See the comment from Georg Ströhlein under 2D Spline Interpolation with ALGLIB
Here are the charts referred to:
drgst__aliasing-1deriv-512.png
drgst__aliasing-2deriv-512
drgst__aliasing-spec-512all
drgst__aliasing-spec-512detail
drgst__aliasing-time-512
Newton Excel Bach, not (just) an Excel Blog 
Nice piece of analysis. For the smooth curve perhaps one could include two extra points at (-1,-1/√2) and (9,1/√2) to reproduce the end segments. The smooth curve actually uses a 2d CRspline which reduces to a 1d CRspline when spacing is regular. If spacing differs by a factor of three or more between consecutive/alternate points, a restriction is imposed which prevents the control points overlapping. The xlrotor linked file will not match exactly in this case, refer to other links and comments on this blog for further details.
I’ll defer details of signal processing to others more in the know, but it may be helpful to tie some of these concepts with material from previous posts. The idea in each case is to take a function and write it as a linear combination of a simpler set of functions or known vectors. Below is a minimal attempt to summarise various function types in a table. In each case there is a dual representation consisting of the coefficients and the degree or dimension of the basis function. For Fourier transforms these correspond to amplitudes and frequencies, respectively, which are plotted in the charts.
A few other related examples can be found in these notes.
Type of FitPropertiesBasis TypeOrthogonal Basis
MultivariateLinearVectorQR/SVD Decomposition
PowerAnalyticPolynomialChebyshev/Legendre
SplineSmoothingBezier/HermiteB-Splines
FourierPeriodicTrigonometricComplex Exponentials
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