scholarly journals Homogeneous polynomials on strictly convex domains

2007 ◽  
Vol 135 (12) ◽  
pp. 3895-3904 ◽  
Author(s):  
Piotr Kot
2021 ◽  
Vol 15 (4) ◽  
Author(s):  
P. Pierzchała ◽  
P. Kot

AbstractIn this paper we study the so-called Radon inversion problem in bounded, circular, strictly convex domains with $${\mathcal {C}}^2$$ C 2 boundary. We show that given $$p>0$$ p > 0 and a strictly positive, continuous function $$\Phi $$ Φ on $$\partial \Omega $$ ∂ Ω , by use of homogeneous polynomials it is possible to construct a holomorphic function $$f \in {\mathcal {O}}(\Omega )$$ f ∈ O ( Ω ) such that $$\displaystyle \smallint _0^1 |f(zt)|^pdt = \Phi (z)$$ ∫ 0 1 | f ( z t ) | p d t = Φ ( z ) for all $$z \in \partial \Omega $$ z ∈ ∂ Ω . In our approach we make use of so-called lacunary K-summing polynomials (see definition below) that allow us to construct solutions with in some sense extremal properties.


1987 ◽  
Vol 196 (3) ◽  
pp. 343-353 ◽  
Author(s):  
Kam-Wing Leung ◽  
Giorgio Patrizio ◽  
Pit-Mann Wong

2004 ◽  
Vol 105 (1) ◽  
pp. 29-42 ◽  
Author(s):  
Bruno Colbois ◽  
Patrick Verovic

1973 ◽  
Vol 74 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Vishwa Chander Dumir ◽  
Dharam Singh Khassa

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.


2014 ◽  
Vol 180 (1) ◽  
pp. 323-380 ◽  
Author(s):  
Oana Ivanovici ◽  
Gilles Lebeau ◽  
Fabrice Planchon

Analysis ◽  
2015 ◽  
Vol 35 (1) ◽  
Author(s):  
Alexander G. Ramm

AbstractLet


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