scholarly journals Some sharp inequalities for approximations of periodic functions in $L_1$ space

1987 ◽  
pp. 4
Author(s):  
V.F. Babenko
Keyword(s):  
Periodic Functions ◽  
Piecewise Constant ◽  
Sharp Estimates ◽  
Alpha Beta

We provide sharp estimates of Jackson's inequalities type for the best $(\alpha, \beta)$-approximations in the space $L_1$ of periodic functions that are representable as the convolution of the kernel $K$ that does not increase the number of sign alternations with functions $\varphi \in C$, by means of convolutions of the kernel $K$ with the functions that are piecewise-constant in the intervals $\bigl( \frac{l \pi}{n}, \frac{(l+1)\pi}{n} \bigr)$.

scholarly journals The best polynomial approximation, derivatives of fractional order, and widths of classes of functions in $L_2$

2016 ◽  
Vol 24 ◽  
pp. 10
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Fractional Order ◽  
Polynomial Approximation ◽  
Fractional Derivatives ◽  
Periodic Functions ◽  
Best Polynomial Approximation ◽  
Alpha Beta ◽  
Derivatives Of

On the classes of $2\pi$-periodic functions ${\mathcal{W}}^{\alpha} (K_{\beta}, \Phi)$, where $\alpha, \beta \in (0;\infty)$, defined by $K$-functionals $K_{\beta}$, fractional derivatives of order $\alpha$, and majorants $\Phi$, the exact values of different $n$-widths have been computed in the space $L_2$.

EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

2021 ◽  
Vol 10 (9) ◽  
pp. 3113-3128
Author(s):  
M.I. Muminov ◽  
Z.Z. Jumaev
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Linear Equations ◽  
Periodic Functions ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
Second Order Differential Equations ◽  
Greatest Integer Function ◽  
Soluble Problem ◽  
Existence Conditions

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

scholarly journals Non-symmetric approximations of functional classes by splines on the real line

2021 ◽  
Vol 13 (3) ◽  
pp. 831-837
Author(s):  
N.V. Parfinovych
Keyword(s):  
Unit Ball ◽  
Real Line ◽  
Polynomial Splines ◽  
Periodic Functions ◽  
Best Approximations ◽  
The Real ◽  
Functional Classes ◽  
Alpha Beta

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.

Sign in / Sign up

Export Citation Format

Share Document