Approximation Theory XIV: San Antonio 2013 by Gregory E. Fasshauer & Larry L. Schumaker

Author:Gregory E. Fasshauer & Larry L. Schumaker
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(ii)If is a regular point, corresponds to the normal direction of at and , then

(iii)If is a regular point, corresponds to the normal direction of at and , then

That is, if , the continuous shearlet transform decays rapidly, asymptotically for , unless corresponds to the normal direction of at , in which case

Theorem 1 generalizes to the case of functions of the form where and the boundary curve contains corner points. In this case, if is a corner point and corresponds to one of the normal directions of at , then the continuous shearlet transform has a decay rate of order , as similar to the situation of regular points. For other values of , however, the asymptotic decay rate depends both on the tangent and the curvature at (cf. [10]).

Theorem 1 was originally proved in [10] for the case and its proof was successively simplified and streamlined in [13]. In the following section, we sketch the main ideas of the proof, highlighting how to extend the proof from [13] to the case .

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