The sharpness of convergence results for q-Bernstein polynomials in the case q &gt; 1

Ostrovska, Sofiya

doi:10.1007/s10587-008-0079-7

Cited by 22 publications

(10 citation statements)

References 14 publications

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“…The lack of positivity makes the investigation of convergence in the case q > 1 essentially more difficult than for 0 < q < 1. There are few papers ( [5], [6], [8], and [9] ) studying systematically the convergence in the case q > 1. If q 1 then qualitative Voronovskaja-type and saturation results for complex q-Bernstein polynomials were obtained in Wang-Wu [8].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Mahmudov

2012

Czech Math J

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Mahmudov

2012

Czech Math J

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“…Phillips and his colleagues have intensively studied these operators and many applications and generalizations have been investigated (see [134] and references therein). Also Ostrovska (see [125,126,129,130]) established some interesting properties on such operators. In [128], she gave a systematic study on these operators on the completion of one of the decade on q-Bernstein polynomials.…”

Section: Q-bernstein Operatorsmentioning

confidence: 99%

Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

372

206

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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).

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“…The lack of positivity makes the investigation of convergence in the case q > 1 essentially more difficult than for 0 < q < 1. Notice that, the complex q-Bernstein type operators in the case q > 1 systematically are studied in [10], [11], [12], [13], [14], and [15].…”

Section: Remarkmentioning

confidence: 99%

Approximation Properties of the q-Balázs–Szabados Complex Operators in the Case $$q\ge 1$$ q ≥ 1

Mahmudov

2016

Comput. Methods Funct. Theory

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This paper deals with approximation properties of the newly defined q-generalization of the Balázs-Szabados operators in the case q ≥ 1. Quantitative estimates of the convergence and Voronovskaja type theorem are given. In particular, it is proved that the rate of approximation by the q-Balázs-Szabados (q > 1) is of order q −n versus 1/n for the classical Balázs-Szabados (q = 1) operators. The results are new even for the classical case q = 1.

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The sharpness of convergence results for q-Bernstein polynomials in the case q > 1

Cited by 22 publications

References 14 publications

Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Applications of q-Calculus in Operator Theory

Approximation Properties of the q-Balázs–Szabados Complex Operators in the Case $$q\ge 1$$ q ≥ 1

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