Markov- and Bernstein-type inequalities for Muntz polynomials and exponential sums in L-p

摘要

The principal result of this paper is the following Markov-type inequality for Muntz polynomials. THEOREM (Newman's Inequality in Lp[a, b] for [a, b] subset of (0, infinity )). Let Lambda := (lambda(j))(j=0)(infinity) bd an increasing sequence of nonnegative real numbers. Suppose lambda(o) = 0 and there exists a delta > 0 so that lambda(j)greater than or equal to delta j for each j. Suppose 0 < a < b and l less than or equal to p less than or equal to infinity. Then there exists a constant c(a, b, delta) depending only on a, b, and delta so that parallel to P'parallel to(Lp[a,b]) less than or equal to c(a, b, delta)(Sigma(f=0)(n) lambda(j)) parallel to P parallel to(Lp[a,b]) for every P is an element of M-n(Lambda), where M-n(Lambda) denotes the linear span of {x(lambda o), x(lambda i), ..., x(lambda n)} over R. When p = infinity this has been shown by P. B. Borwein and the author (1996, J. Approx. Theory 85, 132-139). When [a, b] = [0, 1] and with parallel to P'parallel to(Lp[a,b]) replaced with parallel to xP'(x)parallel to(Lp[a,b]) this was proved by D. Newman (1976, J. Approx. Theory 18, 360-362) for p = infinity and by P. Borwein and the author (1996, Proc. Amer. Math. Soc. 124, 101-109) for 1 less than or equal to p less than or equal to infinity. Note that the interval [0, 1] plays a special role in the study of Muntz spaces M-n(Lambda). A linear transformation y = alpha x + beta does not preserve membership in M-n(Lambda) in general (unless beta = 0). so the analogue of Newman's Inequality on [a,b] for a > 0 does not seem to be obtainable in any straightforward fashion from the [0,b] case. (C) 2000 Academic Press. [References: 14]