### Discrete Periodic Extension Using an Approximate Step Function

Nathan Albin, Sureka Pathmanathan
2014 SIAM Journal on Scientific Computing
The discrete periodic extension is a technique for augmenting a given 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 will accurately approximate the original function in its original interval of definition. The discrete periodic extension is a key construction in the algorithm FC-Gram (Fourier continuation based on Gram polynomials) algorithm.
more » ... e FC-Gram algorithm, in turn, lies at the heart of several recent efficient and highorder-accurate PDE solvers. This paper presents a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler constructions and significantly decreased setup time. 1. Introduction. The purpose of this paper is to offer a straightforward approach for addressing the approximation problem illustrated in Figure 1 . In the figure, the solid curve represents a smooth function f (x) on the interval [0, 1], and the solid circular dots represent N samples of f at uniformly spaced nodes in this interval. The problem addressed in this paper is summarized as follows: Main problem: By using only the first d and last d data points, how does one produce an additional M function values in an interval [1, b] with the property that the trigonometric polynomial interpolant of all N + M samples on [0, b] provides a highly accurate approximation of f (x) in the interval [0, 1]? In the figure, the trigonometric polynomial interpolant is represented by the union of the solid and dashed curves. For illustration purposes, the sizes of N , M , and d in the schematic are smaller than the values for these numbers used in practice. In the numerical comparisons presented in section 5, for example, M = 25, d = 10, and N ≥ 100. The algorithm presented can be viewed conceptually as a method for efficiently producing, for any choice of N ≥ d, a sparse (N +M )×N matrix E per which maps the N samples of the original function to the N + M samples of its periodic extension. The sparseness of the matrix operator derives from the fact that only 2d samples are used to produce the extension, regardless the size of N . As will be shown (see (2.8)), E per has a very simple structure with only N + 2d 2 nonzero entries. Such a matrix, E per , can be quite useful in practice, as it allows the high-precision approximation of a nonperiodic function f and its derivatives by means of FFT-based interpolation of the extended vector. Moreover, due to the sparsity of E per and the efficiency of the FFT, the resulting algorithm will have O(N log N ) complexity. The *