Monotone approximation with linear differential operators

Anastassiou, George A.; Shisha, Oved

doi:10.1016/0021-9045(85)90089-9

Cited by 31 publications

(13 citation statements)

References 2 publications

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“…In [2] we proved the following Theorem 4. Let h, v, r be integers, 0 ≤ h ≤ v ≤ r and let f P C r pIq, with f prq having modulus of smoothness ω s`f prq , δ˘there, s ≥ 1.…”

Section: Theorem 2 Implies the Usefulmentioning

confidence: 85%

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Univariate right fractional polynomial high order monotone approximation

Anastassiou

2016

Demonstratio Mathematica

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Abstract. Let f P C r pr´1, 1sq, r ≥ 0 and let L˚be a linear right fractional differential operator such that L˚pf q ≥ 0 throughout r´1, 0s. We can find a sequence of polynomials Qn of degree ≤ n such that L˚pQnq ≥ 0 over r´1, 0s, furthermore f is approximated right fractionally and simultaneously by Qn on r´1, 1s. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f prq .

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“…In [2] we proved the following Theorem 4. Let h, v, r be integers, 0 ≤ h ≤ v ≤ r and let f P C r pIq, with f prq having modulus of smoothness ω s`f prq , δ˘there, s ≥ 1.…”

Section: Theorem 2 Implies the Usefulmentioning

confidence: 85%

“…In [3] the authors replaced the kth derivative with a linear differential operator of order k. We mention this motivating result. Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

Univariate right fractional polynomial high order monotone approximation

Anastassiou

2016

Demonstratio Mathematica

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“…Date: February 14, 2018. 1 Furthermore, if there also exists a compact subinterval J(∋ 0) of R, and C 2 > 0 such that…”

Section: Introductionmentioning

confidence: 99%

Higher order derivatives of approximation polynomials on R $\mathbb{R}$

Jung

Sakai

2015

J Inequal Appl

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“…There the problem was: given a positive integer r, approximate with rates a given function whose rth derivative is > 0 by polynomials having the same property. This initial problem was generalized by G. A. Anastassiou and O. Shisha in 1985 (see [1]) by replacing the rth derivative with a linear differential operator of order r. The rate of the related uniform convergence was given through the first modulus of continuity.…”

Section: Introductionmentioning

confidence: 99%

Bivariate Monotone Approximation

Anastassiou¹

1991

Proceedings of the American Mathematical Society

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Abstract.Let / be a two variable continuously differentiable real-valued function of certain order on [0, 1] and let L be a linear differential operator involving mixed partial derivatives and suppose that L(f) > 0 . Then there exists a sequence of two-dimensional polynomials Qm n(x, y) with L(Qm n) > 0, so that / is approximated simultaneously and uniformly by Qm n . This approximation is accomplished quantitatively by the use of a suitable twodimensional first modulus of continuity.

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Monotone approximation with linear differential operators

Cited by 31 publications

References 2 publications

Univariate right fractional polynomial high order monotone approximation

Univariate right fractional polynomial high order monotone approximation

Higher order derivatives of approximation polynomials on R $\mathbb{R}$

Bivariate Monotone Approximation

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