Rate of convergence in the Fourier inversion of indicator functions

Article type: Research Article

Authors: González Vieli, Francisco Javier

Affiliations: SB/IACS/CAHRU, École Polytechnique Fédérale de Lausanne, CH‐1015 Lausanne, Switzerland E‐mail: francisco.gonzalez@epfl.ch

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.

Keywords: Fourier inversion, indicator function, rate of convergence, order of contact

Journal: Asymptotic Analysis, vol. 41, no. 2, pp. 179-187, 2005