I am hoping that someone could explain to be a concept of trigonometric interpolation and why the it looks as follows for $\ln(x+1)$. I am not sure of the understanding as to why there is such imperfection when plotted over the interval $[0,2\pi]$. The first result is as follows
Essentially, I used a sample of 16 abscissae values for the function mentioned above. Even more puzzling to me is the second case when there is a change of conditions to the following, $h(t) = f(t) 0\le t < 2\pi$ and $h(t) = f(4\pi - t), 2\pi \le t < 4\pi$, i get a much better results with a much lower error as follows.
Thus, I'm not sure how a slight change relates to such a significant improvement in the end results and why there was a large fall off in my DFT interpolant for the first 2 cases. Could someone please explain ? I would appreciate any advice, thank you.
In the first picture, the periodic function you approximate has a jump at the end of the interval. Look for Gibbs phenomenon to get a name for the oscillations of the Fourier sums at a jump.
In the second, the first half of the approximation seems suspect. The Fourier sums should be as symmetric as the function is. Since the function is continuous and has only two kinks, convergence should be uniform.