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A notion of boundedness for hyperfunctions and Massera type theorems

Yasunori Okada (2012)

Banach Center Publications

For some classes of periodic linear ordinary differential equations and functional equations, it is known that the existence of a bounded solution in the future implies the existence of a periodic solution. In order to think on such phenomena for hyperfunction solutions to linear functional equations, we introduced a notion of bounded hyperfunctions, and translated the problems into the problems on analytic solutions to some equations in complex domains. In this article, after...

Binomial residues

Eduardo Cattani, Alicia Dickenstein, Bernd Sturmfels (2002)

Annales de l’institut Fourier

A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A -hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A .

Characterization of surjective convolution operators on Sato's hyperfunctions

Michael Langenbruch (2010)

Banach Center Publications

Let μ ∈ ( ℝ d ) ' be an analytic functional and let T μ be the corresponding convolution operator on Sato’s space ( ℝ d ) of hyperfunctions. We show that T μ is surjective iff T μ admits an elementary solution in ( ℝ d ) iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are 0 ≠ μ ∈ ( ℝ d ) ' such that T μ is not surjective on ( ℝ d ) .

Extension of CR functions to «wedge type» domains

Andrea D'Agnolo, Piero D'Ancona, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let X be a complex manifold, S a generic submanifold of X R , the real underlying manifold to X . Let Ω be an open subset of S with ∂ ⁡ Ω analytic, Y a complexification of S . We first recall the notion of Ω -tuboid of X and of Y and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ∂ ¯ b to the extendability of C ⁢ R functions on Ω to Ω -tuboids of X . Next, if X has complex dimension 2, we give results on extension...

Fronts d'onde à l'infini des fonctions analytiques réelles

Jean-Louis Lieutenant (1984)

Annales de l'institut Fourier

En adaptant les méthodes algébriques et géométriques qu’utilisent M. Sato, T. Kawai et M. Kashiwara pour obtenir le faisceau des microfonctions, nous construisons de manière fonctorielle, donc intrinsèque, un faisceau 𝒞 t sur la sphère cotangente à un espace vectoriel réel de dimension finie E . Les sections de ce faisceau jouent vis-à-vis des fonctions analytiques sur E un rôle analogue à celui des microfonctions vis-à-vis des hyperfonctions. Nous en déduisons une notions de front d’onde à l’infini...

Generalized Hermitean ultradistributions

C. Andrade, L. Loura (2009)

Mathematica Bohemica

In this paper we define, by duality methods, a space of ultradistributions ω ' ( ℝ N ) . This space contains all tempered distributions and is closed under derivatives, complex translations and Fourier transform. Moreover, it contains some multipole series and all entire functions of order less than two. The method used to construct 𝔾 ω ' ( ℝ N ) led us to a detailed study, presented at the beginning of the paper, of the duals of infinite dimensional locally convex spaces that are inductive limits of finite dimensional...

Liouville type theorem for solutions of linear partial differential equations with constant coefficients

Akira Kaneko (2000)

Annales Polonici Mathematici

We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and...

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