Integration and Preparation Theorems

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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.