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Displaying similar documents to “On absolutely representing systems in spaces of infinitely differentiable functions”

Some characterizations of ultrabornological spaces

Manuel Valdivia (1974)

Annales de l'institut Fourier

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Let U be an infinite-dimensional separable Fréchet space with a topology defined by a family of norms. Let F be an infinite-dimensional Banach space. Then F is the inductive limit of a family of spaces equal to E . The choice of suitable classes of Fréchet spaces allows to give characterizations of ultrabornological spaces using the result above.. Let Ω be a non-empty open set in the euclidean n -dimensional space R n . Then F is the inductive limit of a family of spaces equal to D ( Ω ) . ...

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

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Let P ( λ , D ) = | α | m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander. ...

Characteristic Cauchy problems and solutions of formal power series

Sunao Ouchi (1983)

Annales de l'institut Fourier

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Let L ( z , z ) = ( z 0 ) k - A ( z , z ) be a linear partial differential operator with holomorphic coefficients, where A ( z , z ) = j = 0 k - 1 A j ( z , z ' ) ( z 0 ) j , ord . A ( z , z ) = m > k and z = ( z 0 , z ' ) C n + 1 . We consider Cauchy problem with holomorphic data L ( z , z ) u ( z ) = f ( z ) , ( z 0 ) i u ( 0 , z ' ) = u ^ i ( z ' ) ( 0 i k - 1 ) . We can easily get a formal solution u ^ ( z ) = n = 0 u ^ n ( z ' ) ( z 0 ) n / n ! , bu in general it diverges. We show under some conditions that for any sector S with the opening less that a constant determined by L ( z , z ) , there is a function u S ( z ) holomorphic except on { z 0 = 0 } such that L ( z , z ) u S ( z ) = f ( z ) and u S ( z ) u ^ ( z ) as z 0 0 in S .

A lifting theorem for locally convex subspaces of L 0

R. Faber (1995)

Studia Mathematica

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We prove that for every closed locally convex subspace E of L 0 and for any continuous linear operator T from L 0 to L 0 / E there is a continuous linear operator S from L 0 to L 0 such that T = QS where Q is the quotient map from L 0 to L 0 / E .

Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ( U )

Sean Dineen (1973)

Annales de l'institut Fourier

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This article is devoted to a study of locally convex topologies on H ( U ) (where U is an open subset of the locally convex topological vector space E and H ( U ) is the set of all complex valued holomorphic functions on E ). We discuss the following topologies on H ( U ) : (a) the compact open topology I 0 , (b) the bornological topology associated with I 0 , (c) the ported topology of Nachbin I ω , (d) the bornological topology associated with I ω  ; and ...

Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz (1994)

Studia Mathematica

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( s j k : j , k = 0 , 1 , . . . ) be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which ( s j k ) converges in Pringsheim’s sense. These conditions are satisfied if ( s j k ) is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If ( s j k ) is summable (C,1,1) to a finite limit and there exist constants n 1 > 0 and H such that j k ( s j k - s j - 1 , k - s j - 1 , k + s j - 1 , k - 1 ) - H , j ( s j k - s j - 1 , k ) - H and k ( s j k - s j , k - 1 ) - H whenever j , k > n 1 , then...