We prove a partial result concerning the long–standing problem on limit periodicity of the Jacobi matrix associated with the balanced measure on the Julia set of an expending polynomial. Besides this, connections of the problem with the Faybusovich–Gekhtman flow and many other objects (the Hilbert transform, the Schwarz derivative, the Ruelle and Laplace operators) that, we sure, are of indepen...

Recently A. D. Ionescu (2000) established the endpoint estimate for the Kunze-Stein phenomenon, which states that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L(G) ∗ L(G) ⊆ L2,∞(G). In this paper, we will prove the corresponding result for the Jacobi transform. Our method is analytical, in which we do not use the structure of Lie groups.

We prove a partial result concerning the hard long standing problem on limit periodicity of the Jacobi matrix associated with the balanced measure on the Julia set of an expending polynomial. Besides this, connections of the problem with the Faybusovich–Gekhtman flow and many other objects (the Hilbert transform, the Schwarz derivative, the Ruelle and Laplace operators) that, we sure, they are ...

An ordinary unambiguous integral representation for the finite propagator of a quantum system is obtained by path integrating in phase space. The skeletonization of the canonical action by means of pieces described by a complete solution of the Hamilton-Jacobi equation leads to a simple composition law that reduces the multiple integration to a sole one. Thus the finite quantum propagator can b...

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermor...

Large deviation for Markov processes can be studied by Hamilton– Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton–Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) ho...

This paper presents an adaptive mesh redistribution (AMR) method for solving the nonlinear Hamilton-Jacobi equations and level set equations in twoand threedimensions. Our approach includes two key ingredients: a non-conservative secondorder interpolation on the updated adaptive grids, and a class of monitor functions (or indicators) suitable for the Hamilton-Jacobi problems. The proposed adapt...

In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second...

Large deviation for Markov processes can be studied by Hamilton– Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton–Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) ho...

We observe that polynomial measure modifications for families of univariate orthogonal polynomials imply sparse connection coefficient relations. We therefore propose connecting L2 expansion coefficients between a polynomial family and a modified family by a sparse transformation. Accuracy and conditioning of the connection and its inverse are explored. The connection and recurrence coefficient...