Piecewise monotone interpolation and approximation with Muntz polynomials

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 kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.