Rate of convergence in the Fourier inversion of indicator functions

Abstract

The rate of convergence, at a point x, of the Fourier integral of the indicator function of a regular plane domain U has been investigated by Popov [Russian Math. Surveys 52 (1997), 73–145] and by Pinsky and Taylor [J. Fourier Anal. Appl. 3 (1997), 647–703] for x not on the boundary $\curpartial U$ of U; they have shown that the larger the maximal order of contact of $\curpartial U$ with the circles centered at x, the slower the convergence. We show here that Popov's approach, which uses the method of stationary phase, can be extended to the case x on the boundary $\curpartial U$ , giving exactly the same relation between rate of convergence and maximal order of contact.