On the Best Constants in Markov–type Inequalities Involving Gegenbauer Norms with Different Weights
نویسندگان
چکیده
The paper is concerned with best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L2 with the Gegenbauer weight corresponding to a parameter α , while the derivative is measured in L2 with the Gegenbauer weight for a parameter β . Under the assumption that β −α is an integer, we determine the first order asymptotics of the best constants as the degree of the polynomial goes to infinity. Mathematics subject classification (2010): Primary 41A44; Secondary 15A18, 26D10, 45D05, 47B35.
منابع مشابه
Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions
The first term of the asymptotics of the best constants in Markov-type inequalities for higher derivatives of polynomials is determined in the two cases where the underlying norm is the L norm with Laguerre weight or the L norm with Gegenbauer weight. The coefficient in this term is shown to be the norm of a certain Volterra integral operator which depends on the weight and the order of the der...
متن کاملHardy inequalities with optimal constants and remainder terms ∗
We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L norms. In the higher-order case further L norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only...
متن کاملStability inequalities for Lebesgue constants via Markov-like inequalities
We prove that L∞-norming sets for finite-dimensional multivariate function spaces on compact sets are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials. 2010 AMS subject classification: 41A10, 41A63, 42A15, 65D05.
متن کاملMultivariate 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 کاملConditioned Square Functions for Non-commutative Martingales
Abstract. We prove a weak-type (1,1) inequality involving conditioned square functions of martingales in non-commutative L-spaces associated with finite von Neumann algebras. As application, we determine the optimal orders for the best constants in the non-commutative Burkholder/Rosenthal inequalities from Ann. Probab. 31 (2003), 948-995. We also discuss BMO-norms of sums of non commuting order...
متن کامل