scholarly journals Best Polynomial Approximation in -Norm and -Growth of Entire Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed El Kadiri ◽  
Mohammed Harfaoui
Keyword(s):  
Continuous Function ◽  
Polynomial Approximation ◽  
Extremal Function ◽  
Entire Functions ◽  
Maximum Norm ◽  
Best Polynomial Approximation ◽  
General Growth ◽  
Growth Of Entire Functions ◽  
Approximation Errors ◽  
Positive Capacity

The classical growth has been characterized in terms of approximation errors for a continuous function on by Reddy (1970), and a compact of positive capacity by Nguyen (1982) and Winiarski (1970) with respect to the maximum norm. The aim of this paper is to give the general growth (-growth) of entire functions in by means of the best polynomial approximation in terms of -norm, with respect to the set , where is the Siciak's extremal function on an -regular nonpluripolar compact is not pluripolar.

scholarly journals Generalized Order and Best Approximation of Entire Function in -Norm

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Mohammed Harfaoui
Keyword(s):  
Entire Function ◽  
Polynomial Approximation ◽  
Best Approximation ◽  
Extremal Function ◽  
Entire Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Best Polynomial Approximation ◽  
Growth Of Entire Functions

The aim of this paper is the characterization of the generalized growth of entire functions of several complex variables by means of the best polynomial approximation and interpolation on a compact with respect to the set , where is the Siciak extremal function of a -regular compact .

scholarly journals The minimal growth of entire functions with given zeros along unbounded sets

2020 ◽  
Vol 54 (2) ◽  
pp. 146-153
Author(s):  
I. V. Andrusyak ◽  
P.V. Filevych
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Entire Functions ◽  
Counting Function ◽  
Maximum Modulus ◽  
Positive Function ◽  
Minimal Growth ◽  
Complex Sequence ◽  
Growth Of Entire Functions ◽  
Necessary And Sufficient

Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

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