Approximation from a curve of functions

for all ?-polynomials P,(A, T) of order n. Here I[ lip denotes the norm in Lp [0, 1]. A solution is a best Lp approximation to f . Not unexpectedly, this problem does not always have a solution. The concept of ?-polynomial is extended in section 2 and three existence theorems (Theorems 3, 4 and 5) are established. The first, Theorem 3, is for 1 <p__< 0% I compact and ?(t, x) sufficiently differentiable. The important case when I is not compact is considered in Theorem 4 for 1 <p < oo and Theorem 5 for p = oo. In section 3 we note the interesting fact that if F spans Lp [0, 1] (as in the above examples), then the error of the best Lp approximation, for l < p < o o , strictly decreases as a function of n until it reaches zero. The final section considers Tchebycheff approximation when P,(A, T, x) is analytic in t and a continuous complex valued function of the real variable x. Under the hypothesis that ?(t, x) is strictly totally positive (see [2], e tx, x t qualify)