scholarly journals On approximation of functions by algebraic polynomials in $L^p_{\rho}$ metric

2021 ◽  
Vol 17 ◽  
pp. 48
Author(s):  
S.V. Goncharov
Keyword(s):  
Algebraic Polynomials ◽  
Approximation Of Functions ◽  
Approximation By Algebraic Polynomials

The version has been found, of improved Jackson's theorem analogue, concerning the approximation by algebraic polynomials on the interval in the integral metric for some classes of functions being integrable with the following weight: $\rho(x) = (1-x)^{\alpha} (1+x)^{\beta}$.

scholarly journals Approximation by algebraic polynomials in metric spaces $L_{\psi}$

2013 ◽  
Vol 21 ◽  
pp. 3
Author(s):  
T.A. Agoshkova
Keyword(s):  
Modulus Of Continuity ◽  
Metric Spaces ◽  
Jackson Inequality ◽  
Algebraic Polynomials ◽  
Periodic Functions ◽  
Jackson Theorem ◽  
Approximation By Algebraic Polynomials

In the space $L_{\psi}[-1;1]$ of non-periodic functions with metric $\rho(f,0)_{\psi} = \int\limits_{-1}^1 \psi(|f(x)|)dx$, where $\psi$ is a function of the type of modulus of continuity, we study Jackson inequality for modulus of continuity of $k$-th order in the case of approximation by algebraic polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

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