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Multivariate Davenport Series

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Further Developments in Fractals and Related Fields

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Abstract

We consider series of the form ∑a n {n ⋅x}, where n ∈ Z d and {x} is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.

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Acknowledgements

The authors are grateful to Julien Brémont for pointing out a mistake in a first version of this chapter and to the anonymous referee for the careful reading and many valuable remarks.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud, 91405, Orsay Cedex, France

    Arnaud Durand

  2. Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris-Est - Créteil Val-de-Marne, 61 avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Stéphane Jaffard

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  1. Arnaud Durand
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  2. Stéphane Jaffard
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Correspondence to Arnaud Durand .

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Editors and Affiliations

  1. , LAGA, Université Paris 13, avenue J.B. Clément 99, Villetaneuse, 93430, France

    Julien Barral

  2. , Laboratoire d'Analyse et de Mathématique, Université Paris-Est Créteil - Val de Ma, Avenue du Général de Gaulle 61, Créteil Cedex, 94010, France

    Stéphane Seuret

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Durand, A., Jaffard, S. (2013). Multivariate Davenport Series. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_5

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