Integration and Preparation Theorems

这个项目包括两条调查路线。课题的第一部分研究了实数可构造函数的积分理论，实数可构造函数的定义是实数全局子解析函数及其对数积的和。可构造函数的意义在于它们构成了扩展全局子解析函数的最小函数类，并且在积分下是稳定的。勒贝格空间的性质和多元调和分析将在可构造函数的背景下进行研究，包括振荡积分的渐近估计和可构造函数的傅里叶变换的可积性问题。研究可构造函数积分的主要工具是子解析准备定理。项目的第二部分旨在通过一个纯解析证明来证明子解析准备定理在更一般的拟解析设置中成立。这样做的一个重要动机是为了获得一个更有信息量的准备定理的证明，该准备定理可用于研究实域由拟解析类函数和幂函数展开的可判决性，以便将主要研究者以前关于只处理拟解析类函数的可判决性的工作与Jones和Servi关于只处理幂函数的可判决性的工作结合起来。这个项目的起源源于两个非常经典的问题，这两个问题在数学及其在科学和工程中的应用中普遍存在：1)如何解决方程和不等式，以及与此相关的，如何确定通过使用方程和不等式以及逻辑运算建立的陈述的真假；2)如何计算和研究由积分公式定义的函数的性质。子解析准备定理表明，至少在理论上，一大类方程（包括所有多项式方程）可以用根在一个更自由的意义上求解，除了算术运算和根之外，还允许使用局部定义的解析函数。本项目的目的之一是获得准备定理的一个新的算法证明，顺便提一下，这也将定理推广到更广泛的函数类。除研究方程外，准备定理是研究可构造函数的渐近行为和积分的重要工具。可构造函数是一类包含，特别是所有代数函数的函数。傅里叶变换在数学及其应用的许多领域中都是一个重要的运算，它由一个积分公式定义。该项目的另一个目的是为积分理论奠定基础，该理论可以用来证明可构造函数的傅里叶变换具有简单的性质。

This project consists of two lines of investigation. The first part of the project studies the integration theory of real constructible functions, which by definition are sums of products of real globally subanalytic functions and their logarithms. The significance of the constructible functions is that they form the smallest class of functions that extends the globally subanalytic functions and is stable under integration. Properties of Lebesgue spaces and also multivariate harmonic analysis will be studied in the context of constructible functions, including asymptotic estimates of oscillatory integrals and possibly questions concerning the integrability of Fourier transforms of constructible functions. The main tool employed to study integration of constructible functions is the subanalytic preparation theorem. The second part of the project aims to show through a purely analytic proof that the subanalytic preparation theorem holds in a more general quasianalytic setting. An important motivation for doing so is to obtain a more informative proof of the preparation theorem that could be used to study the decidability of expansions of the real field by functions from quasianalytic classes and power functions in order to combine the principal investigator's previous work on decidability, which dealt only with functions from quasianalytic classes, and the work of Jones and Servi on decidability, which dealt only with power functions.The origins of this project stem from two very classical questions which are pervasive throughout much of mathematics and its applications to science and engineering: 1) how to solve equations and inequalities, and related to this, how to determine the truth or falsity of statements built up through the use of equations and inequalities and also logical operations; 2) how to compute and study properties of functions defined by integral formulas. The subanalytic preparation theorem shows that, at least in theory, a wide class of equations (which includes all polynomial equations) can be solved by radicals in a more liberal sense that allows the use of locally defined analytic functions in addition to arithmetic operations and radicals. One aim of this project is to obtain a new algorithmic proof of the preparation theorem which, incidentally, would also generalize the theorem to wider classes of functions. In addition to studying equations, the preparation theorem is an important tool for studying the asymptotic behavior and integrals of constructible functions, which is a class of functions that contains, in particular, all algebraic functions. The Fourier transform is an important operation used in many areas of mathematics and its applications, and it is defined by an integral formula. Another aim of the project is to the lay groundwork for a theory of integration that could be used to show that Fourier transforms of constructible functions have simple properties.