The $V_a$-deformation of the classical convolution

Volume 78 / 2007

Anna Dorota Krystek Banach Center Publications 78 (2007), 185-199 MSC: Primary 46L53, 46L54; Secondary 60E10. DOI: 10.4064/bc78-0-14


We study deformations of the classical convolution. For every invertible transformation $T:\mu\mapsto T\mu$, we are able to define a new associative convolution of measures by $$ \mu\mathbin{\ast_T} \nu = T^{-1}(T \mu \mathbin{\ast} T \nu). $$ We deal with the $V_a $-deformation of the classical convolution. We prove the analogue of the classical Lévy–Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_a$-deformed classical convolution and give the orthogonal polynomials associated to the limiting measure.


  • Anna Dorota KrystekMathematical Institute
    University of Wroc/law
    Pl. Grunwaldzki 2/4
    50-384 Wroc/law, Poland

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