On uniqueness of the polynomial of best approximation of the function cos kx by trigonometric polynomials in the L metric

AbstractWe derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

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NOTE ON JACKSON AND BERNSTE N TYPE APPROXIMATION THEOREMS IN THE CASE OF APPROXIMATION BY ALGEBRAIC POLYNOMIALS N THE SPACES L AND C

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.36.2000.3-4.8 ◽

2000 ◽

Vol 36 (3-4) ◽

pp. 353-358 ◽

Cited By ~ 1

Author(s):

S. Pawelke

Keyword(s):

Periodic Function ◽

Best Approximation ◽

Trigonometric Polynomials ◽

Algebraic Polynomials ◽

Cient Condition ◽

Approximation Theorems ◽

Type Approximation ◽

The Best Approximation ◽

Approximation By Algebraic Polynomials

We con ider the best approximation E (n,f)by algebraic polynomials of degree at most n for function f in L 1 (-1, 1)or C [-1, 1]and give imple necessary and u .cient condition for E (n,f)=O (n-.),n .,u ing the well-known results in the ca e of ap- proximation of periodic function by trigonometric polynomials.

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On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

Ukrainian Mathematical Journal ◽

10.1007/bf01057518 ◽

1995 ◽

Vol 47 (9) ◽

pp. 1435-1440 ◽

Cited By ~ 1

Author(s):

A. S. Serdyuk

Keyword(s):

Best Approximation ◽

Trigonometric Polynomials ◽

Periodic Functions ◽

The Best Approximation

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To the question of approximation of continuous periodic functions by trigonometric polynomials

Researches in Mathematics ◽

10.15421/247711 ◽

2021 ◽

pp. 39

Author(s):

V.V. Shalaev

Keyword(s):

Best Approximation ◽

Modulus Of Smoothness ◽

Trigonometric Polynomials ◽

Periodic Functions ◽

Bounded Operators ◽

The Best Approximation

In the paper, it is proved that$$1 - \frac{1}{2n} \leqslant \sup\limits_{\substack{f \in C\\f \ne const}} \frac{E_n(f)_C}{\omega_2(f; \pi/n)_C} \leqslant \inf\limits_{L_n \in Z_n(C)} \sup\limits_{\substack{f \in C\\f \ne const}} \frac{\| f - L_n(f) \|_C}{\omega_2 (f; \pi/n)_C} \leqslant 1$$where $\omega_2(f; t)_C$ is the modulus of smoothness of the function $f \in C$, $E_n(f)_C$ is the best approximation by trigonometric polynomials of the degree not greater than $n-1$ in uniform metric, $Z_n(C)$ is the set of linear bounded operators that map $C$ to the subspace of trigonometric polynomials of degree not greater than $n-1$.

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

Ural mathematical journal ◽

10.15826/umj.2020.1.002 ◽

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 $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$.

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