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approx. by splines with nonneg. coeff
Posted:
Oct 21, 2012 4:00 PM


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=(ba)/N and the ordinary piecewise polynomial Bsplines of degree k which are in C^{k1}. We are especially interested in the case k=3. These Bsplines have as support the intervals x_{i2},...,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^{k1} 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



