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.
Keywords: Approximation theory ; Multivariate polynomial approximation ; Markov inequality ; Nikolskii inequality ; Orthogonal polynomials ; Downward closed sets ; Legendre polynomials ; Chebyshev polynomials ; Jacobi polynomials ; Gegenbauer polynomials
Record created on 2014-11-11, modified on 2016-08-09