Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski¹

¹ Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 152 (2002) no. 1, pp. 69-101

Résumé

We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra ${\cal A}(S)$ of complex-valued functions on the free semigroup $S={\rm FS}(\{z,w\})$ on two generators. First, to given probability measures $\mu$, $\nu$ with all moments finite, we associate states $\widehat{\mu}$, $\widehat{\nu}$ on the unital free *-bialgebra $({\cal B},\varepsilon ,{\mit\Delta})$ on two self-adjoint generators $X,X'$ and a projection $P$. Then we introduce and study cumulants which are additive under the convolution $\widehat{\mu}\star \widehat{\nu}=\widehat{\mu}\otimes \widehat{\nu} \circ {\mit\Delta}$ when restricted to the “noncommutative plane” ${\cal B}_{0}={\mathbb C}\langle X, X'\rangle$. We find a combinatorial formula for the Möbius function in the inversion formula and define the moment and cumulant generating functions, $M_{\widehat{\mu}}\{z,w\}$ and $L_{\widehat{\mu}}\{z,w\}$, respectively, as elements of ${\cal A}(S)$. When restricted to the subsemigroups ${\rm FS}(\{z\})$ and ${\rm FS}(\{w\})$, the function $L_{\widehat{\mu}}\{z,w\}$ coincides with the logarithm of the Fourier transform and with the $K$-transform of $\mu$, respectively. Moreover, $M_{\widehat{\mu}}\{z,w\}$ is a “semigroup interpolation” between the Fourier transform and the Cauchy transform of $\mu$. If one chooses a suitable weight function $W$ on the semigroup $S$, the moment and cumulant generating functions become elements of the Banach algebra $l^{1}(S,W)$.

Détail

DOI : 10.4064/sm152-1-5

Keywords: introduce noncommutative extensions fourier transform probability measures its logarithm algebra cal complex valued functions semigroup generators first given probability measures moments finite associate states widehat widehat unital * bialgebra cal varepsilon mit delta self adjoint generators projection introduce study cumulants which additive under convolution widehat star widehat widehat otimes widehat circ mit delta restricted noncommutative plane cal mathbb langle rangle combinatorial formula bius function inversion formula define moment cumulant generating functions widehat widehat respectively elements cal restricted subsemigroups function widehat coincides logarithm fourier transform k transform respectively moreover widehat semigroup interpolation between fourier transform cauchy transform chooses suitable weight function semigroup moment cumulant generating functions become elements banach algebra