Local approximation by a variant of Bernstein–Durrmeyer operators

Abel, Ulrich; Gupta, Vijay; Mohapatra, R. N.

doi:10.1016/j.na.2007.03.026

Cited by 42 publications

(13 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our results will put in evidence the overconvergence phenomenon for the operators (1). The results established here are the extensions of approximation properties with exact quantitative estimates from the real interval [0, 1], to compact disks in the complex plane.…”

supporting

confidence: 59%

See 1 more Smart Citation

Approximation by complex summation-integral type operator in compact disks

Gupta

Yadav

2013

Mathematica Slovaca

Self Cite

View full text Add to dashboard Cite

ABSTRACT. In the present paper we estimate a Voronovskaja type quantitative estimate for a certain type of complex Durrmeyer polynomials, which is different from those studied previously in the literature. Such estimation is in terms of analytic functions in the compact disks. In this way, we present the evidence of overconvergence phenomenon for this type of Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane. In the end, we mention certain applications.

show abstract

supporting

confidence: 59%

“…P r o o f. For p = 0 the relationship is evident from M n (e 0 , z) = 1 and M n (e 1 , z) = nz n+2 (see e.g., [1]). Therefore, let p ∈ N. Using the equality…”

Section: Z)mentioning

confidence: 96%

Approximation by complex summation-integral type operator in compact disks

Gupta

Yadav

2013

Mathematica Slovaca

Self Cite

View full text Add to dashboard Cite

show abstract

“…It can be easily verified that in the case q = 1, the operators defined by (4.27) reduce to the Durrmeyer-type operators recently introduced and studied in [3].…”

Section: Discretely Defined Q-durrmeyer Operatorsmentioning

confidence: 88%

“…It is observed from the above lemma that for q → 1 − , we get the moments of the Baskakov-Kantorovich operators (see, e.g., [3]) K n (e 0 ; x) = 1, K n (e 1 ; x) = x + 1 2n ,…”

Section: Q-analogue Of Baskakov-kantorovich Operatorsmentioning

confidence: 95%

See 1 more Smart Citation

Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

Self Cite

372

206

View full text Add to dashboard Cite

In order to approximate integrable functions on the interval [0, 1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein-Durrmeyer operators. In this chapter we discuss the q analogues of various integral modifications of Bernstein polynomials. The results were discussed in recent papers [45, 62, 86, 89, 92, 94, 121], etc. q-Bernstein-Kantorovich OperatorsRecently, Dalmanoglu [45] proposed the q-Kantorovich-Bernstein operators asIn case q = 1, the operators (4.1) reduce to well-known Bernstein-Kantorovich operators Direct ResultsFor the operators (4.1), Dalmanoglu [45] obtained the following theorems:Theorem 4.1. If the sequence (q n ) satisfies the conditions lim n→∞ q n = 1 and lim n→∞Proof. First, we haveAlso by definition of q-integral q-Bernstein-Kantorovich Operators 115Again by definition of q-integralTo estimate K n,q (t 2 , x), we haveTherefore using [k + 1] q = q[k] q + 1 and using the similar methods as above, we haveReplacing q by a sequence {q n } such that lim n→∞ q n = 1, it is easily seen that K n,q (t i , x), i = 0, 1, 2 converges uniformly to t i . Thus the result follows by Korovkin's theorem. for all f ∈ C[0, a] and δ n = K n,q ((t − x) 2 , x).

show abstract

Approximation by Durrmeyer Type Operators Using Polya Distribution

Sharma,

Sharma

2024

Mathematical Modeling for Computer Applications

View full text Add to dashboard Cite

Local approximation by a variant of Bernstein–Durrmeyer operators

Cited by 42 publications

References 14 publications

Approximation by complex summation-integral type operator in compact disks

Approximation by complex summation-integral type operator in compact disks

Applications of q-Calculus in Operator Theory

Approximation by Durrmeyer Type Operators Using Polya Distribution

Contact Info

Product

Resources

About