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Distributions of truncations of the heat kernel on the complex projective space

Nizar Demni — 2014

Annales mathématiques Blaise Pascal

Let ( U t ) t 0 be a Brownian motion valued in the complex projective space P N - 1 . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of | U t 1 | 2 and of ( | U t 1 | 2 , | U t 2 | 2 ) , and express them through Jacobi polynomials in the simplices of and 2 respectively. More generally, the distribution of ( | U t 1 | 2 , , | U t k | 2 ) , 2 k N - 1 may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group 𝒰 ( N - k + 1 ) yet computations become tedious. We also revisit the approach initiated in [] and based on a...

Spectral distribution of the free Jacobi process associated with one projection

Nizar DemniTaoufik Hmidi — 2014

Colloquium Mathematicae

Given an orthogonal projection P and a free unitary Brownian motion Y = ( Y ) t 0 in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...

Topics on Meixner families

Marek BożejkoNizar Demni — 2010

Banach Center Publications

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.

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