Abstract
In this paper, we consider the deformed Hankel transform \({\mathscr {F}}_{\kappa } \), which is a deformation of the Hankel transform by a parameter \(\kappa >\frac{1}{4}\). We introduce, via modulus of continuity, a function subspace of \(L^p(d\mu _{\kappa })\) that we call deformed Hankel Dini–Lipschitz spaces. In the case \(p = 2\), we provide equivalence theorem: we get a characterization of those spaces by means of asymptotic estimate growth of the norm of their \({\mathscr {F}}_{\kappa }\) transform for \(0< \gamma < 1\) and \(\alpha \ge 0\). As a consequence we have the analogous of generalized Titchmarsh theorem in \( L^{2}(d\mu _{\kappa }) \). Moreover, we introduce the modulus of smoothness related to \({\mathscr {F}}_\kappa \) for which we study some properties on the Sobolev type space.
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Negzaoui, S., Oukili, S. Modulus of Continuity and Modulus of Smoothness related to the Deformed Hankel Transform . Results Math 76, 164 (2021). https://doi.org/10.1007/s00025-021-01474-7
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DOI: https://doi.org/10.1007/s00025-021-01474-7
Keywords
- Lipschitz spaces
- Dini–Lipschitz conditions
- modulus of continuity
- modulus of smoothness
- deformed Hankel transform