Permutation Operators and the Central Limit Theorem Associated with Partial Differential Operators / Operateurs de Permutation et Theoreme de la Centrale Associes a des Operateurs aux Derivees Partielles

For the differential operators % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgs5aenaaBa % aaleaacaaIXaaabeaakiaacQdacqGH9aqpdaWcbaWcbaGaeyOaIyla % baGaeyOaIyRaamiEaaaaaaa!3E05! $$ {\Delta _1}: = {\textstyle{\partial \over {\partial x}}} $$ and % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgs5aenaaBa % aaleaacaaIYaaabeaakiaacQdacqGH9aqpdaWcbaWcbaGaeyOaIy7a % aWbaaWqabeaacaaIYaaaaaWcbaGaeyOaIyRaamiEamaaCaaameqaba % GaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaikdacqaHXoqycqGHRaWk % caaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadk % haaaGaeyOeI0YaaSqaaSqaaiabgkGi2oaaCaaameqabaGaaGOmaaaa % aSqaaiabgkGi2kaadIhadaahaaadbeqaaiaaikdaaaaaaaaa!5054! $$ {\Delta _2}: = {\textstyle{{{\partial ^2}} \over {\partial {x^2}}}} + \frac{{2\alpha + 1}}{r}\frac{\partial }{{\partial r}} - {\textstyle{{{\partial ^2}} \over {\partial {x^2}}}} $$ on ℝ and IR + × ×IR respectively (α∈IR+) one introduces the corresponding permutation operators ℝα and tℝα commuting with the operators Δ1 and % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaleaaleaacq % GHciITdaahaaadbeqaaiaaikdaaaaaleaacqGHciITcaWG4bWaaWba % aWqabeaacaaIYaaaaaaaaaa!3BC8! $$ {\textstyle{{{\partial ^2}} \over {\partial {x^2}}}} $$ respectively. The paper is concerned with problems of harmonic analysis related to the operators Δ1 and Δ2 such as generalized Fourier transforms, Plancherel and Paley-Wiener theorems, generalized translation operators, and products of generalized convolution structures. Within this general framework a central limit theorem is proved. More precisely, sufficient conditions in terms of moments up to the fourth order are given for a triangular system of probability measures on IR + × to converge weakly towards the Gaussian distribution on IR + × . The main results can be considered as contributions to the analysis and probability theory on two-dimensional hypergroups. The French text follows.