Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The appr...
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Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely-many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, [5] we studied the approximation of so-called Banach-valued, (b, epsilon)-holomorphic functions on the infinite-dimensional hypercube [-1, 1]N from m (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive m-widths for classes of such functions, which showed that certain algebraic rates of the form m1/2-1/pare the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from the underlying probability measure for the problem. Specifically, for Hilbert-valued (b, epsilon)-holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for tensor-product Jacobi measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best s-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d. pointwise samples drawn from an underlying probability measure are near-optimal for the recovery of infinite-dimensional, holomorphic functions. (c) 2025 Published by Elsevier Inc.
In this work, we analyze the weighted composition semigroup on the space of holomorphic functions by providing cocycles as its weights. We establish some conditions under which it is analytically extended from the pos...
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In this work, we analyze the weighted composition semigroup on the space of holomorphic functions by providing cocycles as its weights. We establish some conditions under which it is analytically extended from the positive real line to a sector.
We establish Liouville-type results on complete Kahler manifolds admitting a real holomorphic gradient vector field. As an application, we conclude that a bounded holomorphic function on a shrinking gradient Kahler-Ri...
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We establish Liouville-type results on complete Kahler manifolds admitting a real holomorphic gradient vector field. As an application, we conclude that a bounded holomorphic function on a shrinking gradient Kahler-Ricci soliton must be constant, and more generally, the dimension of the space of holomorphic functions with fixed polynomial growth order is finite and bounded by the growth order.
Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H-bU(E). We prove that, under very natural conditions satisfied by many usual classe...
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Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H-bU(E). We prove that, under very natural conditions satisfied by many usual classes of polynomials, the spectrum M-bU(E) of this algebra "behaves" like the classical case of M-b(E) (the spectrum of H-b(E), the algebra of bounded type holomorphic functions). More precisely, we prove that M-bU(E) can be endowed with a structure of Riemann domain over E" and that the extension of each f is an element of H-bU(E) to the spectrum is an U-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.
Abstract: Let $P$ be a $p \times q$ matrix of polynomials in $n$ complex variables. If $\Omega$ is a domain of holomorphy in ${{\mathbf {C}}^n}$ and $u$ is a $q$-tuple of holomorphic functions we show that the...
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Abstract: Let $P$ be a $p \times q$ matrix of polynomials in $n$ complex variables. If $\Omega$ is a domain of holomorphy in ${{\mathbf {C}}^n}$ and $u$ is a $q$-tuple of holomorphic functions we show that the equation $Pv = Pu$ has a solution $v$ which is a holomorphic $q$-tuple in $\Omega$ and which satisfies an ${L^2}$ estimate in terms of $Pu$. Similar results have been obtained by Y.-T. Siu and R. Narasimhan for bounded domains and by L. Höormander for the case $\Omega = {{\mathbf {C}}^n}$.
In this paper we investigated some properties of holomorphic functions (belonging to the kernel of the Dirac operator) defined on domains of the real Cayley-Dickson algebras. For this purpose, we study first some prop...
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In this paper we investigated some properties of holomorphic functions (belonging to the kernel of the Dirac operator) defined on domains of the real Cayley-Dickson algebras. For this purpose, we study first some properties of these algebras, especially multiplication tables for certain elements of the basis. Using these properties, we provided an algorithm for constructing examples of the class of functions under consideration.
Let X be a projective manifold, rho : (X) over bar -> X its universal covering and rho* : Vect(X) -> Vect((X) over tilde) the pullback map for the isomorphism classes of vector bundles. This article establishes ...
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Let X be a projective manifold, rho : (X) over bar -> X its universal covering and rho* : Vect(X) -> Vect((X) over tilde) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map rho* and the properties of the function theory on (X) over tilde. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map rho* is almost an imbedding.
We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and eve...
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We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly c-algebrable for all separable Banach spaces. For specific spaces including lp or duals of Lorentz sequence spaces, we have strongly c-algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.
We discuss the implication \f\ is an element of A(omega)(G) double right arrow f is an element of Lambda(omega)(G), where f is a holomorphic function (resp., a quasiconformal mapping) on a domain Gsubset ofC(n) (resp....
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We discuss the implication \f\ is an element of A(omega)(G) double right arrow f is an element of Lambda(omega)(G), where f is a holomorphic function (resp., a quasiconformal mapping) on a domain Gsubset ofC(n) (resp., Gsubset ofR(n)) and Lambda(omega)(G) is the Lipschitz space associated with a majorant omega. (C) 2003 Elsevier Inc. All rights reserved.
We suggest a generalization of Pontryagin duality from the category of commutative, complex Lie groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In c...
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