Infinite-dimensional singular value decomposition Say, I have an $n\times n$ matrix $w_{ij}$. I can perform a singular value decomposition such that
$$ w_{ij}=\sum_l \sum_n u_{il}\lambda_{ln}v_{jn} $$
with $\lambda_{ln}$ diagonal.  Now, is there such a generalization so that, given a function of two variables $w(\theta_1, \theta_2)$,
$$ w(\theta_1 ,\theta_2)=\int dy \int dx \,u(\theta_1 ,x) \, \lambda(x,y) \, v(\theta_2 ,y) $$
where $\lambda$ plays a similar role like it did in the SVD?  For instance,  say I have the following
$$ \exp {[\alpha \cos(\theta-\phi)]} $$
is it possible to find a decomposition such that
$$ \exp {\alpha \cos(\theta-\phi)}=\iint dx \, dy \, u(\theta,x) \, \lambda(\alpha,x,y) \, v(\phi,y) $$
 A: Just view a matrix as a sampling of a continous variable. I.e. the matrix indices are local integrals in some sense. Assume you can sample the function at arbitrary small intervals ( perform local integrals / mean value integrals ). (That is practically what the riemann integral is, no?). 
At every given resolution you can make an SVD of the matrix containing those local integrals. Now consider the Haar wavelet, where you can build orthogonal refinements. (Think of it as splitting every matrix element into blocks of $2 \times 2$ elements where the old element is the mean value of the new elements). Haar wavelet additions (measuring discrete derivatives) will then be either orthogonal or parallell both within the vectors of the matrices themselves and as matrix-blocks. So you can make a successive refinement of the SVD up to any resolution you like, and because of block properties of matrix multiplication, it will be "nice" too in the sense that you can just find successive refinements.
If this is fruitful, it may be used to perform SVDs fast for large matrices, since we can pretend to measure on the matrix itself as if it were a continous variable.
