My problem is as follows:
I am attempting to use a reproducing kernel and quadratic programming to optimize/interpolate a smooth function from constraints. I am using all local extrema to calculate constraints, and I am trying to estimate a mean value for the signal.
Given a function $f(t)$ as $t \in [0, T]$, $K(t,t_i)=exp(-||t - t_i||^2)$, and $f(t) = f_M(t) = \sum_{i=1}^{M}\alpha_iK(t,t_i)$ (The reproducing kernel property)
Solve the following
$P = K(t_{t \in T}, t_{i \in M})^tK(t_{t \in T}, t_{i \in M})$ (L2 normalization)
Minimize the cost function $C(\alpha) = \alpha^tP\alpha$ such that,
$K_{ineq}\alpha \le u$
$K_{eq}\alpha = e$
My knowledge about reproducing kernels is that even with only a subset of points should be able to interpolate the original function. I am getting successful optimizations with the above quadratic program, but the output function $f_M(t)$ is only fit at all $t_M$ points and is usually near 0.0 for all other points. I assumed that constructing my kernel matrix P using all time points in the series would lead to coefficients $\alpha_j$ that would allow the formula for $f_M(t)$ to be used to evaluate the fit function at all points $T$.
I am stumped. What could be the problem?