The Plancherel-Polya inequalities assert that for 1 < p <∞, and entire functions f of exponential type at most π, Ap ∞ ∑ j=−∞ |f (j)| ≤ ∫ ∞ −∞ |f | ≤ Bp ∞ ∑ j=−∞ |f (j)| . The Marcinkiewicz-Zygmund inequalities assert that for n ≥ 1, and polynomials P of degree ≤ n− 1, Ap n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p ≤ ∫ 1 0 ∣∣P (e2πit)∣∣p dt ≤ B′ p n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p . We show that the sharp constant...

We study random partitions λ = (λ1, λ2, . . . , λd) of n whose length is not bigger than a fixed number d. Suppose a random partition λ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter α > 0. We prove that for all α > 0, in the limit as n → ∞, the joint distribution of scaled λ1, . . . , λd converges to the joint distributi...

We introduce a generalization of the Fourier transform, denoted by FC , on the isotropic cone C associated to an indefinite quadratic form of signature (n1, n2) on R (n = n1 + n2: even). This transform is in some sense the unique and natural unitary operator on L2(C), as is the case with the Euclidean Fourier transform FRn on L 2(Rn). Inspired by recent developments of algebraic representation ...

As a substantial generalization of the technique for constructing canonical and the related nonlinear and q-deformed coherent states, we present here a method for constructing vector coherent states in the same spirit. These vector coherent states may have a finite or an infinite number of components. As examples we first apply the technique to construct vector coherent states using the Planche...

These lecture notes are devoted to the spectral analysis of adjacency operators of graphs and random graphs. With the notion of unimodular random graphs, it is possible to define a natural notion of average spectral measure which corresponds to the density of states in the language of mathematical physics, to the Plancherel measure for Cayley graphs and, for finite graphs, to the empirical meas...

These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomial...

We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann–Hilbert problem and of an integrable integral operator and show that computing the resolvent of a discrete integrable operator can be reduced to solving a corresponding discr...