scholarly journals On the best mean-square approximation by entire functions of finite type on the line

2021 ◽  
Vol 17 ◽  
pp. 36
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Finite Type ◽  
Entire Functions ◽  
Jackson Type ◽  
Mean Square ◽  
Differentiable Functions

Jackson-type inequalities have been obtained for the best mean square approximation of differentiable functions by means of the entire functions of finite type on the line.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

scholarly journals Sharp inequalities of Jackson type in weighted space $L_{2;\rho}({\mathbb{R}}^2)$

2014 ◽  
Vol 22 ◽  
pp. 17
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Best Approximation ◽  
Weighted Space ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
The Best Approximation ◽  
Differentiable Functions ◽  
Functions Of Two Variables

Sharp inequalities of Jackson type, connected with the best approximation by "angles" of algebraic polynomials have been obtained on the classes of differentiable functions of two variables in the metric of space $L_{2;\rho}({\mathbb{R}}^2)$ of the Chebyshev-Hermite weight.

scholarly journals On approximation by trigonometrical polynomials in $L_2$ space

2016 ◽  
Vol 24 ◽  
pp. 89
Author(s):  
O.V. Polyakov
Keyword(s):  
Modulus Of Continuity ◽  
Best Approximation ◽  
Generalized Modulus Of Continuity ◽  
Jackson Type ◽  
The Best Approximation ◽  
Differentiable Functions

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

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