Discrete harmonic analysis seminar - past schedule for the year 2009/2010:
Current page of the seminar is available here.
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Discrete harmonic analysis seminar - past schedule for the year 2009/2010:Current page of the seminar is available here. |
Omówione zostaną twierdzenia transplantacyjne dla wielu typów dyskretnych i ciągłych rozwinięć ortogonalnych. Dyskutowana również będzie idea transferencji transplantacji. Pokażemy także, jak pojęcie sprzężoności dla rozwinięć ortogonalnych może być interpretowane jako uogólniona transplantacja. Należy mieć na uwadze, że przedstawione w odczycie twierdzenia transplantacyjne nie znajdują żadnego zastososowania w tzw. życiu praktycznym, w szczególności w transplantologii.
The talk is devoted to the equivalence of two type direct theorems in Approximation Theory: a) for smooth functions (Favard's estimates ). b) for arbitrary continuous function (Jackson--Stechkin estimates). Specifically, we will show that Jackson--Stechkin inequality with optimal respect to the order of smoothness constants follows from Favard's inequality. The main tool for this is the function $W_{2k}$, measuring the smoothness of integrable periodic function. This characteristic is more delicate than standard modulus of continuity of the $2k$-th order. The function $W_{2k}$ allows us to obtain asymptotically sharp results for approximation by Favard-type operators. For example, we obtain the Jackson--Stechkin inequality for periodic splines with optimal constants.
Opowiem o pewnych wynikach dotyczących świata grup i półgrup kwantowych, których leitmotivem są własności uniwersalne. Zdefiniuję funktor kwantowego uzwarcenia Bohra i pokażę jak, w połączeniu z opisem kwantowych grup SO(3) jako obiektów uniwersalnych, pozwala on podać pełną klasyfikację działań zwartych grup kwantowych, a także pewnych działań dowolnych półgrup kwantowych na algebrze macierzy 2x2.
J. Meixner in 1934 searched for all probability measures on the real line whose system of orthogonal polynomials has a generating function of exponential type. It appeared that only five measures, including the normal, Poisson and gamma distributions, have this property. These measures and their orthogonal polynomials (Hermite, Charlier, Laguerre...) play a fundamental role in probability theory. The Meixner measures are infinitely divisible and corresponding Lévy processes, which include Brownian motion, Poisson process and gamma process, are widely used in applications. In this talk we will identify a Meixner class of Lévy processes within the framework of free probability. These stochastic processes have non-commuting freely independent increments and exhibit many properties similar to those in the classical case. Their orthogonal polynomials of infinitely many non-commuting variables have a generating function of the resolvent type. This is a joint work with M. Bozejko.
Wielomiany Kerova opisuja znormalizowane charaktery grup permutacji. Korzystajac z interpretacji kombinatorycznej wspolczynnikow w wielomianie Kerova omowimy pytania Lassalle'a dotyczace ich struktury.
We will introduce a class of random walks on the infinite symmetric group for each of which we will be able to estimate (thanks to a very symple structure of the spectrum of the corresponding Laplacian) the transition function of the random walk under consideration.
We discuss the asymptotics of sums of matricially free random variables called random pseudomatrices and compare it with that of random matrices with block-identical variances. The concept of matricial freeness can be viewed as a `matricial version of freeness'. For both random matrices and random pseudomatrices we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the limit joint distributions of symmetric blocks are the same for objects of both types.
In this joint work with Simeon Reich and David Shoikhet, we study the structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded or convex) domain in a complex Banach space, by means of ergodic theory and a nonlinear numerical range.
We show that representations of the unitary groups asymptotically behave like random matrices. For example: the decomposition of Kronecker tensor product of two irreducible representations behaves like a sum of two independent unitarily invariant random matrices with prescribed eigenvalues. Our main tool is to view representations as random matrices with quantum entries.