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## Studia Mathematica

2002 | 152 | 1 | 69-101
Tytuł artykułu

### Noncommutative extensions of the Fourier transform and its logarithm

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We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra 𝓐(S) of complex-valued functions on the free semigroup S = FS({z,w}) on two generators. First, to given probability measures μ, ν with all moments finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (ℬ,ε,Δ) on two self-adjoint generators X,X' and a projection P. Then we introduce and study cumulants which are additive under the convolution μ̂* ν̂ = μ̂ ⊗ ν̂ ∘ Δ when restricted to the "noncommutative plane" ℬ₀ = ℂ⟨X, X'⟩. We find a combinatorial formula for the Möbius function in the inversion formula and define the moment and cumulant generating functions, $M_{μ̂}{z,w}$ and $L_{μ̂}{z,w}$, respectively, as elements of 𝓐(S). When restricted to the subsemigroups FS({z}) and FS({w}), the function $L_{μ̂}{z,w}$ coincides with the logarithm of the Fourier transform and with the K-transform of μ, respectively. Moreover, $M_{μ̂}{z,w}$ is a "semigroup interpolation" between the Fourier transform and the Cauchy transform of μ. If one chooses a suitable weight function W on the semigroup S, the moment and cumulant generating functions become elements of the Banach algebra l¹(S,W).
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Tom
Numer
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69-101
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wydano
2002
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• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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