Given f in C[a, b], the space of real-valued continuous functions on the interval [a, b], the best approximation in the uniform norm by an element of x,,, the polynomials degree <n, is attained by a polynomial whose properties are governed by the Tchebycheff theory (cf. [2]). If a better approximation is desired, then one may enlarge the approximating space to q,+ i, a linear space of dimension n + 2. In this paper we consider an enlargement of x,, to a space designated as V,. That V, might be a useful space for approximation purposes was suggested to us by Norman H. Painter. Let F be a function which is continuous and strictly positive on (--00, co). For a real number a, let V,(a) be the space of functions of the form F(ax)P(x), where P belongs to 71,. Let V, be the union (over all real a) of the V,(a). The situation may be generalized to consider general Haar systems, rather than just rc,. While some of our results will be valid in that context, we shall be content to state and prove our results for the special Haar system 71,. Although V, is not a linear space, it is the union of linear spaces, VJa), each of which has the Haar property. Moreover, V,(O) = TC,, and the underlying parameter space of I’,, has dimension n + 2. Hence, as a space of approximating functions, it could be compared reasonably with rc,, + , . In this regard V,, has both advantages and disadvantages. Naturally, if f, the function to be approximated, is “F-like,” then V, is preferable. More concretely, consider the case F(x) = ex. Then a function 4 (f0) of V, may have up to n zeros in [a, b], and this is true also of all its derivatives. A function of 7r,+, may have up to n + 1 zeros, but its derivatives (f0) will have less. Thus if f has several changes of curvature, an approximating function from V, may be preferable to one from K,+ *. On the other hand, any f can be interpolated by a function of rr,+ 1 at any n + 2 points. Such 333 0021-9045184 $3.00