Entire Functions of Exponential Type in Approximation Theory D
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Complex Analysis (Princeton Lectures in Analysis, Volume
COMPLEX ANALYSIS Ibookroot October 20, 2007 Princeton Lectures in Analysis I Fourier Analysis: An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces Princeton Lectures in Analysis II COMPLEX ANALYSIS Elias M. Stein & Rami Shakarchi PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Control Number 2005274996 ISBN 978-0-691-11385-2 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 5 7 9 10 8 6 To my grandchildren Carolyn, Alison, Jason E.M.S. To my parents Mohamed & Mireille and my brother Karim R.S. Foreword Beginning in the spring of 2000, a series of four one-semester courses were taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. The present series of books is an elaboration of the lectures that were given. While there are a number of excellent texts dealing with individual parts of what we cover, our exposition aims at a different goal: pre- senting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected. -
Duality Results in Banach and Quasi-Banach Spaces Of
DUALITY RESULTS IN BANACH AND QUASI-BANACH SPACES OF HOMOGENEOUS POLYNOMIALS AND APPLICATIONS VIN´ICIUS V. FAVARO´ AND DANIEL PELLEGRINO Abstract. Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasi- norms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided. Contents 1. Introduction 1 2. Main results 2 3. Prerequisites for the applications 9 3.1. Hypercyclicity results 11 3.2. Existence and approximation results 12 4. Applications 12 4.1. Lorentz nuclear and summing polynomials: the basics 12 4.2. New hypercyclic, existence and approximation results 14 References 22 Mathematics Subject Classifications (2010): 46A20, 46G20, 46A16, 46G25, 47A16. Key words: Banach and quasi-Banach spaces, homogeneous polynomials, entire functions, convolution operators, Lorentz sequence spaces. 1. Introduction arXiv:1503.01079v2 [math.FA] 28 Jul 2016 In the 1960s, many researchers began the study of spaces of holomorphic functions defined on infinite dimensional complex Banach spaces. In this context spaces of n-homogeneous polynomials play a central role in the development of the theory. Several tools, such as topological tensor products and duality theory, are useful and important when we are working with spaces of homogeneous polynomials. Many duality results on spaces of homogeneous polynomials and their applications have appeared in the last decades (see for instance [5, 8, 9, 10, 11, 12, 15, 20, 22, 26, 28, 29, 33, 34, 41, 46] among others). -
On Entire Functions of Exponential Type and Indicators of Analytic Functionals
ON ENTIRE FUNCTIONS OF EXPONENTIAL TYPE AND INDICATORS OF ANALYTIC FUNCTIONALS BY C. O. KISELMAN The University of Stockholm, Stockholm, Sweden and The Institute for Advanced Study, Princeton, N.J., U.S.A. x O. Summary We shall be concerned with the indicator p of an analytic functional # on a complex manifold U: p(v) = li--m 1 log I/~(et~)l, where V is an arbitrary analytic function on U. More specifically, we shall consider the smallest upper semicontinuous majorant p~ of the restriction of p to a subspace ~ of the analytic functions. An obvious problem is then to characterize the set of functions p~ which can occur as regularizations of indicators. In the case when U =(~a and :~ is the space of all linear functions on 0 a, this set can be described more easily as the set of functions lira lira 1 log [u(tO)l (0.1) 0---'-~ t--~+r162t of n complex variables ~ 6C a where u is an entire function of exponential type in Ca. We shall prove that a function in Ca is of the form (0.1) for some entire function u of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projective n-space which are Stein manifolds. Our objective in Sections 4 and 5 is to study the relation between properties ofp ~ and existence and uniqueness of :~-supports of #, i.e. carriers of/~ which are convex with respect to :~ in a certain sense and which are minimal with this property (see Section 1 for defiai- I This work was supported by the Swedish Natural Science Research Council and Air Force Office of Scientific Research grant AF-AFOSR-359-64. -
Laplace Transform 1 Laplace Transform
Laplace transform 1 Laplace transform The Laplace transform is a widely used integral transform with many applications in physics and engineering. Denoted , it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s).[1] It is named after Pierre-Simon Laplace, who introduced the transform in his work on probability theory. The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In such analyses, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. -
One-Sided Band-Limited Approximations of Some Radial
ONE-SIDED BAND-LIMITED APPROXIMATIONS OF SOME RADIAL FUNCTIONS FELIPE GONC¸ALVES, MICHAEL KELLY AND JOSE MADRID Abstract. We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of [12] to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given. 1. Introduction In this paper we address a class of problems that have come to be known as Beurling-Selberg extremal problems. The most well-known example of such a problem is due to Selberg himself [39, 45, 51]. Given an interval I ⊂ R and δ > 0, Selberg constructed an integrable function M(x) that satisfies (i) Mˆ (ξ) = 0 if |ξ| >δ where Mˆ (ξ) is the Fourier transform (see §2) of M(x), (ii) M(x) ≥ χI (x) for each x ∈ R, and (iii) M(x) has the smallest integral1 among all functions satisfying (i) and (ii). The key constraint (which is common in Beurling-Selberg problems) is condition (ii), that M(x) majorizes the characteristic function of I. Such problems are sometimes called one-sided approximation problems. Most work on these sorts of problems have been focused on solving Selberg’s problem but with χI (x) replaced by a different single variable function, such as e−λ|x|. -
About the Growth of Entire Functions Solutions of Linear Algebraic Q
ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE JEAN-PIERRE RAMIS About the growth of entire functions solutions of linear algebraic q-difference equations Annales de la faculté des sciences de Toulouse 6e série, tome 1, no 1 (1992), p. 53-94 <http://www.numdam.org/item?id=AFST_1992_6_1_1_53_0> © Université Paul Sabatier, 1992, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitu- tive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Annales de la Faculté des Sciences de Toulouse Serie 6, Vol. I, nO 1, 1992 About the growth of entire functions solutions of linear algebraic q-difference equations JEAN-PIERRE RAMIS(1) Qu’un cui de dame damascene. GUILLAUME APOLLINAIRE, La chanson du mal-aimé. RESUME. - Nous prouvons un q-analogue d’un résultat de Valiron . entière solution d’une différentielle [VI,, 2]. Une fonction f, equation lineaire algébrique D f = b, a une croissance exponentielle d’ordre fini k > 0 et de type fini. De plus les valeurs optimales de k sont des nombres rationnels appartenant a un ensemble fini explicitement calculable a partir du polygone de Newton de D. Ce result at a d’abord été prouve par Valiron (1926). Notre principal résultat est un q-analogue (q étant un nombre complexe, avec |q| > 1) :: une fonction entière f, solution d’une equation aux q- différences lineaire algébrique S f = b, a une croissance q-exponentielle d’ordre fini k > 0 et de type fini.