approx. by splines with nonneg. coeff

[Peter Spellucci]

we consider an equidistant grid on the real axis x_i, i in Z, with grid size
h and its restriction
a=x_0<..<x_N=b h=(b-a)/N and the ordinary piecewise polynomial B-splines of
degree k which are in C^{k-1}.
We are especially interested in the case k=3. These B-splines have
as support the intervals x_{i-2},...,x_{i+2} and are positive there .
(support of B_0 and B_1 extends below a, of course and similar at b).
Next we consider all linear combinations

sum_{i=0}^N a_i B_i(x) with a_i >=0 for all i

which are of course nonnegative and C^{k-1} on [a,b]
question: given a function u ,
nonnegative and of bounded variation in [a,b], can we approximate this
arbitrarily close (in some useful norm, L2 would suffice) by such
a combination?
Without the sign constraint on the coefficients this is clear.
Maybe the question is stupid, but I found nowhere any discussion of this.

Thanks in advance
P. Spellucci