Abstract In this paper, we study some local spectral properties of operators having form JTJ , where J is a conjugation on Hilbert space H and $$T\in L(H)$$ T ? L ( H ) . We also the relationship betwee...

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = ∑m i=1 X i Xi + X0 + f , where the Xj denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of Xi in L. For any ε > 0 we show that an inequality of the form ‖u‖δ,δ ≤ C(‖u‖0,ε + ‖(K + iy)u‖0,0) holds for suitable δ and C which are...

We investigate the spectral asymptotic properties of the stationary dynamical system ξt = φ(T (X0)). This process is given by the iterations of a piecewise expanding map T of the interval [0, 1], invariant for an ergodic probability μ. The initial state X0 is distributed over [0, 1] according to μ and φ is a function taking values in R. We establish a strong law of large numbers and a central l...

A Borel measure μ in R is called a spectral measure if there exists a set Λ ⊂ R such that the set of exponentials {exp(2πiλ · x) : λ ∈ Λ} forms an orthogonal basis for L(μ). In this paper we prove some properties of spectral measures. In particular, we prove results that highlight the 3/2-rule.

We examine the spectra of boolean functions obtained as the sign of a real polynomial of degree d. A tight lower bound on various norms of the lower d levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables on linear threshold functions. Some conjectures are posed concerning upper and lower bounds on in...

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hy...

In this paper, we prove that the probability kernel of a random walk on a trinomial tree converges to the density of a Brownian motion with drift at the rate O(h), where h is the distance between the nodes of the tree. We also show that this convergence estimate is optimal in which the density of the random walk cannot converge at a faster rate. The proof is based on an application of spectral ...