A simple method is presented by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials into Bernstein polynomials.

In [1] and [2], Bernstein and Reznikov have introduced a new way of estimating the coefficients in the spectral expansion of φ2, where φ is a Maass cusp of norm 1 on a quotient Y = Γ\H of the Poincaré upper half-plane with finite volume. The question of obtaining the precise exponential decay of those coefficients had been posed by Selberg, and first solved by Good [5] (for holomorphic forms) a...

We prove that the exponential localization of a frame with respect to an orthonormal basis in a Hilbert space is not sufficient to get a Bernstein inequality. In other words, the fact that a function belongs to an approximation space of the frame cannot be characterized in terms of the sparseness of its frame coefficients.

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = xe, where α > −1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coeffic...

0. Introduction; notation This paper lists the essential facts about the representation of polynomials in m variables as Bernstein polynomials. An expanded version may appear elsewhere. While univariate Bernstein polynomials are well studied see, e.g., Lorentz’ classical book Lorentz (1953), the multivariate version has only attracted attention sporadically. Lorentz’ book devotes just one page ...