Monotone approximation of functions by trigonometric polynomials

1983 ◽  
Vol 34 (3) ◽  
pp. 669-675
Author(s):  
A. Yu. Shadrin
Keyword(s):  
Trigonometric Polynomials ◽  
Approximation Of Functions ◽  
Monotone Approximation

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 $r є (3,4)$.

scholarly journals EESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM

2020 ◽  
Vol 6 (1) ◽  
pp. 16
Author(s):  
Gabdolla Akishev
Keyword(s):  
Lorentz Space ◽  
Best Approximation ◽  
Trigonometric Polynomials ◽  
Exact Order ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation ◽  
Anisotropic Norm

In this paper, we consider the anisotropic Lorentz space \(L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})\) of periodic functions of \(m\) variables. The Besov space \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) by trigonometric polynomials under different relations between the parameters \(\bar{p}, \bar\theta,\) and \(\tau\).The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function \(f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})\) to belong to the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\) in the case \(1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},\) in terms of the best approximation and prove its unimprovability on the class \(E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}\) \({n=0,1,\ldots\},}\) where \(E_{n}(f)_{\bar{p},\bar{\theta}}\) is the best approximation of the function \(f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\) by trigonometric polynomials of order \(n\) in each variable \(x_{j},\) \(j=1,\ldots,m,\) and \(\lambda=\{\lambda_{n}\}\) is a sequence of positive numbers \(\lambda_{n}\downarrow0\) as \(n\to+\infty\). In the second section, we establish order-exact estimates for the best approximation of functions from the class \(B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}\) in the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).

scholarly journals Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

2021 ◽  
Vol 103 (3) ◽  
pp. 54-67
Author(s):  
A.E. Jetpisbayeva ◽  
◽  
A.A. Jumabayeva ◽  
Keyword(s):  
Best Approximation ◽  
Three Dimensional ◽  
Continuous Functions ◽  
Trigonometric Polynomials ◽  
Dimensional Case ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

On the Jackson-Type Inequality for the Best Sp-Approximations of Functions by Trigonometric Polynomials

2017 ◽  
Vol 8 ◽  
pp. 27-35
Author(s):  
Alexander N. Shchitov
Keyword(s):  
Best Approximation ◽  
Type Inequality ◽  
Trigonometric Polynomials ◽  
Sharp Constant ◽  
Jackson Type ◽  
Moduli Of Continuity ◽  
Approximation Of Functions ◽  
The Best Approximation

We find the sharp constant in the Jackson-type inequality between the value of the best approximation of functions by trigonometric polynomials and moduli of continuity of m-th order in the spaces Sp, 1 ≤ p < ∞. In the particular case we obtain one result which in a certain sense generalizes the result obtained by L.V. Taykov for m = 1 in the space L2 for the arbitrary moduli of continuity of m-th order (m Є N).

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