scholarly journals On Best Approximations in Hyperconvex Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Reny George ◽  
Zoran D. Mitrović ◽  
Hassen Aydi
Keyword(s):  
Best Approximation ◽  
Approximation Theorem ◽  
Best Approximations ◽  
The Best Approximation

In this manuscript, we present further extensions of the best approximation theorem in hyperconvex spaces obtained by Khamsi.

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.

scholarly journals Some Inequalities in Functional Analysis, Combinatorics, and Probability Theory

10.37236/330 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Chunrong Feng ◽  
Liangpan Li ◽  
Jian Shen
Keyword(s):  
Probability Theory ◽  
Functional Analysis ◽  
Finite Field ◽  
Best Approximation ◽  
Approximation Theorem ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
The Best Approximation

The main purpose of this paper is to show that many inequalities in functional analysis, probability theory and combinatorics are immediate corollaries of the best approximation theorem in inner product spaces. Besides, as applications of the de Caen-Selberg inequality, the finite field Kakeya and Nikodym problems are also studied.

scholarly journals Best approximations for the Laguerre-type Weierstrass transform on[0,∞[×ℝ

2005 ◽  
Vol 2005 (17) ◽  
pp. 2757-2768 ◽  
Author(s):  
E. Jebbari ◽  
F. Soltani
Keyword(s):  
Hilbert Spaces ◽  
Best Approximation ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Extremal Functions ◽  
Best Approximations ◽  
The Best Approximation ◽  
Weierstrass Transform ◽  
Kernel Hilbert Spaces

We use reproducing kernel Hilbert spaces to give the best approximation for Laguerre-type Weierstrass transform. Estimates of extremal functions are also discussed.

THE ORDER OF THE BEST APPROXIMATION OF FUNCTIONS IN LEBESQUE SPACE

2020 ◽  
pp. 43-51
Author(s):  
A.A. Jumabayeva ◽  
◽  
A.E. Zhetpisbayeva ◽  
Keyword(s):  
Best Approximation ◽  
Upper Bounds ◽  
Trigonometric Polynomials ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
Hyperbolic Cross ◽  
The Best Approximation ◽  
Functions Of Two Variables ◽  
Step Hyperbolic Cross

The article considers the LP(T2) Lebesque space of periodec functions of two variables. The problems of approximation of functions of two variables by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses are stydied. Value EQγn(f)p=inft∈(Qγn)⌈f−t⌉p,i≤p≤∞ the best approximation of the function f(x) by trigonometric polynomials with “numbers” of harmonics from a step hyperbolic cross of Qγn The article consists of two sections. The first section contains some well-known statements necessary to prove the main results. In the second section, exact estimates of the best approximations of certain functions are established. These estimates make it possible to estimate the upper bounds of the best approximations for certain classes of functions. As approximation apparatuses, trigonometric polynomials with a spector from a stepwise hyperbolic cross are used. The questions considered in this work belong to the circle of questions studied in the works of K. I. Babenko, S. A.Telyakovsky, I. S.Bugrova, N.S.Nikolsky.

scholarly journals Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space

2019 ◽  
Vol 4 (4) ◽  
pp. 72-78
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajrayacharya
Keyword(s):  
Normed Space ◽  
Best Approximation ◽  
Normed Linear Space ◽  
Inner Product ◽  
Birkhoff Orthogonality ◽  
Isosceles Orthogonality ◽  
Best Approximations ◽  
Linear Spaces ◽  
The Best Approximation ◽  
Pythagorean Orthogonality

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

scholarly journals On the best approximation, by polynomials, of some classes of functions

1998 ◽  
Vol 6 ◽  
pp. 92
Author(s):  
O.V. Motornaia
Keyword(s):  
Best Approximation ◽  
Conjugate Functions ◽  
Algebraic Polynomials ◽  
Best Approximations ◽  
The Best Approximation ◽  
Approximation By Polynomials

We obtain asymptotically exact estimates of the best approximations of classes of conjugate functions by algebraic polynomials in the spaces $C$ and $L_1$.

Intracranial volume-pressure relationship in man

1982 ◽  
Vol 56 (4) ◽  
pp. 524-528 ◽  
Author(s):  
Joseph Th. J. Tans ◽  
Dick C. J. Poortvliet
Keyword(s):  
Best Approximation ◽  
Continuous Monitoring ◽  
Fluid Pressure ◽  
Constant Term ◽  
Volume Index ◽  
Intracranial Volume ◽  
Content Type ◽  
The Best Approximation ◽  
Ventricular Fluid Pressure ◽  
Fine Print

✓ The pressure-volume index (PVI) was determined in 40 patients who underwent continuous monitoring of ventricular fluid pressure. The PVI value was calculated using different mathematical models. From the differences between these values, it is concluded that a monoexponential relationship with a constant term provides the best approximation of the PVI.

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