Interpolation of Smooth Functions and the Ulam-Zahorski Problem
by
Jack B. Brown
Auburn University

The " Ulam-Zahorski Problem" is a problem about interpolation between functions f : R ® R from the following classes:

A ® Coo ® ... ® Cn ® Dn ® ... ® C1 ® D1 ® C,

where "®" means " subset of " or " implies ", A = { real analytic functions }, C^oo = { infinitely differentiable functions }, Cⁿ = { n-times continuously differentiable functions }, Dⁿ = { n-times differentiable functions }, and C = { continuous functions }.

The orginal problem stated by Ulam in the early 1940's [Scottish Book Problem 17.1] asked whether every continuous function agrees with some real analytic function on some uncountable set. Zahorski showed in 1947 that the answer is "no" and he raised a number of related problems. The problem evolved into the question of whether a function in one of the function classes indicated above necessarily agrees on an uncountable set with some function in one of the smaller classes and, in particular, to determine exactly what is the smallest such class for which this is true. The speaker will discuss the theorems and examples described by many authors which have almost provided the "complete" solution to this general problem. However, it will be seen during the lecture that there are still some open problems to be solved to fill the last gaps in the complete solution.

Some of the theorems which were proved in the above process suggested that certain "differentiability variants" of Blumberg's Restriction Theorem (about continuous restrictions of arbitrary functions to dense subsets of R) should hold. Some of these variants will be discussed, including a C¹-Interploation variant of Blumberg's Theorem recently proved by the speaker. The question of whether a C^oo-Restriction variant of Blumberg's Theorem holds is still open.

Finally, the Ulam-Zahorski Problem as it applies to multivariate functions will be discussed.

Date received: March 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagj-08.