Holomorphic extension theorem for tempered ultrahyperfunctions

In recent analysis we have defined and studied holomorphic functions in tubes inℂnwhich generalize the HardyHpfunctions in tubes. In this paper we consider functionsf(z),z=x+iy, which are holomorphic in the tubeTC=ℝn+iC, whereCis the finite union of open convex conesCj,j=1,…,m, and which satisfy the norm growth of our new functions. We prove a holomorphic extension theorem in whichf(z),z ϵ TC, is shown to be extendable to a function which is holomorphic inT0(C)=ℝn+i0(C), where0(C)is the convex hull ofC, if the distributional boundary values in𝒮′off(z)from each connected componentTCjofTCare equal.

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A holomorphic extension theorem from a fractal hypersurface in ℂ2

Bulletin of the Brazilian Mathematical Society New Series ◽

10.1007/s00574-007-0066-x ◽

2007 ◽

Vol 38 (4) ◽

pp. 635-647

Author(s):

Ricardo Abreu Blaya ◽

Juan Bory Reyes ◽

Tania Moreno García

Keyword(s):

Extension Theorem ◽

Holomorphic Extension

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An extension theorem for holomorphic mappings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410005739x ◽

1980 ◽

Vol 88 (1) ◽

pp. 125-127

Author(s):

J. C. Wood

Keyword(s):

Complex Structure ◽

Extension Theorem ◽

Holomorphic Extension ◽

Holomorphic Mappings ◽

Necessary Condition ◽

Holomorphic Map ◽

Complex Linear ◽

Smooth Extension ◽

Smooth Map ◽

Topological Obstruction

Let X, Y be complex manifolds with smooth (C∞) boundaries ∂X, ∂Y. We give conditions which ensure that a smooth map Φ: ∂X → ∂Y has an extension to a holomorphic map X → Y. Let J denote the complex structure on X. We say that Φ satisfies the ‘tangential Cauchy-Riemann equation ∂¯bΦ = 0’if the differential dΦ restricted to the complex subspace Tp(∂X) ∩ JTp(∂X) of the tangent space Tp(∂X) is complex linear at all points p ∈ ∂X. Clearly this is a necessary condition for the existence of a holomorphic extension. A further necessary condition is that there exists no topological obstruction to extension, hence we assume that a smooth extension φ: X → Y is given and we shall look for a holomorphic map f: X → Y with the same boundary values.

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A Holomorphic Extension Theorem using Clifford Analysis

Complex Analysis and Operator Theory ◽

10.1007/s11785-009-0037-x ◽

2009 ◽

Vol 5 (1) ◽

pp. 113-130 ◽

Cited By ~ 1

Author(s):

Ricardo Abreu Blaya ◽

Juan Bory Reyes ◽

Dixan Peña Peña ◽

Frank Sommen

Keyword(s):

Clifford Analysis ◽

Extension Theorem ◽

Holomorphic Extension

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Holomorphic extension theorem

Proceedings of Symposia in Pure Mathematics - Differential Geometry ◽

10.1090/pspum/027.2/9918 ◽

1975 ◽

pp. 89-89

Author(s):

Peter Kiernan

Keyword(s):

Extension Theorem ◽

Holomorphic Extension

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Holomorphic extension of generalizations ofHpfunctions. II

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171287000012 ◽

1987 ◽

Vol 10 (1) ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Richard D. Carmichael

Keyword(s):

Extension Theorem ◽

Holomorphic Functions ◽

Holomorphic Extension ◽

In a previous article we have obtained a holomorphic extension theorem (edge of the wedge theorem) concerning holomorphic functions in tubes inℂnwhich generalize the HardyHpfunctions for the cases1<p≤2. In this paper we obtain a similar holomorphic extension theorem for the cases2<p<∞.

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Relativized realizability in intuitionistic arithmetic of all finite types

Journal of Symbolic Logic ◽

10.2307/2271946 ◽

1978 ◽

Vol 43 (1) ◽

pp. 23-44 ◽

Cited By ~ 25

Author(s):

Nicolas D. Goodman

Keyword(s):

Extension Theorem ◽

General Notion ◽

Conservative Extension ◽

First Order ◽

New Information ◽

Reflection Theorem ◽

Order Arithmetic ◽

Special Case ◽

Dependent Choice ◽

Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

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