Dolomites Research Notes on Approximation

We consider a very general case of vector spaces of multivariate polynomials equipped with some norms. Between them we single out a class of spectral norms, that satisfy the condition ∥Pk∥ = ∥P∥k for all positive integer k. In spaces of polynomials one can consider some linear operators that are usually unbounded, for example derivations, inclusions and multiplying by a fixed polynomial. Bounds for norms of derivatives of polynomials are related to Markov type inequality and Markov’s exponent. We introduce a new concept of an asymptotic Markov’s exponent and show that it is equal to Markov’s exponent for a wide class of norms. However it is not true for all norms in the space of polynomials. We give some examples to show this. We prove an important and very useful inequality, which says that Markov’s exponent for a norm with Nikolskii’s property related to a compact set E is not less than Markov’s exponent for the supremum norm on the set E. As a consequence we obtain a lower bound for the optimal exponent in Markov’s inequality considered with Lp norms and other norms possessing a Nikolskii type property. Our result was used in the paper of Tomasz Beberok published in the Dolomites Research Notes on Approximation and it seems to be useful for future research. One of the main theorems shows a nice application of the Dedania theorem.