scholarly journals On the Relationship of a Convolution Type Transformation and the Best Approximation of Periodic Functions

2021 ◽  
pp. 5-15
Author(s):  
Yusufali Khasanov ◽  
◽  
Yosuman Kasimova ◽  
Keyword(s):  
Best Approximation ◽  
Convolution Type ◽  
Periodic Functions ◽  
The Best Approximation ◽  
Type Transformation ◽  
Approximation Of Periodic Functions ◽  
Relationship Of ◽  
The Relationship

scholarly journals Automated Kinematic Analysis of Closed-Loop Planar Link Mechanisms

Machines ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 41
Author(s):  
Tatsuya Yamamoto ◽  
Nobuyuki Iwatsuki ◽  
Ikuma Ikeda
Keyword(s):  
Kinematic Analysis ◽  
Closed Loop ◽  
Analysis Procedure ◽  
Linkage Mechanism ◽  
A Algorithm ◽  
Analysis Process ◽  
Type Transformation ◽  
Extraction Algorithm ◽  
Relationship Of ◽  
The Relationship

The systematic kinematic analysis method for planar link mechanisms based on their unique procedures can clearly show the analysis process. The analysis procedure is expressed by a combination of many kinds of conversion functions proposed as the minimum calculation units for analyzing a part of the mechanism. When it is desired to perform this systematic kinematics analysis for a specific linkage mechanism, expert researchers can accomplish the analysis by searching for the procedure by themselves, however, it is difficult for non-expert users to find the procedure. This paper proposes the automatic procedure extraction algorithm for the systematic kinematic analysis of closed-loop planar link mechanisms. By limiting the types of conversion functions to only geometric calculations that are related to the two-link chain, the analysis procedure can be represented by only one type transformation function, and the procedure extraction algorithm can be described as a algorithm searching computable 2-link chain. The configuration of mechanism is described as the “LJ-matrix”, which shows the relationship of connections between links with pairs. The algorithm consists of four sub-processes, namely, “LJ-matrix generator”, “Solver process”, “Add-link process”, and “Over-constraint resolver”. Inputting the sketch of the mechanism into the proposed algorithm, it automatically extracts unique analysis procedure and generate a kinematic analysis program as a MATLAB code based on it. Several mechanisms are analyzed as examples to show the usefulness of the proposed method.

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

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 $n-1$.

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

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