A Minimax Formula for the Best Natural C([0,1])-Approximant by Nondecreasing Functions

Abstract

Let f be a function in C ( [ 0 , 1 ] ) . We denote by f p the best approximant to f in L p ( [ 0 , 1 ] ) by nondecreasing functions. It is well known that the limit f ∗ := lim p → ∞ f p exists and f ∗ is a best approximant to f in C ( [ 0 , 1 ] ) by nondecreasing functions. In this paper we show an explicit formula for the function f ∗ and we prove some additional minimization properties of f ∗ .