For a matrix $X \in \mathbb{R}^{m\times n}$, we have the SVD decomposition $$ X = U D V^\top, $$ where $U\in\mathbb{R}^{m\times r},\ V\in\mathbb{R}^{n\times r}$ are orthonormal matrices and $D=\text{diag}(\sigma_1,\dots,\sigma_r)$ is a diagonal matrix.
Now, let's consider the vector form $ X = \sum_{k=1}^r \sigma_k u_k v_k^\top $, where $u_k$, $v_k$ represent the $k$th column of $U$ and $V$, respectively. Then, the $(i,j)$th element of $X$ is $$ x_{ij} = \sum_{k=1}^r \sigma_k u_{ik} v_{jk}. $$ By regarding indexing as a mapping from integers to function values, I'm wondering if this continuous (functional) counterpart of SVD exists: $$ x(s,t) = \sum_{k=1}^\infty \sigma_k u_k(s) v_k(t), $$ such that $(u_k)$ and $(v_k)$ are orthonormal function series.
Does the factorization above exist? If yes, has anyone studied the approximation to the left side with a truncated series of the right side? If no, does any bivariate-to-univariate factorization exist?