The spectral radius of a multivariate sampling operator Original Research Article

Let Z denote the set of integers, let s be a natural number, and let Is=[0,2π)×cdots, three dots, centered×[0,2π). Let M be a s×s matrix with integer entries and let h=∑nset membership, variantZshne−i(n·θ) be a complex valued multivariate trigonometric polynomial on Is, where θ=(θ1,…,θs)set membership, variantIs, n=(n1,…,ns)set membership, variantZs, and n·θ=n1θ1+cdots, three dots, centered+nsθs. The sum in the function h is taken over a finite set. Consider a multivariate sampling operator S=Sh(M) on the Hilbert space L2(Is) defined by the actionimageSh(M):f(θ)maps toh(θ)f(θM),where fset membership, variantL2(Is). In our paper we give an upper bound on the spectral radius of the operator S=Sh(M), in particular, we prove (under a certain technical assumption)imagewhereh2(z)=∑nset membership, variantZspnz−n.