On the Lebesgue Function for Polynomial Interpolation

Abstract

Properties of the Lebesgue function associated with interpolation at the Chebyshev nodes ${{\{ \cos [(2k - 1)\pi } {(2n)}}],\, k = 1,2, \cdots ,n\} $ are studied. It is proved that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval. An exact estimate for the smallest maximum is obtained. This estimate together with Rivlin's estimate for the largest maximum leads to the conclusion that the deviation between any two local maxima doesn't exceed ${1 / 2}$. It is shown that for the extended Chebyshev nodes this deviation is less than 0.201. Analogous results are obtained for the set of nodes based on the roots of the Chebyshev polynomials of the second kind.