I am working on a research problem which leads to the following optimization problem: \begin{equation} \hat{M} =\arg\max_M \|\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi k \omega\right)\|^2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1) \end{equation} where, ${\mathbf y}_k$ is a complex vector and a function of integer $k$, $\omega$ is a real scalar (it does not change the problem if you assume that $\omega$ is some known number say one), and $M$ is an integer scalar.
The problem that I have is using discrete convolution formula, i.e. $z[n] = x[n]\star h[n] \triangleq \sum_{k=0}^\infty x[k]h[n-k]$ to rewrite the term $\|\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi k \omega\right)\|^2$ as the norm of convolution of two functions.
If ${\mathbf y}_k$ was a scalar, i.e. $y_k$, the problem would be very straightforward. We could simply use $$\sum_{k=0}^{M-1} {y}_k \exp\left(-j 2\pi k \omega\right) = y[n]u[n]\star \exp\left(-j 2\pi n \omega\right)u[n-M+1] $$ at $n=0$, where $u[n]$ is the discrete unit step function. My question is as follows: Is there a technique to use the convolution notation to rewrite (1)?