Frames from Functional Calculus

Abstract

We introduce frames generated by applying families of functions to a diagonal operator through functional calculus, with diagonal entries forming a Carleson interpolating sequence. First, for separated spectra contained in a Stolz domain and an angular sector, we prove that systems of fractional operator powers remain frames whenever the step size satisfies the natural angular restriction. This extends earlier results for positive real spectra and permits noninteger powers. We also show that the angular restriction is sharp in general. Second, we give an $\ell^2$-perturbation criterion for frames generated by more general continuous functions, assuming a density condition on a curve containing the spectrum. The results broaden the class of structured frames available in dynamical sampling and related operator-orbit problems.

Disclosure

“NSERC Discovery Grant RGPIN-2024- 04232. We also wish to thank Ole Christensen and Marzieh Hasannasab for pointing out the compact-perturbation argument to us and for further useful discussions. Finally, the authors acknowledge the use of generative AI for assistance in reviewing and improving this manuscript. Statements and Declarations Conflict of interest statement. The authors declare that there are no conflicts of interest. Data availability. No datasets wer”

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2026-08-23T19:02:50.076643Z

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