Dynamical Representation of Frames in Tensor Product of Hardy Spaces

Dynamical Sampling of frames and tensor products are important topics in harmonic analysis. This paper combines the concepts of dynamical sampling of frames and the Carleson condition in the tensor product of Hardy spaces. Initially we discuss the preservation of the frame property under the tensor product on the Hilbert spaces. Then we discuss the iterative representation of frames in tensor product of Hardy spaces. The key ingredient of this paper is the so-called Carleson condition on the sequence $\{ \lambda_k \}_{k=1}^{\infty} \otimes\{ \gamma_l \}_{l=1}^{\infty}$ in the open unit disc $\mathbb{D}_1 \otimes \mathbb{D}_2$. Our proof is motivated by the result of Shapiro and Shields.