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Hankel Transforms of General Monotone Functions

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Book cover Topics in Classical and Modern Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel–Olivier test for real-valued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.

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Acknowledgement

This research was partially funded by the ERC starting grant No. 713927, the ISF grant No. 447/16, the CERCA Programme of the Generalitat de Catalunya, Centre de Recerca Matemàtica, and the MTM-2014-59174-P grant.

Author information

Authors and Affiliations

  1. Bar-Ilan University, Ramat-Gan, Israel

    Alberto Debernardi

  2. Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain

    Alberto Debernardi

Authors
  1. Alberto Debernardi
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Editor information

Editors and Affiliations

  1. Georgia Southern University, Statesboro, GA, USA

    Martha Abell

  2. Georgia Southern University, Statesboro, GA, USA

    Emil Iacob

  3. Georgia Southern University, Statesboro, GA, USA

    Alex Stokolos

  4. Georgia Southern University, Statesboro, GA, USA

    Sharon Taylor

  5. ICREA, Passeig de Lluís Companys, Barcelona, Spain ; Centre de Recerca Matemàtica, Barcelona, Spain

    Sergey Tikhonov

  6. Georgia Southern University, Statesboro, GA, USA

    Jiehua Zhu

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Debernardi, A. (2019). Hankel Transforms of General Monotone Functions. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_5

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