How to approximate a smooth function

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.

• Lanczos Resampling Feb 10, 2014 at 22:05
• @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? Feb 10, 2014 at 22:17
• I don't use MatLab. Sorry. Feb 10, 2014 at 23:31
• Do you have any hint on what the function might look like? It sure seems to be something along the line $e^{-\alpha (x - \beta)^2} \sin (\gamma x + \delta)$... Feb 11, 2014 at 1:03
• @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! Feb 11, 2014 at 18:36

• @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. Feb 10, 2014 at 23:55