Reconstruction of Bandlimited Functions from Unsigned Samples

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index $\nu > -1$ conditioned never to collide with each other, in which if $-1 < \nu < 0$ the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function $J_{\nu}$ is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in J. Stat. Phy

Determinantal process starting from an orthogonal symmetry is a Pfaffian process

When the number of particles $N$ is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index $\nu > -1$ (BESQ$^{(\nu)}$) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The $2 \times 2$ skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, $N \delta_0$, and by the equivalence between the noncolliding BESQ$^{(\nu)}$ and that of the noncolliding squared generalized meander starting from $N \delta_0$.Comment: v2: AMS-LaTeX, 17 pages, no figure, corrections made for publication in J.Stat.Phy

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson's model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is $\beta/2$ times the inverse of particle distance. When $\beta=2$, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson's model with $\beta=2$ and $N$ particles, \X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N < \infty, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function \Ai(z) is an entire function with zeros all located on the negative part of the real axis $\R$. We consider Dyson's model with $\beta=2$ starting from the first $N$ zeros of \Ai(z), $0 > a_1 > ... > a_N$, $N \geq 2$. In order to properly control the effect of such initial confinement of particles in the negative region of $\R$, we put the drift term to each Brownian motion, which increases in time as a parabolic function : $Y_j(t) = X_j(t) + t^2/4 + \{d_1 + \sum_{\ell=1}^N (1/a_{\ell})\}t, 1 \leq j \leq N$, where d_1=\Ai'(0)/\Ai(0). We show that, as the $N \to \infty$ limit of \Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty), we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of \Ai(z) on the negative $\R$ is occupied by one particle, to the stationary state \mu_{\Ai}. The stationary state \mu_{\Ai} is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on $\R$ and in which the Tracy-Widom distribution describes the rightmost particle position.Comment: AMS-LaTeX, 33 pages, no figure, v4: minor corrections made for publication in J. Stat. Phy