Let {pm(wα)}m be the sequence of the polynomials orthonormal w.r.t. the Sonin-Markov weight wα(x) = e−x 2 |x|. The authors study extended Lagrange interpolation processes essentially based on the zeros of pm(wα)pm+1(wα), determining the conditions under which the Lebesgue constants, in some weighted uniform spaces, are optimal.

The Hermitian Lanczos method for Hermitian matrices has a wellknown connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of R. This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomi...

Abstract. Let Jμ[X; q, t] be the integral form of the Macdonald polynomial and set H̃μ[X; q, t] = tJμ[X/(1− 1/t); q, 1/t ], where n(μ) = ∑ i(i− 1)μi. This paper focusses on the linear operator ∇ defined by setting ∇H̃μ = tq ′)H̃μ. This operator occurs naturally in the study of the Garsia-Haiman modules Mμ. It was originally introduced by the first two authors to give elegant expressions to Frobeni...

In [20] a generalization of Bernstein polynomials and Bézier curves based on umbral calculus has been introduced. In the present paper we describe new geometric and algorithmic properties of this generalization including: (1) families of polynomials introduced by Stancu [19] and Goldman [12], i.e., families that include both Bernstein and Lagrange polynomial, are generalized in a new way, (2) a...

We obtain by elementary methods necessary and sufficient conditions for a k-dimensional cubature formula to hold for all polynomials of degree up to 2m− 1 when the nodes of the formula have Lagrange polynomials of degree at most m. The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m− 1 evaluated at the node plus an orthogona...

Experimental observations of rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable, and indeed sometimes be much more stable than rootfinding of polynomials expressed in even the Bernstein basis. This paper details some of those experiments and provides a theoretical justification for this. We prove that a ne...

We review complexity results for minimizing polynomials over the standard simplex and unit hypercube. In addition, we derive new results on the computational complexity of approximating the minimum of some classes of functions (including Lipschitz continuous functions) on the standard simplex. The main tools used in the analysis are Bernstein approximation and Lagrange interpolation on the simp...

This paper, as the sequel to previous work, develops numerical schemes for fractional diffusion equations on a two-dimensional finite domain with triangular meshes. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization by the use of high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles. Stability analysis and error estimates...

Vector-valued L p-convergence of orthogonal series and Lagrange interpolation. Abstract We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the vector-valued expansion problem of L p-functions in terms of Jacobi polynomi...

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to systems with constraints, the procedure of finding a Hamiltonian for a degenerate Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairau...