Now I have a target smooth function f which is infinitely differentiable over $R^d$, $f \in C^{\inf}(R^d)$. $f = \Sigma c_ig_i(x)$, where $c_i$s are unknown coefficients and $g_i(x)$ is a smooth function defined on the distance from $x$ to some point $x_i \in R^d$. For instance, $g_1(x)=exp(-(x-x_1)^2)$ in 1D and $g_1(x)=exp(-||x-x_1||_2^2)$ in 2D. I am supposed to approximate $f$ as more accurately as possible. Furthermore, I only care about some prescribed point like $x_1$, so as long as the approximation is best around $x_1$ locally, the result is optimal. This means I only need to obtain a function $f_{appr}$ such that $||f_{appr}-f||$ is small around $x_1$.

I have been trying to use Lagrange interpolation with least square solution, which employs polynomails.

Is it possible to make use of Fourier analysis or regression method? I am not good at sine and cosine function theory, nor inverse problem area. So any comments?

Attached are two cases which show the exact function $f$ values in 1D.

enter image description here

  • $\begingroup$ Lanczos Resampling $\endgroup$ – Felix Marin Feb 10 '14 at 22:05
  • $\begingroup$ @FelixMarin That is inspiring. Thanks! I just added two pictures to illustrate my function. The function is always localized. The max is over 1, is that okay? BTW, is there any Matlab function for Lanczos Resampling? $\endgroup$ – Appalachian Math Feb 10 '14 at 22:17
  • $\begingroup$ I don't use MatLab. Sorry. $\endgroup$ – Felix Marin Feb 10 '14 at 23:31
  • $\begingroup$ Do you have any hint on what the function might look like? It sure seems to be something along the line $e^{-\alpha (x - \beta)^2} \sin (\gamma x + \delta)$... $\endgroup$ – vonbrand Feb 11 '14 at 1:03
  • $\begingroup$ @vonbrand Thank you! The function is a linear combination of basis functions like $e^{−α(x−β)2}$ and that's why it looks like what you said. You are smart! $\endgroup$ – Appalachian Math Feb 11 '14 at 18:36

The standard solution for that is to use spline interpolation. Perhaps the nicest solution to the equations is the one given by Wilf in his generatingfunctionology.

  • $\begingroup$ Thanks @vonbrand I downloaded the book and which part I'm supposed to look at? With spline interpolation, should I obtain first and second derivative values for each node except the end points? $\endgroup$ – Appalachian Math Feb 10 '14 at 22:07
  • $\begingroup$ Do you know how I can reply to somebody with @? I don't seem to use it correctly. $\endgroup$ – Appalachian Math Feb 10 '14 at 22:21
  • $\begingroup$ @user2820584 Just put the "at" symbol in front of their name. $\endgroup$ – AnonSubmitter85 Feb 10 '14 at 23:51
  • $\begingroup$ @user2820584 If you are using Matlab, you don't have to worry about calculating any of that. Just specify the 'spline' option to the interp1() function. $\endgroup$ – AnonSubmitter85 Feb 10 '14 at 23:55
  • $\begingroup$ @vonbrand Thank you so very much for your time!!! You have a great day! $\endgroup$ – Appalachian Math Feb 11 '14 at 0:44

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