Is it possible to find a function g(x) ∈ L1([0,1]) so that:
∫x^n g(x)dx = δ_{n1}
For n = 0, 1, 2,..., N where N is finite? (The right hand side denotes the Kronecker delta.)
It is even possible to find a polynomial of degree N with this property. The monomials form a basis for the (N+1)-dimensional space of polynomials with degree at most N. The subspace spanned by has dimension N. Let f(x) be a nonzero element in its one-dimensional orthogonal complement with respect to the inner product and let .