scholarly journals A Characterization of Smoothness in Terms of Approximation by Algebraic Polynomials in LP

1995 ◽  
Vol 81 (1) ◽  
pp. 13-22 ◽  
Author(s):  
V.A. Operstein
Keyword(s):  
Algebraic Polynomials ◽  
Approximation By Algebraic Polynomials

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.

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

2000 ◽  
Vol 36 (3-4) ◽  
pp. 353-358 ◽  
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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