Carleman Approximation by Entire Functions on the Union of Two Totally Real Subspaces of Cn

1994 ◽  
Vol 37 (4) ◽  
pp. 522-526
Author(s):  
Per E. Manne
Keyword(s):  
Entire Functions ◽  
Continuous Functions ◽  
Real Dimension ◽  
Carleman Approximation ◽  
Polynomially Convex ◽  
Cauchy Riemann ◽  
Totally Real

AbstractLet L1, L2 ⊂ Cn be two totally real subspaces of real dimension n, and such that L1 ∩ L2 = {0}. We show that continuous functions on L1 ∪L2 allow Carleman approximation by entire functions if and only if L1 ∪L2 is polynomially convex. If the latter condition is satisfied, then a function f:L1 ∪L2 —> C such that f|LiCk(Li), i = 1,2, allows Carleman approximation of order k by entire functions if and only if f satisfies the Cauchy-Riemann equations up to order k at the origin.

A Characterization of Totally Real Carleman Sets and an Application to Products of Stratified Totally Real Sets

2016 ◽  
Vol 118 (2) ◽  
pp. 285 ◽  
Author(s):  
Benedikt S. Magnusson ◽  
Erlend Fornæss Wold
Keyword(s):  
Entire Functions ◽  
Carleman Approximation ◽  
Totally Real

We give a characterization of stratified totally real sets that admit Carleman approximation by entire functions. As an application we show that the product of two stratified totally real Carleman sets is a Carleman set.

scholarly journals BANACH SPACE VALUED CAUCHY–RIEMANN EQUATIONS WITH TOTALLY REAL BOUNDARY CONDITIONS

2004 ◽  
Vol 06 (04) ◽  
pp. 601-635 ◽  
Author(s):  
KATRIN WEHRHEIM
Keyword(s):  
Banach Space ◽  
Boundary Conditions ◽  
Closed Subspace ◽  
Lagrangian Submanifolds ◽  
Regularity Result ◽  
Cauchy Riemann ◽  
Natural Boundary Conditions ◽  
Definition Of ◽  
Regularity Results ◽  
Totally Real

The main purpose of this paper is to give a general regularity result for Cauchy–Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic Lp-regularity results hold true under one crucial assumption: The Banach space is isomorphic to a closed subspace of an Lp-space. (Equivalently, the totally real submanifold is modelled on a closed subspace of an Lp-space.) Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah–Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy–Riemann equation with totally real boundary condition.

scholarly journals Carleman approximation by holomorphic automorphisms of ℂ n

2018 ◽  
Vol 2018 (738) ◽  
pp. 131-148 ◽  
Author(s):  
Frank Kutzschebauch ◽  
Erlend Fornæss Wold
Keyword(s):  
Smooth Maps ◽  
Carleman Approximation ◽  
Totally Real ◽  
Holomorphic Automorphisms

Abstract We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of \mathbb{C}^{n} .

scholarly journals Polynomially convex embeddings of odd-dimensional closed manifolds

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Purvi Gupta ◽  
Rasul Shafikov
Keyword(s):  
Lower Bound ◽  
Continuous Functions ◽  
Topological Approach ◽  
Analytic Methods ◽  
Complex Dimension ◽  
Polynomially Convex ◽  
Orientable Manifold

Abstract It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

Multidimensional Volterra integral equations of convolution type

1979 ◽  
Vol 27 (3) ◽  
pp. 305-312 ◽  
Author(s):  
P. G. Laird
Keyword(s):  
Entire Functions ◽  
Continuous Solution ◽  
Continuous Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Convolution Type ◽  
Convolution Product ◽  
Exponential Polynomials ◽  
Linear Partial Differential Equations ◽  
All Solutions

AbstractIn this article, it is shown that the Volterra integral equation of convolution type w − w⊗g = f has a continuous solution w when f, g are continuous functions on Rx and ⊗ denotes a truncated convolution product. A similar result holds when f, g are entire functions of several complex variables. Also simple proofs are given to show when f, g are entire, f⊗g is entire, and, if f⊗g=0, then f = 0 or g = 0. Finally, the set of exponential polynomials and the set of all solutions to linear partial differential equations are considered in relation to this convolution product.

Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions

1975 ◽  
Vol 27 (1) ◽  
pp. 75-87 ◽  
Author(s):  
Andrew Adler ◽  
R. Douglas Williams
Keyword(s):  
Entire Functions ◽  
Continuous Functions ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Maximal Ideals ◽  
Primary Ideals ◽  
Special Case ◽  
The Ideal

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.

scholarly journals REMARKS ON SOME CLASSES OF POSITIVE CONTINUOUS FUNCTIONS IN C^n

2021 ◽  
pp. 9-15
Author(s):  
А. І. Bandura
Keyword(s):  
General Theory ◽  
Wide Class ◽  
Entire Functions ◽  
Logarithmic Derivative ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Complex Conjugate ◽  
Zero Points

Here we prove two propositions providing sufficient conditions of belonging positive continuous functions in  to classes  and  These auxiliary classes plays important role in theory of entire functions of bounded L-index in direction and bounded L-index in joint variables, where   are continuous functions. They help to constuct general theory of bounded index for very wide class of entire functions, because for every entire functions with bounded multiplicities of zero points there exists a corresponding function  or  providing boundedness of L-index in direction or boundedness  L–index in joint variables respectively. Our result requires uniform boundedness of logarithmic derivative in all variables  and  for belonging the function to class Q^n.  Another result requires uniform boundedness of logarithmic derivative in directions  and  for belonging the function to class Q^n_b  where  is the complex conjugate vector to  b.

scholarly journals Screen transversal cauchy Riemann lightlike submanifolds

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1581-1599
Author(s):  
Burçin Doḡan ◽  
Bayram Şahin ◽  
Erol Yaşar
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
New Class ◽  
Metric Connection ◽  
Cauchy Riemann ◽  
Lightlike Submanifolds ◽  
Totally Real ◽  
Lightlike Submanifold

We introduce a new class of lightlike submanifolds, namely, Screen Transversal Cauchy Riemann (STCR)-lightlike submanifolds, of indefinite K?hler manifolds. We show that this new class is an umbrella of screen transversal lightlike, screen transversal totally real lightlike and CR-lightlike submanifolds. We give a few examples of a STCR lightlike submanifold, investigate the integrability of various distributions, obtain a characterization of such lightlike submanifolds in a complex space form and find new conditions for the induced connection to be a metric connection. Moreover, we investigate the existence of totally umbilical (STCR)-lightlike submanifolds and minimal (STCR)-lightlike submanifolds. The paper also contains several examples.

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