Scalability of Tensor Products of Multiple Frames in Hilbert Spaces

This paper studies the scalability of tensor products of several frames, which may come from different Hilbert spaces. Although the scalability of single frames and tensor products of two frames has been explored, the case with more than two frames has not been studied and solved yet. We establish sufficientand necessary conditions under which such tensor products are scalable, and we describe these conditions using operator theory, spectral analysis, and numerical methods. To clarify the issue, we provide examples and counterexamples. Finally, we briefly mention how this can be applied in signal processing and sparse representation.