Related papers: An explicit calculation of the Ronkin function
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also…
Special values of the Lommel functions allow the calculation of Fresnel like integrals. These closed form expressions along with their asymptotic values are reported.
In this paper we derive the bi-orthogonality relations, diagonal term evaluations and evaluation formulas for the non-symmetric Koornwinder polynomials. For the derivation we use certain representations of the (double) affine Hecke algebra…
In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. In non generic cases it is an iterated integral. In previous papers one of the authors gives a precise description of the Principal Poincar\'e…
In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also…
First we prove some kernel representations for the covariance of two functions taken on the same random variable and deduce kernel representations for some functionals of a continuous one-dimensional measure. Then we apply these formulas to…
Final representation of all those measures $\mu$ for which algebraic polynomials are dense in $L_p(R, d\mu)$ is found. The weighted analogue of the Weierstrass polynomial approximation theorem and a new version of the M. Krein's theorem…
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.
We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct…
This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally polynomial Markov processes. The polynomial structure allows for the…
Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and…
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are ({\alpha},m)-convex.
In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated…
We establish formulae for the derivative of the Poincar\'e constant of Gibbs measures on both compact domains and all of $\R^d$. As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the…
We derive a priori estimates for second order derivatives of solutions to a wide calss of fully nonlinear elliptic equations on Riemannian manifolds. The equations we consider naturally appear in geometric problems and other applications…
A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise…
The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\`ere equation. The approximation theory of…