Minimal polynomials for compact sets of the complex plane

1996 ◽  
Vol 12 (4) ◽  
pp. 481-488 ◽  
Author(s):  
F. Peherstorfer
Keyword(s):  
Complex Plane ◽  
Compact Sets ◽  
Minimal Polynomials

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.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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