For a class of anisotropic integrodifferential operators $ℬ$ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations $ℬ$ u = f on [0,1]n with possibly large n. Under certain conditions on $ℬ$, the scheme is of essentially optimal and dimension independent complexity $𝒪$(h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements...

Let (T1,…,TN) be an N-tuple of commuting contractions on a separable, complex, infinite-dimensional Hilbert space ℋ. We obtain the existence of a commuting N-tuple (V1,…,VN) of contractions on a superspace K of ℋ such that each $V_j$ extends $T_j$, j=1,…,N, and the N-tuple (V1,…,VN) has a decomposition similar to the Wold-von Neumann decomposition for coisometries (although the $V_j$ need not be coisometries). As an application, we obtain a new proof of a result of Słociński (see [9])