Blending‐type approximation by Lupaş–Durrmeyer‐type operators involving Pólya distribution

Abstract

In the present work, we construct a new sequence of positive linear operatorsinvolving Pólya distribution. We compute a Voronovskaja type and a Grüss–Voronovskaja type asymptotic formula as well as the rate of approximation by using the modulus of smoothness and for functions in a Lipschitz type space. Lastly, we provide some numerical results, which explain the validity of the theoretical results and the effectiveness of the constructed operators.

Convergence on sequences of Szász-Jakimovski-Leviatan type operators and related results

“…where Θ τ,λ (y) defined by Theorem 4.2 and M ϕ be any positive constant. by infimum over all of ψ ∈ C 2 B [0, ∞) and from equality (15), we get…”

“…Recently, Szász-Mirakyan operators have been introduced by researchers in various spaces to get the approximation process, for instance, we refer the readers to see Szász-Mirakyan-Baskakov operators [2], Szász-Jakimovski-Leviatan type Appell-Polynomials [3,4,28,29,30], Szász-Dunkl type generalization [5,6,19,20], Szász-Durrmeyer type operators [36,37], moreover, Bernstein Kantorovich operators [7], Baskakov-Durrmeyer-Stancu operators [8], Kantorovich type q-Bernstein-Stancu Operators [21], Korovkin and Voronovskaya types approximation [22], Bézier bases with Schurer polynomials [31], generalized Bernstein-Schurer operators [32], Stancu variant of Bernstein-Kantorovich operators [26], family of Bernstein-Kantorovich operators [27] and references therein. In addition, we prefer to see the recent article [15,16,23,24,25] and [11,12,41]. Most recent Sucu [39] introduced the Szász-Mirakyan operators by applying the generated exponential function of Hermite type polynomials, expressed in form of the confluent hypergeometric function (see [38]) given by…”

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

“…Some other work in this sense was discussed by Gupta et al [7][8][9], Agrawal et al [10,11], Razi [12], Wang et al [13], Finta [14,15], Deo et al [16], Abel et al [17], and Kajla et al [18].…”

Construction of q-Baskakov Operators by Wavelets and Approximation Properties

“…It is well known that for every continuous function g the Bernstein polynomials Bernstein (1912) converge uniformly to g(x) for all x 2 ½0; 1. The Bernstein polynomials are defined by The Sza ´sz Sza ´sz (1950) and Baskakov Baskakov (1957) operators were constructed to approximate the continuous functions defined on the unbounded interval ½0; 1Þ: The Baskakov operators are define by (see also Al-Abied et al (2021), Cai and Aslan (2021), Cai et al (2022), Kajla et al (2021), Khan et al (2022), Kilicman et al (2020), Heshamuddin et al (2022), Mohiuddine et al (2020Mohiuddine et al ( , 2021, Mohiuddine and O ¨zger (2020), Ayman Mursaleen et al (2022), Rao et al (2021)…”