Pointwise estimation of comonotone approximation

Dzyubenko, G. A.

doi:10.1007/bf01058880

Cited by 5 publications

(4 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Estimates (5) and (7) are in [10] where a "flip" of DeVore and Yu's [7] polynomial was used to prove (5). Using the approach of this paper one can modify the polynomial in [10] to get (4) but we construct a "new" polynomial for Theorem 1 below.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comonotone approximation with interpolation at the ends on an interval

Dzyubenko¹

2006

Analys in Theo Applic

Self Cite

View full text Add to dashboard Cite

Let a function f ∈ C[−1, 1], changes its monotonisity at the finite collection Y := {y1, · · · , ys} of s points yi ∈ (−1, 1). For each n ≥ N (Y ), we construct an algebraic polynomial Pn, of degree ≤ n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, andwhere N (Y ) is a constant depending only on Y, c(s) is a constant depending only on s and ω2 (f, t) is the second modulus of smoothness of f.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Using the approach of this paper one can modify the polynomial in [10] to get (4) but we construct a "new" polynomial for Theorem 1 below. Finally, Wu and Zhou [11] proved that (8) (and hence (4)-(7) as well) is invalid with ω k , k > 2.…”

Section: Introductionmentioning

confidence: 99%

Comonotone approximation with interpolation at the ends on an interval

Dzyubenko¹

2006

Analys in Theo Applic

Self Cite

View full text Add to dashboard Cite

show abstract

“…More recently, R. A. DeVore and X. M. Yu [4] and G. A. Dzyubenko [6] have shown that one can get also pointwise estimates, namely,…”

mentioning

confidence: 98%

Nearly Comonotone Approximation

Leviatan

Shevchuk

1998

Journal of Approximation Theory

View full text Add to dashboard Cite

We discuss the degree of approximation by polynomials of a function f that is piecewise monotone in [&1, 1]. We would like to approximate f by polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show here that in that case the polynomials can achieve the rate of | 3 . On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allows | 4 estimates. Academic Press

show abstract

“…Comonotone approx. with s 2, validity of f ÔxÕ¡P n ÔxÕ cÔk, r, sÕρ r n ÔxÕω k f ÔrÕ , ρ n ÔxÕThe results for r 0 are by Shvedov[Shv81a], Zhou[Zho93], and Dzyubenko[Dzy94]. All other results are due to Dzyubenko, Gilewicz and Shevchuk[DGS98].We summarize all references for both tables in one table of references.…”

mentioning

confidence: 99%

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

Kopotun¹,

Leviatan²,

Shevchuk³

2019

The SMAI Journal of Computational Mathematics

View full text Add to dashboard Cite

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same.The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-q-monotone approximation with q ≥ 3 is completely different from comonotone and coconvex cases.Additionally, we show that, for each function f from ∆ (1) , the set of all monotone functions on [−1, 1], and every α > 0, we have lim sup n→∞ inf Pn∈Pn∩∆ (1)

show abstract

Pointwise estimation of comonotone approximation

Cited by 5 publications

References 5 publications

Comonotone approximation with interpolation at the ends on an interval

Comonotone approximation with interpolation at the ends on an interval

Nearly Comonotone Approximation

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

Contact Info

Product

Resources

About