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I usually use cubic spline or chebyshev interpolation since both are generally pretty good. For a certain problem, I can't use chebyshev nodes, and my sample points are neither spaced equally. This is no problem for spline interpolation, but I have difficulties with the boundaries of my domain. One of the boundaries is extremely important for my problem, and I need interpolation to be pretty good there. I tried to use clamped splines where I give the first derivative, but it seems like it's not precise enough; it probably can't be approximated by a linear curve. There's also the problem that the derivative on the boundary is hard to get. I use central difference, but extrapolate using a fourth order polynomial for the sample point outside the domain. It's not perfect, but it doesn't work bad. Unfortunately, depending on the parameters of my problem, mostly sample s pacing, it can be quite bad. Does anyone have any idea on a better interpolation scheme near the boundaries? One which would preferably not suffer from the Runge effect. Asking for higher derivatives would be problematic, since they are hard to get near the boundaries. Unless someone has a better way to get them than me.

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  • $\begingroup$ What are the sample points near the important boundary? And how come you know your approximation is bad? Do you know the exact solution? $\endgroup$ – Oppenede Jul 9 '18 at 7:10

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