Discrete Triebel-Lizorkin spaces and expansive matrices

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices A and B, it is shown that ḟ^₏, ₐ (A) = ḟ^₏, ₐ (B) for all R and p, q (0, ] if and only if the set \Aʲ B^{-j: j Z\} is finite, or in the trivial case when p = q and | (A) |^ + 1/2 - 1/p = | (B) |^ + 1/2 - 1/p. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.