The Markoff-Duffin-Schaeffer inequalities abstracted

Abstract

The kth Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1 </= x </= 1, of the kth derivative of a normalized polynomial of degree n. The bound is the corresponding maximum of the Chebyshev polynomial of degree n, T = cos(n cos(-1)x). The requisite normalization is over the values of the polynomial at the n + 1 points where T achieves its extremal values. The inequality is an equality only if the polynomial equals T or -T. The proof uses complex variable theory. This paper deals with a well-known generalization of polynomials-namely, functions satisfying some of the oscillation and approximation properties of ordinary polynomials. In particular, the generalized Chebyshev polynomial exhibits the extremal oscillations characteristic of the classical Chebyshev polynomial. It is shown that the direct analogs of the Markoff-Duffin-Schaeffer inequalities hold in this abstract setting and that they are included as a special case. Moreover, the proof is more elementary.