Mathematical Sciences: Investigations in Analysis

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The study of functions defined on real n-space (i.e. the situation in which a quantity depends jointly on some number n of variable quantities) has been one of the basic endeavors of mathematics for centuries, with frequent feed- back from applications. For example, suppose we have an inhomogeneous body situated in 3-space; density within the body may be considered as a function of position. In practice, e.g. in computed tomography, the density function is unknown and must be explored indirectly. The mathematical procedure for doing this is to "transform" the unknown function into another function that one can more easily see. (A CAT scanner is an expensive and elaborate implementation of a particular sort of transform.) Mathematical investigation of a given transform typically addresses questions such as the following. How much information is lost in forming the new function from the old? What kind (from very smooth to extremely chaotic) of new function does one obtain from what kind of old function? If you change the old function just a little, does the new, transformed function also change just a little, or drastically? (In mathematical language, is the transform continuous, or not?) Professor Oberlin's project has mostly to do with the question of continuity for a family of transforms indexed by real numbers between 1 and infinity. At the two extremes, one has, respectively, the Riesz potential and the spherical maximal operator, both of whose behavior is well-understood. The problem is, what happens in between? This is a hard question. One can get a feel for the situation by means of computer experiments, but these are necessarily inconclusive. Intricate and clever arguments by the mathematician himself are required to settle matters finally.