory 12 (2018), 155–167.10.1007/s11785-016-0613-9Suche in Google Scholar
[7] Acar, T.—Mohiuddine, S. A.—Mursaleen, M.: Approximation by (p, q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory 12(6) (2018), 1453–1468.10.1007/s11785-016-0633-5Suche in Google Scholar
[8] A. D, Gadzhiev.: Theorem of the type of P. P. Korovkin type theorems, Mat. Zametki 20(5) (1976), 781–786.10.1007/BF01146928Suche in Google Scholar
[9] Aral, A.—Gupta, V.—Agarwal, R. P.: Application of q-Calculus in Operator Theory, Springer, Berlin, 2013.10.1007/978-1-4614-6946-9Suche in Google Scholar
[10] Aral, A.—Gupta, V.: Application of (p, q)-gamma function to Szász Durrmeyer operators, Publ. Inst. Math. 102(116) (2017), 211–220.10.2298/PIM1716211ASuche in Google Scholar
[11] Cai, Q. B.—Zhou, G. R.: On (p, q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput. 276 (2016), 12–20.10.1016/j.amc.2015.12.006Suche in Google Scholar
[12] Cheng, W. T.—Zhang, W. H.: On (p, q)-Analogue of Gamma operators, J. Funct. Space (2019), Art. ID 9607517.10.1155/2019/9607517Suche in Google Scholar
[13] Cheng, W. T.— Zhang, W. H.—Cai, Q. B.: (p, q)-gamma operators which preserve x2, J. Inequal. Appl. 108 (2019).Suche in Google Scholar
[14] Devore, R. A.—Lorentz, G. G.: Constructive Approximation, Springer, Berlin, 1993.10.1007/978-3-662-02888-9Suche in Google Scholar
[15] Govil, N. K.— Gupta, V.: Approximation properties of Phillips operators. In: Mathematics Without Boundaries, Springer, New York, 2014, pp. 221-243.Suche in Google Scholar
[16] Gupta, V.: (p, q)-Szász-Mirakyan-Baskakov operators, Complex Anal. Oper. Theory 12(1) (2018), 17–25.10.1007/s11785-015-0521-4Suche in Google Scholar
[17] Gupta, V.: Rate of convergence by the Bézier of Phillips operators for bounded variation functions, Taiwanese J. Math. 8 (2004), 183–190.10.11650/twjm/1500407620Suche in Google Scholar
[18] Gupta, V.— Agarwal, R. P.: Convergence Estimates in Approximation Theory, Springer, New York, 2014.10.1007/978-3-319-02765-4Suche in Google Scholar
[19] Gupta, V.—Rassias, T. M.—Agrawal, P. N.—Acu, A. M.: Recent Advances in Constructive Approximation Theory, Springer, New York, 2018.10.1007/978-3-319-92165-5Suche in Google Scholar
[20] Gupta, V.—Srivastava, G. S.: On the rate of convergence of Phillips operators for functions of bounded variation, Annal. Soc. Math. Polon. Comment. Math. 36 (1996), 123–130.Suche in Google Scholar
[21] I, Yüksel.: Approximation by q-Phillips operators, Hacet. J. Math. Stat. 40(2) (2011), 191–201.Suche in Google Scholar
[22] Iiarslan, H. G. I.— Acar, T.: Approximation by bivariate (p, q)-Baskakov-Kantorovich operators, Georgian Math. J. 25(3) (2018), 397–407.10.1515/gmj-2016-0057Suche in Google Scholar
[23] Liu, Q.—Qi, R. S.: Stancu type generalization of the q-Phillips operators, Numer. Algorithms 72 (2016), 181–193.10.1007/s11075-015-0040-4Suche in Google Scholar
[24] Lenze, B.: On Lipschitz-type maximal functions and their smoothness spaces, Indag. Math. 50(1) (1988), 53–63.10.1016/1385-7258(88)90007-8Suche in Google Scholar
[25] Mahmudov, N.—Gupta, V.—Kaffaoǧlu, H.: On certain q-Phillips operators, Rocky Mountain J. Math. 42(4) (2012), 1291–1312.10.1216/RMJ-2012-42-4-1291Suche in Google Scholar
[26] Malik, N.—Gupta, V.: Approximation by (p, q)-Baskakov-Beta operators, Appl. Math. Comput. 293 (2017), 49–53.10.1016/j.amc.2016.08.005Suche in Google Scholar
[27] May, C. P.: On Phillips operators, J. Approx. Theory 20 (1977), 315–322.10.1016/0021-9045(77)90078-8Suche in Google Scholar
[28] Mohiuddine, S. A.—Acar, T.—Alotaibi, A.: Durrmeyer type (p, q)-Baskakov operators preseving linear functions, J. Math. Inequal. 12(4) (2018), 961–973.10.7153/jmi-2018-12-73Suche in Google Scholar
[29] Mursaleen, M.—Khan, F.—Khan, A.: Some approximation results by (p, q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput. 264 (2015), 392–402; Erratum: Appl. Math. Comput. 269 (2016), 744–746.10.1016/j.amc.2015.03.135Suche in Google Scholar
[30] Mursaleen, M.—Khan, F.—Khan, A.: On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882; Erratum: Appl. Math. Comput. 278 (2016), 70–71.10.1016/j.amc.2015.04.090Suche in Google Scholar
[31] Phillips, R. S.: An inversion formula for semi-groups of linear operators, Ann. Math. 59 (1954), 325–356.10.2307/1969697Suche in Google Scholar
[32] Sadjang, P. N.: On two (p, q)-analogues of the Laplace transform, J. Differ. Equ. Appl. 23(9) (2017), 1562–1583.Suche in Google Scholar
[33] Sharma, H.—Maurya, R.—Gupta, C.: Approximation properties of Kantorovich type modifications of (p, q)-Meyer-König-Zeller operators, Constr. Math. Anal. 1(1) (2018), 58–72.10.33205/cma.436071Suche in Google Scholar
[34] Yuksel, I.—Ispir, N.: Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl. 52(10–11) (2006), 1463–1470.10.1016/j.camwa.2006.08.031Suche in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- RNDr. Kvetoslava Dvořáková passed away
- On the Riesz structures of a lattice ordered abelian group
- On Diophantine equation x4 + y4 = n(u4 + v4)
- On a Waring-Goldbach problem involving squares and cubes
- D(n)-quadruples in the ring of integers of ℚ(√2, √3)
- Geometry of ℙ2 blown up at seven points
- Preservation of Rees exact sequences
- Pointwise multipliers between weighted copson and cesàro function spaces
- Some properties associated to a certain class of starlike functions
- Asymptotic properties of noncanonical third order differential equations
- Existence and regularity results for unilateral problems with degenerate coercivity
- Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
- Some approximation properties of a kind of (p, q)-Phillips operators
- Best proximity points for a new type of set-valued mappings
- Weakly demicompact linear operators and axiomatic measures of weak noncompactness
- Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
- A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
- On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
- Generalized Meir-Keeler type contractions and discontinuity at fixed point II
Artikel in diesem Heft
- Regular papers
- RNDr. Kvetoslava Dvořáková passed away
- On the Riesz structures of a lattice ordered abelian group
- On Diophantine equation x4 + y4 = n(u4 + v4)
- On a Waring-Goldbach problem involving squares and cubes
- D(n)-quadruples in the ring of integers of ℚ(√2, √3)
- Geometry of ℙ2 blown up at seven points
- Preservation of Rees exact sequences
- Pointwise multipliers between weighted copson and cesàro function spaces
- Some properties associated to a certain class of starlike functions
- Asymptotic properties of noncanonical third order differential equations
- Existence and regularity results for unilateral problems with degenerate coercivity
- Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
- Some approximation properties of a kind of (p, q)-Phillips operators
- Best proximity points for a new type of set-valued mappings
- Weakly demicompact linear operators and axiomatic measures of weak noncompactness
- Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
- A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
- On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
- Generalized Meir-Keeler type contractions and discontinuity at fixed point II
