Descriptive Set Theory and Harmonic Analysis
نویسندگان
چکیده
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. authors gave a series of eight lectures (short course) on the topic of the title. This survey article consists basically of the lecture notes for that course distributed to the participants of that conference. We have purposely tried in this printed version to preserve the informal style of the original notes. Let us say first a few things about the content of these lectures. Our aim has been to present some recent work in descriptive set theory and its applications to an area of harmonic analysis. Typical uses of descriptive set theory in analysis are most often through regularity properties of definable sets, like measurability, the property of Baire, capacitability, etc., which are used to show that certain problems have solutions that behave nicely. In the theory we will present, definability itself, in fact the precise analysis of the "definable complexity" of certain sets, will be the main concern. It will be through such knowledge that we will be able to infer important structural properties of various objects which will then be used to solve analysis problems. The first lecture provides a short historical introduction to the subject of uniqueness for trigonometric series, which is the area of harmonic analysis whose problems are the origin of this work. As is well known, it was Cantor who proved the first major result in this subject in 1870, and it was his subsequent work here that led him to the creation of set theory. The next three lectures describe the recently developed definability theory of a-ideals of closed sets, which is the main tool through which descriptive set theory is applied to the analysis problems we are interested in. Proofs or sketches of proofs are given here for most of the main facts, especially those that are used later on. In the last four lectures, we first present an outline of the analytical theory of uniqueness of trigonometric series that is needed here. In order to give a bit of the flavor of …
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ورودعنوان ژورنال:
- J. Symb. Log.
دوره 57 شماره
صفحات -
تاریخ انتشار 1992