scholarly journals Polynomial approximation with varying weights on compact sets of the complex plane

1998 ◽  
Vol 126 (11) ◽  
pp. 3283-3292
Author(s):  
Igor E. Pritsker
Keyword(s):  
Complex Plane ◽  
Polynomial Approximation ◽  
Compact Sets

scholarly journals Co-convexial reflector curves with applications

1987 ◽  
Vol 10 (2) ◽  
pp. 233-240
Author(s):  
Abdallah M. Al-Rashed ◽  
Neyamat Zaheer
Keyword(s):  
Complex Plane ◽  
Convex Compact ◽  
Compact Sets ◽  
Full Generality ◽  
Line Segments ◽  
General Family ◽  
Family Of Curves ◽  
Closed Line ◽  
Van Vleck ◽  
Convex Compact Sets

The concept of reflector curves for convex compact sets of reflecting type in the complex plane was introduced by the authors in a recent paper (to appear in J. Math. Anal. and Appln.) in their attempt to solve a problem related to Stieltjes and Van Vleck polynomials. Though, in the said paper, certain convex compact sets (e.g. closed discs, closed line segments and the ones with polygonal boundaries) were shown to be of reflecting type, it was only conjectured that all convex compact sets are likewise. The present study not only proves this conjecture and establishes the corresponding results on Stleltjes and Van Vleck polynomials in its full generality as proposed earlier by the authors, but it also furnishes a more general family of curves sharing the properties of confocal ellipses.

scholarly journals On convergence in capacity

1976 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Burnett Meyer
Keyword(s):  
Complex Plane ◽  
Logarithmic Capacity ◽  
Set Convergence ◽  
Transfinite Diameter ◽  
Compact Sets ◽  
Convergence In Measure ◽  
The Given ◽  
Convergence In Capacity

The (logarithmic) capacity or transfinite diameter is originally defined for compact sets in the complex plane. An extension may be made by defining the capacity of a given arbitrary set in the plane as the supremum of the capacities of all compact sets contained in the given set. Convergence in capacity is defined analogously to convergence in measure. It is shown in this paper that properties of convergence in capacity are also analogous to those of convergence in measure.

scholarly journals Inequalities for products of polynomials I

2009 ◽  
Vol 104 (1) ◽  
pp. 147 ◽  
Author(s):  
I.E. Pritsker ◽  
S. Ruscheweyh
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Compact Sets ◽  
Sharp Constants ◽  
Best Constant ◽  
Connected Sets

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane.

Sign in / Sign up

Export Citation Format

Share Document