Bernstein Operators and Their Properties

Bustamante, Jorge

doi:10.1007/978-3-319-55402-0

Cited by 51 publications

(28 citation statements)

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“…Here we deal with indices = 2 16 assuming that 8. First let us prove asymptotic formula (16). We shall study separately two cases fitting them with standard asymptotics of form (11).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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On growth rate of coefficients in Bernstein polynomials for the standard modulus function on a symmetric interval

Petrosova¹,

Tikhonov²,

Sherstyukov³

2018

Ufimskii Matematicheskii Zhurnal

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The subject of the paper is closely related to one general direction in the approximation theory, within which the growth rate of the coefficients of algebraic polynomials is studied for uniform approximations of continuous functions. The classical Bernstein polynomials play an important role here. We study in detail a model example of Bernstein polynomials for the standard modulus function on a symmetric interval. The question under consideration is the growth rate of of the coefficients in these polynomials with an explicit algebraic representation. It turns out that in the first fifteen polynomials the growth of the coefficients is almost not observed. For the next polynomials the situation changes, and coefficients of exponential growth appear. Our main attention is focused on the behaviour of the maximal coefficient, for which exact exponential asymptotics and corresponding two-sided estimates are established (see Theorem 2). As it follows from the obtained result, the maximal coefficient has growth 2 /2 / 2 , where is the index of the Bernstein polynomial. It is shown that the coefficients equidistant from the maximal one have a similar growth rate (for details, see Theorem 3). The group of the largest coefficients is located in the central part of the Bernstein polynomials but at the ends the coefficients are sufficiently small. The behavior of the sum of absolute values of all coefficients is also considered. This sum admits an explicit expression that is not computable in the sense of traditional combinatorial identities. On the base of a preliminary recurrence relation, we succeeded to obtain the exact asymptotics for the sum of absolute values of all coefficients and to give the corresponding two-sided estimates (see Theorem 4). The growth rate of the sum is 2 /2 / 3/2. In the end of the paper, we compare this result with a general Roulier estimate and new related problems are formulated.

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“…Here we deal with indices = 2 16 assuming that 8. First let us prove asymptotic formula (16). We shall study separately two cases fitting them with standard asymptotics of form (11).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…While analyzing the proof of Theorem 2, there can arise the idea not to split the proof of estimate (17) into two cases but in view of established asymptotics (16), to introduce one resulting sequence…”

Section: Proof Of Theoremmentioning

confidence: 99%

On growth rate of coefficients in Bernstein polynomials for the standard modulus function on a symmetric interval

Petrosova¹,

Tikhonov²,

Sherstyukov³

2018

Ufimskii Matematicheskii Zhurnal

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“…Firstly, T. Popoviciu ([21,22]) gave a solution in this direction considering the first modulus of continuity. Later these results have been improved by Lorentz [18] and Sikkema [23] and the recent monograph of Bustamante [3]. New technique introduced in order to study the order of approximation was given by Esser [7], who used the first estimates with ω 2 .…”

Section: Introductionmentioning

confidence: 99%

Better Numerical Approximation by Durrmeyer Type Operators

2019

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“…In the final section, we derive a Voronovskaja type theorem for the geometric series of Bernstein operators. For a general theory on Bernstein operators see the papers [9], [6], [5]. For specific problems regarding Voronovskaja theorem we indicate the papers [4], [7].…”

Section: Introductionmentioning

confidence: 99%