I am struggling to find the dual problem for a given primal problem:
$$ (1) \qquad \underset {x}{\text{minimize}} \hspace{0.4cm} f_0(x) \\ \qquad s.t. \hspace{0.3cm} y = F_n x $$ where $x$ is continuous-time signal which is a weighted superposition of spikes: $$ (2) \qquad x = \sum_j a_j \delta_{t_j} . $$ $F_n$ is the linear map collecting the lowest $n=2f_c+1 $ frequency coefficients ($f_c$ is an integer). In fact the constraint of the above problem is a matrix notation to relate the data $y$ and the object $x$ in the following equation: $$ (3) \qquad y(k) = \int_0^1 e^{-i2\pi kt}x dt = \sum_ja_je^{-i2\pi kt_j} \hspace{0.3cm} ,\hspace{1cm} k \in \mathbb{Z},|k|\leq f_c . $$
The constraint is a little bit strange so that I can not form the Lagrange dual function. I kind of don't know what to do.