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.
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.