Given a stochastic process X = {Xt, t ∈ R+} with finite moments of convenient order, a time–space harmonic polynomial relative to X is a polynomial Q(x, t) such that the process Mt =Q(Xt, t) is a martingale with respect to the filtration associated with X . Major examples are the Hermite polynomials relative to a Brownian motion, the Charlier polynomials relative to a Poisson process and the La...

Twenty years ago Yau, 56], generalized the classical Liouville theorem of complex analysis to open manifolds with nonnegative Ricci curvature. Speciically, he proved that a positive harmonic function on such a manifold must be constant. This theorem of Yau was considerably generalized by Cheng-Yau (see 15]) by means of a gradient estimate which implies the Harnack inequality. As a consequence o...

Adaptive optics is implemented in a harmonic generation microscope using a wavefront sensorless correction scheme. Both the second- and third-harmonic intensity signals are used as the optimization metric. Aberration correction is performed to compensate both system- and specimen-induced aberrations by using an efficient optimization routine based upon Zernike polynomial modes. Images of live m...

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline “interpolation” (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the Lp-error for adaptive approximation by harm...

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric...

We present several polynomial congruences about sums with central binomial coefficients and harmonic numbers. In the final section we collect some new congruences involving Fibonacci and Lucas numbers.

We establish various inequalities relating the coefficients of a polynomial with the separation of its roots. Applications are given to oscillatory integrals and sublevel sets in euclidean harmonic analysis as well as exponential sums and polynomial congruences in number theory. These applications depend on precise structural statements of sublevel sets for polynomials with coefficients in a ge...

We prove a Fourier restriction result for general polynomial curves in Rd . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications ...

We study the Plateau problem restricted to polynomial surfaces using techniques coming from the theory of Computer Aided Geometric Design. The results can be used to obtain polynomial approximations to minimal surfaces. The relationship between harmonic Bézier surfaces and minimal surfaces with free boundaries is shown.