Two algorithms for periodic extension on uniform grids

On the adaptive spectral approximation of functions using redundant sets and frames

“…The weighted least squares problem yields a vector with small norm y , hence c = Dy may be expected to exhibit c k ∼ k −α decay. This approach is similar to the method of [30] and to the first algorithm in [25]. One difference is that, in [25], the ill-conditioned system is solved using iterative refinement.…”

“…This approach is similar to the method of [30] and to the first algorithm in [25]. One difference is that, in [25], the ill-conditioned system is solved using iterative refinement. A disadvantage of this weighted scheme is that one has to decide a priori on the decay rate α, which assumes knowledge of the smoothness properties of f .…”

“…Say the optimal value in an approximation problem lies between N l = 16 and N h = 32, and say, N opt = 18. If the bisection algorithm notices that convergence criteria are not satisfied for N = (N l + N h )/2 = 24, then it will continue to look for an optimal value in the interval [25,32]. Whatever it returns will be larger than 18.…”

“…[30] proposes the solution of a weighted least squares problem for univariate Fourier extension, and adapts the fast method of [29] for an efficient implementation. Two algorithms were proposed by [25], again for Fourier extension, that achieve coefficient decay. The first algorithm uses weighted least squares, similar to [30].…”

“…The first algorithm uses weighted least squares, similar to [30]. It is stated that in case a good Fourier extension exists, i.e., an extension with rapidly decaying Fourier coefficients, then it will be found numerically [25,Lemma 2.3]. The second method is based on Hermite interpolation (interpolation of function values and derivatives at the endpoints).…”