This paper is concerned with the characterization of α-modulation spaces by Banach frames, i.e., stable and redundant nonorthogonal expansions, constituted of functions obtained by a suitable combination of translation, modulation and dilation of a mother atom. In particular, the parameter α ∈ [0,1] governs the dependence of the dilation factor on the frequency. The result is achieved by exploi...

We construct orthonormal bases compatible with bi-variate homogeneous $$\alpha $$ -modulation spaces and the associated of Triebel–Lizorkin type. The construction is based on generating a separable -covering using carefully selected tensor products univariate brushlet functions regard to this covering. show that systems form an unconditional for -spaces