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-index set. The proofs of these inequalities rely on a general result concerning the summation of tensorized polynomials over arbitrary downward closed multi-index sets.

[1]  D. Wilhelmsen,et al.  A Markov inequality in several dimensions , 1974 .

[2]  G. Migliorati,et al.  Adaptive Polynomial Approximation by Means of Random Discrete Least Squares , 2013, ENUMATH.

[3]  Fabio Nobile,et al.  Analysis of Discrete $$L^2$$L2 Projection on Polynomial Spaces with Random Evaluations , 2014, Found. Comput. Math..

[4]  Szilárd Gy. Révész,et al.  On Bernstein and Markov-Type Inequalities for Multivariate Polynomials on Convex Bodies , 1999 .

[5]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[6]  Albert Cohen,et al.  Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs , 2015 .

[7]  Nira Dyn,et al.  Multivariate polynomial interpolation on lower sets , 2014, J. Approx. Theory.

[8]  T. J. Rivlin The Chebyshev polynomials , 1974 .

[9]  P. Borwein,et al.  Polynomials and Polynomial Inequalities , 1995 .

[10]  Jasper V. Stokman,et al.  Orthogonal Polynomials of Several Variables , 2001, J. Approx. Theory.

[11]  W. Gautschi Orthogonal Polynomials: Computation and Approximation , 2004 .

[12]  Giovanni Migliorati,et al.  Polynomial approximation by means of the random discrete L2 projection and application to inverse problems for PDEs with stochastic data , 2013 .

[13]  M. Ganzburg Polynomial inequalities on measurable sets and their applications , 2001 .

[14]  Albert Cohen,et al.  High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs , 2013, Foundations of Computational Mathematics.

[15]  Albert Cohen,et al.  On the Stability and Accuracy of Least Squares Approximations , 2011, Foundations of Computational Mathematics.

[16]  M. Ganzburg Polynomial Inequalities on Measurable Sets and Their Applications II. Weighted Measures , 2000 .

[17]  Yuan Xu,et al.  Orthogonal Polynomials of Several Variables: Preface , 2001 .

[18]  Fabio Nobile,et al.  Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L2 Projection on Polynomial Spaces , 2013, SIAM J. Sci. Comput..

[19]  András Kroó,et al.  On Bernstein-Markov-type inequalities for multivariate polynomials in Lq-norm , 2009, J. Approx. Theory.