NEWMANS INEQUALITY FOR MUNTZ POLYNOMIALS ON POSITIVE INTERVALS

The principal result of this paper is the following Markov-type inequa lity for Muntz polynomials. THEOREM (Newman's Inequality on [a,b] subs et of (0,infinity)). Let Lambda:=(lambda(j))(j=0)(infinity) be an incr easing sequence of nonnegative real numbers. Suppose lambda(0)=0 and t here exists a delta > 0 so that lambda(j) greater than or equal to del ta(j) for each j. Suppose 0 < a < b. Then there exists a constant c(a, b,delta) depending only on a, b, and delta so that \\P'\\[a,b]less tha n or equal to c(a,b,delta)((j=0)Sigma(n) lambda(j)) \\P\\([a,b]) for e very P is an element of M(n)(Lambda) where M(n)(Lambda) denotes the li near span of {x(lambda 0), x(lambda 1),...,x(lambda n)} over R. When [ a,b]=[0,1] and with \\P'\\([a,b]) replaced with parallel to xP'(x)para llel to([a,b]) this was proved by Newman. Note that the interval [0,1] plays a special role in the study of Muntz spaces M(n)(Lambda). A lin ear transformation y=alpha x+beta does not preserve membership in M(n) (Lambda) in general (unless beta=0). So the analogue of Newman's Inequ ality on [a,b] for a>0 does not seem to be obtainable in any straightf orward fashion from the [0,b] case. (C) 1996 Academic Press, Inc.