Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski. "Noncommutative extensions of the Fourier transform and its logarithm." Studia Mathematica 152.1 (2002): 69-101. <http://eudml.org/doc/284402>.

@article{RomualdLenczewski2002,
abstract = {We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (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 (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).},
author = {Romuald Lenczewski},
journal = {Studia Mathematica},
keywords = {free semigroup; unital free *-bialgebra; convolution; noncommutative plane; Möbius function; inversion formula; -transform; Cauchy transform; moment; cumulant generating functions},
language = {eng},
number = {1},
pages = {69-101},
title = {Noncommutative extensions of the Fourier transform and its logarithm},
url = {http://eudml.org/doc/284402},
volume = {152},
year = {2002},
}

TY - JOUR
AU - Romuald Lenczewski
TI - Noncommutative extensions of the Fourier transform and its logarithm
JO - Studia Mathematica
PY - 2002
VL - 152
IS - 1
SP - 69
EP - 101
AB - We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (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 (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).
LA - eng
KW - free semigroup; unital free *-bialgebra; convolution; noncommutative plane; Möbius function; inversion formula; -transform; Cauchy transform; moment; cumulant generating functions
UR - http://eudml.org/doc/284402
ER -