Discrete periodic extension using an approximate step function (Q2875001)

Let \(h>0\) be a given step size for a set of \(N+M\) uniformly spaced nodes NEWLINE\[NEWLINE \{0=x_1,x_2,\dots,x_N=1,x_{N+1},x_{N+2},\dots,x_{N+M}=b-h\}. NEWLINE\]NEWLINE Let \(f\) be a smooth function on \([0,1]\) with samples \({\mathbf f}=(f_j)=(f(x_j))\in \mathbb R^N\). A discrete periodic extension of \({\mathbf f}\) is a vector \(\tilde {\mathbf f}=(\tilde f_j)\in\mathbb R^{N+M}\) with the properties that \(\tilde f_j=f_j\) for \(j=1,2,\dots,N\) and the trigonometric polynomial interpolant \(\mathfrak L_{N+M}^b\tilde f\) of the points \(\{(x_j,\tilde f_j)\}_{j=1}^{N+M}\) provides an accurate approximation of \(f(x)\) in the interval \([0,1]\).NEWLINENEWLINEThe aim of this paper is to present a simplified discrete periodic extension, which may act as a replacement for the more complex FC-Gram algorithm.