Abstract
The convergence set of a divergent formal power series f ( x 0 , ⋯ , x n ) is the set of all “directions” ξ ∈ P n along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in P n is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye (J Reine Angew Math 283:86–98, 1976 ), Lelong (Proc Am Math Soc 2:11–19, 1951 ), Levenberg and Molzon (Math Z 197:411–420, 1988 ), and of Ribon (Ann Scuola Norm Sup Pisa Cl Sci (5) 3:657–680 2004 ) are obtained.
Highlights
An important problem in applied mathematics is the problem of restoration of a function from its integrals over a given collection of sets
We prove that a subset of C is a convergence set if and only if it is a quasi--connected set
In R3 the integrals of a function over a set of one dimensional lines allow one to restore the function by using the inverse of the Radon transform
Summary
An important problem in applied mathematics is the problem of restoration of a function from its integrals over a given collection of sets. An example of such a problem is tomography. In R3 the integrals of a function over a set of one dimensional lines allow one to restore the function by using the inverse of the Radon transform. This has wide applications in medicine, namely CAT scan. At the end of this chapter we prove that every quasisimply-connected set in C is a convergence set
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