Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces Lp,q(Rd+1)

In this paper, we study the nonuniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue spaces. We first show that shift-invariant subspaces in mixed Lebesgue spaces Lp,q(Rd+1) can be well-defined. Then we propose that the sampling problem in shift-invariant subspaces of mixed Lebesgue spaces is well-posed. At last, the nonuniform samples {f(xj,yk):k,j∈J} of a function f belonging to a shift-invariant subspace of mixed Lebesgue spaces are proposed, and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X={(xj,yk):k,j∈J} is sufficiently dense.