Universal Taylor Series On Convex Subsets Of \MathbbC^N

2013/02/17 by Nicholas J. Daras, Daras, Nicholas J., Vassili Nestoridis +1
Mathematics · #32A05 #32A30 #41A58 #41A99 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30K05 #msc:32A05 #msc:32A30 #msc:40A05 #msc:41A58 #msc:41A99 #primary 30K05 #secondary 40A05

paper · pdf · doi:10.48550/arxiv.1302.4106

16 pages

arxiv created 2013/02/17 · arxiv updated 2013/02/19

Abstract

We prove the existence of holomorphic functions f defined on any open convex subset \rm Ω⊂ \mathbb Cⁿ, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from \rm Ω and on denumerably many convex compact sets in \mathbb Cⁿ\backslash \rm Ω which may meet the boundary ∂ \rm Ω. If the universal approximation is only required on convex compact sets disjoint from \rm Ω, then f may be chosen to be smooth on ∂ \rm Ω, that is f∈ A^∞(\rm Ω). Those are generic universalities.

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