Multivariate Markov Polynomial Inequalities and Chebyshev Nodes
نویسنده
چکیده
This article considers the extension of V. A. Markov’s theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.
منابع مشابه
Multivariate Markov-type and Nikolskii-type inequalities for polynomials associated with downward closed multi-index sets
We present novel Markov-type and Nikolskii-type inequalities for multivariate polynomials associated with arbitrary downward closed multi-index sets in any dimension. Moreover, we show how the constant of these inequalities changes, when the polynomial is expanded in series of tensorized Legendre or Chebyshev or Gegenbauer or Jacobi orthogonal polynomials indexed by a downward closed multi-inde...
متن کاملMultivariate polynomial interpolation on Lissajous-Chebyshev nodes
In this contribution, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets linked to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes related to these curves, we derive a discrete orthogonality structure on these node sets. Using this discrete orthogonality structure, we can deriv...
متن کاملA bivariate Markov inequality for Chebyshev polynomials of the second kind
This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author h...
متن کاملFast, exact and stable reconstruction of multivariate algebraic polynomials in Chebyshev form
We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1...
متن کاملOptimal Inequalities in Probability Theory: A Convex Optimization Approach
We address the problem of deriving optimal inequalities for P(X E S), for a multivariate random variable X that has a given collection of moments, and S is an arbitrary set. Our goal in this paper is twofold: First, to present the beautiful interplay of probability and optimization related to moment inequalities, from a modern, optimization based, perspective. Second, to understand the complexi...
متن کامل