An SVD-like Decomposition of Functions with Finite 2-induced Norm

The Singular Value Decomposition (SVD) of linear functions facilitates the calculation of their 2-induced norm and row and null spaces, hallmarks of linear control theory. In this work, we present a function representation that, similar to SVD, provides an upper bound on the 2-induced norm of any function where such a bound exists, while also facilitating the computation of generalizations of the notions of row and null spaces for these functions. Borrowing from the notion of "lifting" in Koopman operator theory, we construct a finite-dimensional lifting of inputs that relaxes the unitary property of the right-most matrix in traditional SVD, V^*, to be an injective, norm-preserving mapping to a slightly higher-dimensional space.