Zero Sets of Functions From Non-Quasi-Analytic Classes

It is well known that any closed subset of the line is the zero set of a <!-- MATH ${C^\infty }$ --> -function. One can also specify the orders of the zeros at the isolated points. The present paper improves this result by replacing the class of <!-- MATH ${C^\infty }$ --> -functions by any non-quasi-analytic class of <!-- MATH ${C^\infty }$ --> -functions.