The L~(p,q)-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces L~(p,q)(R~(d+1))

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the L^p,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces L^p,q(R^d+1). We first show that the shifts φ(·-k) (k ∈ Z^d+1) are L^p,q-stable if and only if for any ξ ∈ R^d+1, ∑_k∈Z^d+1|φ (ξ + 2πk)|² > 0. Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces L^p,q(R^d+1) to be L^p,q-stable which improves some known results.