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

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: $\rho(x) = (1-x)^{\alpha} (1+x)^{\beta}$.

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

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.

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

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.