Bounds on a polynomial

Abstract

Methods for computing th e maximum and minimum of a polynomial with real coefficients in the inte rval [0 , 1] are de scribed, and certain bounds are given.

Global Two-Scalar Velocimetry and Development of Low-Order Models for Use in Optical-Phase Correction in a Plane Mixing Layer

“…We use (27) to evaluate the azimuthal {<f>) derivatives of 6, and fourth-degree Lagrange polynomials to evaluate radial derivatives at points on the characteristics. This allows us to extract u with less error in approximating the derivatives of 8 than would be incurred if interpolated values of 8 were numerically differentiated on a rectangular grid.…”

“…To more faithfully mimic extraction of u from experimental data, the temperature was computed at locations on a rectangular grid using (27), and its derivatives were approximated by finite differences. The resolution of the grid is denoted by (I g ,J g ), where l g and J g are the number of x and y grid points, respectively.…”

“…The x and y derivatives can be obtained from £ and <f> derivatives by the chain rule. Recalling that 0 W U.<Ä>=2 a n (£)sinrt77^+g(£,<£), (27) we locally approximate the coefficient functions a"(£) and their first three radial derivatives by fourth-degree Lagrange polynomials fitting the values a'". To illustrate this, consider integration along a characteristic through a point (x 0 ,y 0 ), as shown in Fig.…”

“…However, this method for obtaining the necessary derivatives of the scalar provides more accurate information than one would expect in experiment, since the azimuthal derivatives are computed from an analytical expression rather than by numerical differentiation of spatially discrete data. For this reason, the scalar was also evaluated at locations on a rectangular grid using (27), as discussed in Sec. V A.…”

Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form

“…If 2 ≤ k μ the bound on the right hand side of (11) can be improved slightly, see [10, formula (17)]. For later use we note an extension of [10,Theorem 4].…”