I need to show that a given surface elevation $\zeta(x,y,t)$ defined on a closed region $D(x,y,t): 0<x<L_x,0<y<L_y,0<t<T$ and not periodic on D:
$$ ζ(x,y,t) = \sum_{n=1}^{\infty} a_ncos(\mathbf{k_n}\cdot \mathbf{x} -\omega(|\mathbf{k_n}|)t)+b_nsin(\mathbf{k_n}\cdot \mathbf{x} -\omega(|\mathbf{k_n})|t) $$
(with $\omega(|\mathbf{k_n}|) =\sqrt{g|\mathbf{k_n}|}$ the dispersion relation and $\mathbf{k_n}$ the wave vector) can be represented by a possibly infinite series based on the function of the type
$$cos(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t)\\ sin(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t) $$
where in general $\mathbf{k_m} \neq \mathbf{k_n} \forall n,m$. This set of functions are defined to be periodic in the space $\mathbf{k_{m,i,j}} = [m_i2\pi/L_x,m_j2\pi/L_y]$ but they are clearly not periodic in time for a given region D.
What are the condition under which the series $$ \sum_{m=1}^{\infty} a_m cos(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t)+ b_m sin(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t)$$ is convergent in the mean to $\zeta(x,y,t)$?
The set of function $cos(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t),sin(\mathbf{k_m}\cdot \mathbf{x} -\omega(|\mathbf{k_m}|)t)$ are not orthogonal but maybe linear indipendet. How to prove it? And is it a sufficient condition for a convergent series?