A bandlimited step function for use in discrete periodic extension

Abstract

A new methodology is introduced for use in discrete periodic extension of non-periodic functions. The methodology is based on a band-limited step function, and utilizes the computational efficiency of FC-Gram (Fourier Continuation based on orthonormal Gram polynomial basis on the extension stage) extension database. The discrete periodic extension is a technique for augmenting a set of uniformly-spaced samples of a smooth function with auxiliary values in an extension region. If a suitable extension is constructed, the interpolating trigonometric polynomial found via an FFT(Fast Fourier Transform) will accurately approximate the original function in its original interval. The discrete periodic extension is a key construction in the FC-Gram algorithm which is successfully implemented in several recent efficient and high-order PDEs solvers. This thesis focuses on a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler implementation and significantly decreased setup time.