Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. T...
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Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called (b,epsilon)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varvec{b},\varepsilon )$$\end{document}-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.
Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the...
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Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.
Starting from a characterization of holomorphic functions in terms of a suitable mean value property, we build some nonlinear asymptotic characterizations for complex-valued solutions of certain nonlinear systems, whi...
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Starting from a characterization of holomorphic functions in terms of a suitable mean value property, we build some nonlinear asymptotic characterizations for complex-valued solutions of certain nonlinear systems, which have to do with the classical Cauchy-Riemann equations. From these asymptotic characterizations, we derive suitable asymptotic mean value properties, which are used to construct appropriate vectorial dynamical programming principles. The aim is to construct approximation schemes for the so-called contact solutions, recently introduced by N. Katzourakis, of the nonlinear systems here considered.
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to m...
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ISBN:
(数字)9783110838350
ISBN:
(纸本)9783110041507
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied *** the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. Titles in planning includeFlavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures, and Applications to Parabolic Problems (2019)Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019)Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational Models for Fractional Calculus, second edition (2020)Mariusz Lemańczyk, Ergodic Theory: Spectral Theory, Joinings, and Their Applications (2020)Marco Abate, holomorphic Dynamics on Hyperbolic Complex Manifolds (2021)Miroslava Antić, Joeri Van der Veken, and Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds of Al
The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable numbe...
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The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived.
Let f be a holomorphic function in the disk U = {z : vertical bar z vertical bar < 1}, vertical bar f(z)vertical bar < 1 in U, let f(+/- 1) = +/- 1 in the sense of angular limits, and let the angular Schwarzian ...
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Let f be a holomorphic function in the disk U = {z : vertical bar z vertical bar < 1}, vertical bar f(z)vertical bar < 1 in U, let f(+/- 1) = +/- 1 in the sense of angular limits, and let the angular Schwarzian derivatives S-f (+/- 1) exist. We establish an upper bound for the sum S-f (-1)+ S-f (1) under the assumption that the image f(U) does not contain open arcs of the pencil of circles arg[(1 + w)/(1 - w)] = theta, -pi/2 < theta < phi, with endpoints w = +/- 1 and Re f '' (1) + f' (1)(1 - f' (1)) = - Re f '' (-1) + f '(-1)(1 - f ' (-1)) = 0. This bound depends on phi and f' (+/- 1) only.
This paper continues investigation of conditions involving values shared by holomorphic functions and their total derivatives which imply the normality for a family of holomorphic functions concerning the total deriva...
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This paper continues investigation of conditions involving values shared by holomorphic functions and their total derivatives which imply the normality for a family of holomorphic functions concerning the total derivatives in C-n. Consequently, we obtain normality criterion of a family F of holomorphic functions f, where each function shares complex values with their linear total differential polynomials L-D(k)(f) in C-n.
Let f be a holomorphic function on the strip {z is an element of C : -alpha 0, belonging to the class H(alpha,-alpha;epsilon) defined below. It is shown that there exist holomorphic functions omega (1) on {z is an el...
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Let f be a holomorphic function on the strip {z is an element of C : -alpha < Im z < alpha}, where alpha > 0, belonging to the class H(alpha,-alpha;epsilon) defined below. It is shown that there exist holomorphic functions omega (1) on {z is an element of C : 0 < Im z < 2 alpha} and omega (2) on {z is an element of C : -2 alpha < Im z < 2 alpha}, such that omega (1) and omega (2) have boundary values of modulus one on the real axis, and satisfy the relations omega (1)(z) = f(z - alphai)omega (2)(z - 2 alphai) and omega (2)(z + 2 alphai) = (f) over bar (z + alphai)omega (1)(z) for 0 < Im z < 2 alpha, where (f) over bar (z) := <(f(<(z)over bar>))over bar>. This leads to a 'polar decomposition' f(z) = u(f)(z + alphai)g(f)(z) of the function f(z), where u(f)(z + alphai) and g(f)(z) are holomorphic functions for -alpha < Im z < alpha, such that \u(f)(x)\ = 1 and g(f)(x) greater than or equal to 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.
We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments ma...
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We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of pi. We also prove that if f and g are functions in the Nevanlinna class, and if vertical bar f vertical bar = vertical bar g vertical bar on the unit circle and on a circle inside the unit disc, then f = g up to the multiplication of a unimodular constant.
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