Frame properties induced by iteration of a multiplication operator on Hardy spaces

2025/09/05 by Jahangir Cheshmavar, Cheshmavar, Jahangir
Mathematics · #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2509.04879

openalex publication_date 2025/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the study of frame properties arising from iterates of linear operators, it was previously established that the multiplication operator T_ϕx(t) = ϕ(t)x(t) cannot generate a frame in L²(a,b) (Results Math, 2019). In this note, we examine the behavior of such operators on the Hardy space H²(D), the Hilbert space of holomorphic functions on the open unit disk D with square integrable boundary values. We show that, in contrast to the L²-setting, the iterates of T_ϕ, for ϕ∈ H^∞(D), exhibit fundamentally different frame properties in H²(D), leading to new structural insights and results.

Citations

• On abstract results of operator representation of frames in Hilbert spaces
• Study on operator representation of frames in Hilbert spaces
• Dynamical sampling with additive random noise
• Dynamical Sampling on Finite Index Sets

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