Tangential Markov Inequalities on Transcendental Curves

Abstract

Abstract. We show that on the curves γ:=(x,e^t(x)) , x∈ [a,b] , where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials P _N (x,y) of degree at most N in each variable x,y: ||(P _N (x,e^t(x)))'|| _[a,b] ≤ CN⁴||P _N || _γ , and the exponent four is sharp. On the other hand, the corresponding tangential Markov factors on curves y=x^α with irrational α grow exponentially in the degree of the polynomials.