Convex Polynomial Approximation in the Uniform Norm: Conclusion

2005 ◽  
Vol 57 (6) ◽  
pp. 1224-1248 ◽  
Author(s):  
K. A. Kopotun ◽  
D. Leviatan ◽  
I. A. Shevchuk
Keyword(s):  
Convex Function ◽  
Polynomial Approximation ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
Type Estimate ◽  
Open Problems ◽  
Open Questions

AbstractEstimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Some negative results for the interpolation monotone approximation of functions having a fractional derivative

2020 ◽  
pp. 122-127
Author(s):  
T. O. Petrova ◽  
I. P. Chulakov
Keyword(s):  
Fractional Derivative ◽  
Convex Function ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Negative Results ◽  
Positive Functions ◽  
Approximation Of A Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

Simultaneous Approximations of Polynomials and Derivatives and Their Applications to Neural Networks

2008 ◽  
Vol 20 (11) ◽  
pp. 2757-2791 ◽  
Author(s):  
Yoshifusa Ito
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Probability Measures ◽  
Compact Sets ◽  
Network Parameters ◽  
Whole Space ◽  
Hidden Layer ◽  
Approximating Polynomials

We have constructed one-hidden-layer neural networks capable of approximating polynomials and their derivatives simultaneously. Generally, optimizing neural network parameters to be trained at later steps of the BP training is more difficult than optimizing those to be trained at the first step. Taking into account this fact, we suppressed the number of parameters of the former type. We measure degree of approximation in both the uniform norm on compact sets and the Lp-norm on the whole space with respect to probability measures.

Constrained Approximation with Jacobi Weights

2016 ◽  
Vol 68 (1) ◽  
pp. 109-128 ◽  
Author(s):  
Kirill Kopotun ◽  
Dany Leviatan ◽  
Igor Shevchuk
Keyword(s):  
Polynomial Approximation ◽  
Spline Approximation ◽  
Modulus Of Smoothness ◽  
Jacobi Weights ◽  
Jacobi Weight ◽  
Constrained Approximation ◽  
Monotone Polynomial Approximation

AbstractIn this paper, we prove that for ℓ = 1 or 2 the rate of best ℓ- monotone polynomial approximation in the Lp norm (1 ≤ p ≤) weighted by the Jacobi weight with , is bounded by an appropriate (ℓ + 1)-st modulus of smoothness with the same weight, and that this rate cannot be bounded by the (ℓ + 2)-nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

Artificial Life as a Tool for Biological Inquiry

1993 ◽  
Vol 1 (1_2) ◽  
pp. 1-13 ◽  
Author(s):  
Charles Taylor ◽  
David Jefferson
Keyword(s):  
Cultural Evolution ◽  
Artificial Life ◽  
Living Systems ◽  
Grand Challenge ◽  
Open Problems ◽  
Maintenance Of Sex ◽  
Open Questions ◽  
Shifting Balance ◽  
The Origin Of Life ◽  
Natural Life

Artificial life embraces those human-made systems that possess some of the key properties of natural life. We are specifically interested in artificial systems that serve as models of living systems for the investigation of open questions in biology. First we review some of the artificial life models that have been constructed with biological problems in mind, and classify them by medium (hardware, software, or “wetware”) and by level of organization (molecular, cellular, organismal, or population). We then describe several “grand challenge” open problems in biology that seem especially good candidates to benefit from artificial life studies, including the origin of life and self-organi- zation, cultural evolution, origin and maintenance of sex, shifting balance in evolution, the relation between fitness and adaptedness, the structure of ecosystems, and the nature of mind.

scholarly journals Sharp inequalities of Jackson type in weighted space $L_{2;\rho}({\mathbb{R}}^2)$

2014 ◽  
Vol 22 ◽  
pp. 17
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Best Approximation ◽  
Weighted Space ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
The Best Approximation ◽  
Differentiable Functions ◽  
Functions Of Two Variables

Sharp inequalities of Jackson type, connected with the best approximation by "angles" of algebraic polynomials have been obtained on the classes of differentiable functions of two variables in the metric of space $L_{2;\rho}({\mathbb{R}}^2)$ of the Chebyshev-Hermite weight.

scholarly journals One counterexample for convex approximation of function with fractional derivatives, r>4

2018 ◽  
pp. 53-56
Author(s):  
T. Petrova
Keyword(s):  
Convex Function ◽  
Algebraic Polynomial ◽  
Fractional Derivatives ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Positive Functions ◽  
Approximation Of A Function ◽  
Approximation Of Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

scholarly journals Inequalities of Jackson type for ridge and polynomial approximation of harmonic functions

2013 ◽  
Vol 21 ◽  
pp. 40
Author(s):  
V.F. Babenko ◽  
D.A. Levchenko
Keyword(s):  
Unit Disk ◽  
Polynomial Approximation ◽  
Harmonic Functions ◽  
Jackson Type

New sharp Jackson type inequalities for ridge and polinomial approximation of some classes of functions harmonic inside unit disk are obtained.

scholarly journals On the Approximation Properties of q − Analogue Bivariate λ -Bernstein Type Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Edmond Aliaga ◽  
Behar Baxhaku
Keyword(s):  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Bernstein Type ◽  
Matlab Software ◽  
Type Space ◽  
Mixed Modulus Of Smoothness ◽  
Mixed Modulus ◽  
Boolean Sum

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

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