TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 218, 1976 PIECEWISE MONOTONE INTERPOLATION AND APPROXIMATION WITH MUNTZ POLYNOMIALS BY ELI PASSOW, LOUIS RAYMON AND OVED SHISHA ABSTRACT. The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of "Muntz polynomials" on a real interval is demonstrated: (i) approximation of a continuous function by a "copositive" Muntz polynomial; (ii) approximation of a continuous function by a "comonotone" Muntz polynomial; (iii) approximation of a continuous function with a monotone fcth differ- ence by a Muntz polynomial with a monotone fcth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polyno- mials that are monotone on each of the intervals determined by the points of in- terpolation. The strong interrelationship of these problems is shown implicitly in the proofs. The following related questions have been settled: I iMonotone Approximation). Let fix) he a continuous function with the property that the /th difference u¿f> 0 on [0, 1] where / is some nonnegative integer. Must there be for a given e > 0 a corresponding polynomial p(x) with p0)(x) > 0 on [0, 1] such that ||/-p|| = sup \f(x)-p(x)\<el *e[o,i] II (Comonotone Approximation). Let f(x) he a continuous function with a finite number of nodes on [0, 1] ; i.e., suppose 0 = x0<xx < ••• <xk = 1 and that f(x) is alternately nondecreasing and nonincreasing on the intervals (0, xx), (xx, x2), . . . , (xk_x, xk). For a given e > 0 must there be a correspond- ing polynomial p(x) that has the same monotonicity as fix) on each of the inter- vals (*,_!, xj), i = 1, 2,.... k, and such that ||/-p|| < e? Received by the editors October 31, 1973. AMS (MOS) subject classifications (1970). Primary 41A05, 41A10, 41A30, 41A25. Key words and phrases. Approximation, interpolation, Muntz polynomials, restricted approximation, restricted interpolation, monotone approximation, piecewise monotone inter- polation. The authors wish to acknowledge D. Myers for pointing out errors in thelmanuscript. 1 0_ Copyright © 1976, American-Mathematical Society License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use