Dynamical Representation of Frames in Tensor Product of Hardy Spaces

TL;DR

This paper explores how frames in Hardy spaces behave under tensor products, focusing on the Carleson condition and iterative representations, extending harmonic analysis techniques to tensor product spaces.

Contribution

It introduces conditions for preserving frame properties in tensor products of Hardy spaces and develops an iterative representation framework based on the Carleson condition.

Findings

Frame property preservation under tensor products

Iterative representation of frames in Hardy spaces

Role of Carleson condition in tensor product frames

Abstract

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.