Displaying similar documents to “Continuation of holomorphic solutions to convolution equations in complex domains”

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in ℂ N

S. Melikhov, Siegfried Momm (1995)

Studia Mathematica

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G ⊂ ℂ N is compact and convex it is known for a long time that the nonzero constant coefficients linear partial differential operators (of finite or infinite order) are surjective on the space of all analytic functions on G. We consider the question whether solutions of the inhomogeneous equation can be given in terms of a continuous linear operator. For instance we characterize those sets G for which this is always the case.

Multisummability for some classes of difference equations

Boele L. J. Braaksma, Bernard F. Faber (1996)

Annales de l'institut Fourier

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This paper concerns difference equations y ( x + 1 ) = G ( x , y ) where G takes values in C n and G is meromorphic in x in a neighborhood of ∞ in C and holomorphic in a neighborhood of 0 in C n . It is shown that under certain conditions on the linear part of G , formal power series solutions in x - 1 / p , p ∈ N , are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions. ...

Continuous linear right inverses for convolution operators in spaces of real analytic functions

Michael Langenbruch (1994)

Studia Mathematica

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We determine the convolution operators T μ : = μ * on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).

Non-holomorphic functional calculus for commuting operators with real spectrum

Mats Andersson, Bo Berndtsson (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider n -tuples of commuting operators a = a 1 , ... , a n on a Banach space with real spectra. The holomorphic functional calculus for a is extended to algebras of ultra-differentiable functions on ℝ n , depending on the growth of ∥ exp ( i a · t ) ∥ , t ∈ ℝ n , when | t | → ∞ . In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.