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For a parameter dimension d∈ N, we consider the approximation of many-parametric maps u:[-1,1]d→R by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse Eρ1×⋯×Eρd⊂Cd of semiaxis sums ρi> 1 containing [-1,1]d. We establish the exponential rate O(exp(-bN1/(d+1))) of expressive power in terms of the total NN size N and of the input dimension d of the ReLU NN in W1,∞([-1,1]d). The constant b> 0 depends on (ρj)j=1d which characterizes the coordinate-wise sizes of the Bernstein-ellipses for u. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called “rectified power unit” activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to d-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals., Constructive Approximation, 55 (1), ISSN:0176-4276, ISSN:1432-0940

It is well known that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. If $$X({\varOmega })$$ is a space of holomorphic functions we extend these notions to $$X({\varOmega })$$ -domain of holomorphy and weak $$X({\varOmega })$$ -domain of holomorphy. For several function spaces $$X(\varOmega )$$ , satisfying weak assumptions, we prove that the notions of $$X(\varOmega )$$ -domain of holomorphy and weak $$X(\varOmega )$$ -domain of holomorphy are equivalent and that in this case the set of non-extendable functions in $$X({\varOmega })$$ is a dense $$G_\delta $$ -subset of $$X({\varOmega })$$ . Similar results are obtained for the stronger notion of total unboundedness. Finally we provide examples of new spaces $$X({\varOmega })$$ , where all the above hold. Mainly they are localized versions of classical function spaces and combinations of them.

If $\Omega$ is a domain of holomorphy in $\Bbb C^n$, having a compact topological closure into another domain of holomorphy $U\subset \Bbb C^n$ such that $(\Omega,U)$ is a Runge pair, we construct a function $F$ holomorphic in $\Omega$ which is singular at every boundary point of $\Omega$ and such that $F$ is in $L^p(\Omega)$, for any $p\in (0,+\infty)$.

A fundamental objective in the study of analytic functions of a complex variable is the following: Suppose we know that a given function is holomorphic (i.e., satisfies the Cauchy–Riemann conditions) in some regions of the complex plane. How can we extend this domain of holomorphy, i.e., find an analytic continuation of the function to regions outside the original one?

Let A be a C*-algebra with identity and real rank zero. Suppose a complex-valued function is holomorphic and bounded on the intersection of the open unit ball of A and the identity component of the set of invertible elements of A . We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of A . The author previously deduced this from a more general fact about Banach algebras.

In this paper, we apply Meyer’s holomorphic wavelets to study holomorphic function spaces systematically. The holomorphic function spaces under our setting include the classical function spaces as well as extensions of the latter. Holomorphic wavelets pay a role to connect real analysis and complex analysis.

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold is the pullback of a unique such geometry on the envelope of holomorphy of the domain. We use this result to classify the Hopf manifolds which admit holomorphic reductive geometries, and to classify the Hopf manifolds which admit holomorphic parabolic geometries. Every Hopf manifold which admits a holomorphic parabolic geometry with a given model admits a flat one. We classify flat holomorphic parabolic geometries on Hopf manifolds. For every generalized flag manifold there is a Hopf manifold with a flat holomorphic parabolic geometry modelled on that generalized flag manifold.

We investigate analyticity properties of correlation functions in conformal field theories (CFT) in the Wightman formulation. The goal is to determine domain of holomorphy of permuted Wightman functions. We focus on crossing property of three-point functions. The domain of holomorphy of a pair of three-point functions is determined by appealing to Jost's theorem and by adopting the technique of analytic completion. This program paves the way to address the issue of crossing for the four-point functions on a rigorous footing., 13 pages

We prove that, if $D $ is a normal open subset of a Stein space $X$ of pure dimension such that $D$ is locally Stein at every point of $\partial D \setminus X_{\operatorname{sg}}$, then, for every holomorphic vector bundle $E$ over $D$ and every discrete subset $\Lambda $ of $D \setminus X_{\operatorname{sg}}$ whose set of accumulation points lies in $\partial D \setminus X_{\operatorname{sg}}$, there is a holomorphic section of $E$ over $D$ with prescribed values on $\Lambda$. We apply this to the local Steinness problem and domains of holomorphy.

A well-known result due to Polya for a function given by its holomorphic germ at is extended to the case of a piecewise holomorphic function on an arbitrary compact set in . This result is applied to the problem of the existence of compact sets that have the minimum transfinite diameter in the external field of the logarithmic potential of a negative unit charge among all compact sets such that a certain multivalued analytic function is single-valued and piecewise holomorphic on their complement. Bibliography: 13 titles.

International audience; We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the transformation that maps this Hamil-tonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$ with $$n\ge 3$$ near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

For a tuple A = (A1,A2, …,An) of elements in a unital algebra B over ℂ, its projective spectrum P(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn−1, such that A(z) = z1A1+z2A2+…+znAn is not invertible in B. The first half of this paper proves that if B is Banach then the resolvent set Pc(A) consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.

In this paper we consider continuous functions given on the boundary of a ball B in , , and having a one-dimensional property of holomorphic extension along the families of complex lines, passing through a finite number of points of B. We study the problem of existence of holomorphic extension of such functions in the ball B.

In this paper, we present some inequalities for the non-tan-gential derivative of f(z). For the function f(z) = z + b p+1 z p+1 +b p+2 z p+2 + ··· defined in the unit disc, with ℜ f ′ (z)λf ′ (z)+1−λ > β,0 ≤ β β, 0 ≤ β < 1, 0 ≤ λ < 1.Consider the functiong(z) =

In this paper, we give a brief introduction of several complex variables around analytic continuation, including some basic ideas, methods and recent progress, emphasizing the relation with one complex variable.

Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

Extension of the classical Mandelbrojt’s criterion for normality of a family of zero-free holomorphic functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form is uniformly bounded away from zero is normal.

A boundary analog of the Forelli theorem for real-analytic functions is established, i.e., it is demonstrated that each real-analytic function f defined on the boundary of a bounded strictly convex domain D in the multidimensional complex space with the one-dimensional holomorphic extension property along families of complex lines passing through a boundary point and intersecting D admits a holomorphic extension to D as a function of many complex variables.

In this paper, we study the local structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded or convex) domain in a complex Banach space, using ergodic theory of linear operators and the nonlinear numerical range introduced by L. A. Harris. We provide several constructions of holomorphic retractions and a generalization of Cartan’s Uniqueness Theorem. We also estimate the deviation of a holomorphic mapping from its linear approximation, the Frechet derivative at a fixed point.

We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\). We show that the set of non-extendable functions not only contains a \(G_{\delta }\)-dense subset of \(X({\varOmega })\), but it is itself a \(G_{\delta }\)-dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\)-domain of holomorphy but not a \(A({\varOmega })\)-domain of holomorphy.

We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space of holomorphic functions whose derivatives up to some order extend continuously to the closure of the domain endowed with its natural topology. Generically, every function in this space is proven to be non - extendable, although bounded.

The concept of “domain of holomorphy” is central to the function theory of several complex variables. The celebrated solution of the Levi problem tells us that a connected open set (a domain) is a domain of holomorphy if and only if it is pseudoconvex. For us, in the present book, pseudoconvexity is Levi pseudoconvexity; this is defined in terms of the positive semi-definiteness of the Levi form.

Extension of classical Mandelbrojt’s criterion for normality to several complex variables is given. Some inequalities for holomorphic functions which omit values 0 and 1 are obtained.

In this paper, we focus on relations between holomorphic 1-forms and holomorphic 1-cochains on a closed Riemann surface. Holomorphic 1-cochains are defined by Wilson in 2008 using a combinatorial Hodge theory. A holomorphic 1-form is characterized by its periods. So is a holomorphic 1-cochain. We consider relations between a holomorphic 1-form and a holomorphic 1-cochain which have the same periods and show that holomorphic 1-cochains provide an approximation of holomorphic 1-forms.

In this paper, we completely determine the structure of the holomorphic automorphism group of a generalized Hartogs triangle and obtain natural generalizations of some results due to Landucci and Chen-Xu. These give affirmative answers to some open problems posed by Jarnicki and Pflug.

We propose a mathematical limit of L1-stable weak asymptotic methods. A family of L1-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L2-bounds in a domain of holomorphy in ℂn are obtained with the aid of L. Hormander’s method of L2-bounds for the \(\bar \partial\) operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not necessarily hold for unbounded sets, answering a question posed by Hirschowitz in 1972. We also consider extensions to open subsets of the bidual, present some Banach-Stone type results and show some properties of the spectrum when the domain is the unit ball of $\ell_p$., 25 pages

Let D be an arbitrary domain in , n > 1, and be an open piece of the boundary. Suppose that M is connected and is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood of . Let f : be a holomorphic correspondence such that the cluster set (M) is contained in a smooth closed real-algebraic hypersurface M' in of finite type. It is shown that if f extends continuously to some open peace of M, then f extends as a holomorphic correspondence across M. As an application, we prove that any proper holomorphic correspondence from a bounded domain D in with smooth real-analytic boundary onto a bounded domain D' in with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood of .

Pade-Chebyshev approximants are considered for multivalued analytic functions that are real-valued on the unit interval . The focus is mainly on non-linear Pade-Chebyshev approximants. For such rational approximations an analogue is found of Stahl's theorem on convergence in capacity of the Pade approximants in the maximal domain of holomorphy of the given function. The rate of convergence is characterized in terms of the stationary compact set for the mixed equilibrium problem of Green-logarithmic potentials. Bibliography: 79 titles.

It is proven that the Taylor series of functions holomorphic in C ∞ ∖ { 1 } generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown that such sets necessarily are polar sets.