Approximation properties: modified Szász-Durrmeyer type operators via General-Appell polynomials

Nand Kishor Jha, Nadeem Rao, Brijesh Kumar Sinha

Abstract


We present a new sequence of modified Szász-Durrmeyer type of sequence of operators via general-Appell Polynomials to investigate the approximation properties of Lebesgue integrable functions (L1[0, ∞)). In addition, we are study estimates in view of test functions and central moments. Next, convergence rate is discussed using the Korovkin theorem and Voronovskaja type theorem. Moreover, direct approximation results via modulus of continuity of first and second order, Peetre’s K-functional, Lipschitz type space, and the rth order Lipschitz type maximal functions are investigated. In subsequent section, we present weighted approximation results and statistical approximation theorems are discussed.


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References


K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2 (1885), 633-639.

S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow. 13 (1912/1913), no. 1|2.

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Research Nat. Bur. Standards 45 (1950), 239-245.

F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.

N. Raza, M. Kumar, Approximation with Chlodowsky variant of Kantorovich-Stancu-operators employing associated λ-polynomials, J. Inequal. Appl. 2024 (2024), no. 1, 1-19.

N. Raza, M. Kumar, M. Mursaleen, On approximation operators involving Tricomi function, Bull. Malaysian Math. Sci. Soc. 47 (2024), no. 5, 154.

R. Aslan, Some approximation properties of Riemann-Liouville type fractional Bernstein-Stancu-Kantorovich operators with order of α, Iran. J. Sci. 49 (2025), no. 2, 481-494.

R. Aslan, Approximation by α -Chlodowsky-Durrmeyer Operators, Math. Methods Appl. Sci. (2026), 1-17.

F. Özger, On new Bézier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. Al. Math. Stat. 69 (2020), no. 1, 376-393.

F. Özger, M.T. Ersoy, O.Z. Özger, Existence of solutions: Investigating Fredholm integral equations via a fixed-point theorem, Axioms 13 (2024), 261.

Q.B. Cai, R. Aslan, Note on a new construction of Kantorovich form q-Bernstein operators related to shape parameter λ, Comput. Model. Eng. Sci. 130 (2022), no. 3, 1479-1493.

Q.B. Cai, R. Aslan, F. Özger, H.M. Srivastava, Approximation by a new Stancu variant of generalized (λ, μ)-Bernstein operators, Alex. Eng. J. 107 (2024), 205-214.

N.L. Braha, Some summability methods for Dunkl-Gamma-type operators including Appell polynomials, Math. Methods Appl. Sci. (2024).

N.L. Braha, T. Mansour, M. Mursaleen, Some Approximation Properties of Parametric Baskakov-Schurer-Szász Operators Through a Power Series Summability Method, Complex Anal. Oper. Theory 18 (2024), no. 3, 71.

M. Ayman Mursaleen, M. Nasiruzzaman, N. Rao, On the Approximation of Szász-Jakimovski-Leviatan beta type integral operators enhanced by Appell polynomials, Iranian J. Sci. 49 (2025), 1013-1022.

M. Ayman-Mursaleen, E. Alshaban, M. Nasiruzzaman, Approximation to Family of α-Bernstein operators

using Shifted knot properties, J. Inequal. Appl. 2025 (2025), 107.

M. Ayman Mursaleen, pproximation and Convergence Analysis of Blending-Type q-Baskakov Operators Using Wavelet Transformations, Filomat 39 (2025), no. 31, 11117-11130.

K.J. Ansari, M. Mursaleen, S. Rahman, Approximation by Jakimovski-Leviatan operators of Durrmeyer type involving multiple Appell polynomials, Rev. R. Acad. Ciene. Exactas Fis. Nat. Ser. A Math. RACSAM 113 (2019), 1007-1024.

K. J. Ansari, F. Özger, O. Z. Özger, Numerical and theoretical approximation results for Schurer-Stancu operators with shape parameter λ, Comput. Appl. Math. 41 (2022), no. 4, 181.

S.A. Mohiuddine, T. Acar, A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Methods Appl. Sci. 40 (2017), 7749-7759.

S.A. Mohiuddine, N. Ahmad, F. Özger, A. Alotaibi, B. Hazarika, Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran. J. Sci. Technol. Trans. 45 (2021), no. 2, 593-605.

M. Mursaleen, K.J. Ansari, A. Khan, Approximation properties and error estimation of q-Bernstein shifted operators, Numer. Algorithms 84 (2020), 207-227.

M. Mursaleen, A. Naaz, A. Khan, Improved approximation and error estimations by King type (p, q)-Sz_asz-Mirakjan Kantorovich operators, Appl. Math. Comput. 348 (2019), 2175-185.

A. Khan, M. Mansoori, K. Khan, M. Mursaleen, Phillips-type q-Bernstein operators on triangles, J. Funct. Spaces 2021 (2021), 1-13.

M. Nasiruzzaman, N. Rao, M. Mursaleen, A.S.S. Albalawi, A.K. Yadav, Approximation by a new sequence of generalized α-Szász-Gamma operators, Filomat 39 (2025), no. 19, 6771-6782.

M. Nasiruzzaman, R.T. Alqahtani, S.A. Mohiuddine, Approximation Properties of a New Class of Beta-Type Szász-Mirakjan Operators, J. Math. 2025 (2025), no. 1, 1-17.

N. Rao, N.K. Jha, M. Shahzad, On a new sequence of Bernstein-Chlodowsky operators and approximation properties, J. Math. Sci. 298 (2025), no. 1, 1-12.

N. Rao, M. Farid, N.K. Jha, A study of (σ,μ)-Stancu-Schurer as a new generalization and approximations, J. Inequal. Appl. 2025 (2025), no. 1, 104.

N.K. Jha, N. Rao, Hermite polynomial connecting Szász integral Chlodowsky operators and applications, Iranian J. Sci. 50 (2025), no. 2, 1-12.

N. Raza, M. Kumar, M. Mursaleen, Approximation with Szász-Chlodowsky operators employing general-Appell polynomials, J. Inequal. Appl. 2024 (2024), Article 26.

R.A. DeVore, G.G. Lorentz, Constructive Approximation, Grudlehren der Mathematischen Wissenschaften Fundamental principales of Mathematical Sciences, Springer-Verlag, Berlin, 1993.

O. Shisha, B. Mond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA 60 (1968), 1196-1200.

P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR 90 (1953), 961-964.

B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl Akad Indag Math. 50 (1988), 53-63.

A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P. P. Korovkin, Dokl. Akad. Nauk SSSR 218 (1974), no. 5.

A.D. Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki 20 (1976), 781-786; Transl. in Math. Notes 5-6 (1978), 995-998.

A.D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2007), no. 1, 129-138.




DOI: https://doi.org/10.52846/ami.v53i1.2123