Abstract
A theorem of Beurling states that if f satisfies \({||\Delta ^n _{h}f||_{\infty}\leq \rho ^n \cdot ||f||_{\infty}}\) , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.
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Sjödin, T. Beurling’s analyticity theorem for quantum differences. manuscripta math. 127, 369–380 (2008). https://doi.org/10.1007/s00229-008-0213-8
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DOI: https://doi.org/10.1007/s00229-008-0213-8