Spectral gap for hyperbounded operators

Let be a probability space, and a symmetric linear contraction operator on with and . We prove that is the optimal sufficient condition for to have a spectral gap. Moreover, the optimal sufficient conditions are obtained, respectively, for the defective log-Sobolev and for the defective Poincaré inequality to imply the existence of a spectral gap. Finally, we construct a symmetric, hyperbounded, ergodic contraction -semigroup without a spectral gap.

     

Mass points of measures on the unit circle and reflection coefficients

Measures on the unit circle and orthogonal polynomials are completely determined by their reflection coefficients through the Szego recurrences. We find the conditions on the reflection coefficients which provide the lack of a mass point at . We show that the result is sharp in a sense.     

Spectral gap on path spaces with infinite time-interval

Under certain curvature condition, the existence of spectral gap is proved on path spaces with infinite time-interval. Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060), Beijing Normal University and the State Education Commission of China.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.     

Positive Lyapunov exponents for a class of ergodic orthogonal polynomials on the unit circle

Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C¹ function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent.     

Spectral gap for open Jackson networks

We generalize the decomposition method of the finite Markov chains for Poincaré inequality in Jerrum et al. (Ann. Appl. Probab., 14, 1741-1765 (2004)) to the reversible continuous-time Markov chains. And inductively, we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure. The upper bound of the spectral gap is also presented.     

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (μ₀,μ₁) is a coherent pair of measures on the unit circle, then μ₀ is a semi-classical measure. Moreover, we obtain that the linear functional associated with μ₁ is a specific rational transformation of the linear functional corresponding to μ₀. Some examples are given.     

Spectral gap for jump processes by decomposition method

By using a decomposition method, we give a criterion for the spectral gap of the reversible general jump process. This criterion enables us to obtain the lower bound for the spectral gap via Lyapunov drift condition. Some examples are presented to illustrate the results.      

Spectral gap and logarithmic Sobolev constant for continuous spin systems

The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.     

Spectral transformations of measures supported on the unit circle and the Szegő transformation

In this paper we analyze spectral transformations of measures supported on the unit circle with real moments. The connection with spectral transformations of measures supported on the interval [?1,1] using the Szeg? transformation is presented. Some numerical examples are studied.     

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials B_n(^.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle

Processes of knowledge construction are investigated. A learner is constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle. The analysis is based on the dynamically nested epistemic action model for abstraction in context. Different tasks are offered to the learner. In his effort to perform the different tasks, he has the opportunity to understand the process used to create unit circle representations of trigonometric expressions. The theoretical framework of abstraction in context is used to analyse the evolution of the learner's construction of knowledge in the transition from ‘triangle’ trigonometry to ‘circle’ trigonometry.     

Absolute continuity of the Sobolev type functions on metric spaces

Under study is the absolute continuity of the functions satisfying the Poincaré inequality on s-regular metric spaces.     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.