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The polynomial hull of unions of convex sets in n

Ulf BacklundAnders Fällström — 1996

Colloquium Mathematicae

We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form ( z 1 , z 2 , z 3 ) 3 : | z 1 | 2 + | z 2 | 2 + | z 3 | 2 m 1 , such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.

On the algebra of A k -functions

Ulf BacklundAnders Fällström — 2006

Mathematica Bohemica

For a domain Ω n let H ( Ω ) be the holomorphic functions on Ω and for any k let A k ( Ω ) = H ( Ω ) C k ( Ω ¯ ) . Denote by 𝒜 D k ( Ω ) the set of functions f Ω [ 0 , ) with the property that there exists a sequence of functions f j A k ( Ω ) such that { | f j | } is a nonincreasing sequence and such that f ( z ) = lim j | f j ( z ) | . By 𝒜 I k ( Ω ) denote the set of functions f Ω ( 0 , ) with the property that there exists a sequence of functions f j A k ( Ω ) such that { | f j | } is a nondecreasing sequence and such that f ( z ) = lim j | f j ( z ) | . Let k and let Ω 1 and Ω 2 be bounded A k -domains of holomorphy in m 1 and m 2 respectively. Let g 1 𝒜 D k ( Ω 1 ) , g 2 𝒜 I k ( Ω 1 ) and h 𝒜 D k ( Ω 2 ) 𝒜 I k ( Ω 2 ) . We prove that the...

Spectrum of certain Banach algebras and ∂̅-problems

Linus CarlssonUrban CegrellAnders Fällström — 2007

Annales Polonici Mathematici

We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂̅-problems with certain estimates can be solved. We show that the projection of the spectrum onto ℂⁿ equals Ω̅ and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.

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