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.)

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- Feb 13th 2011, 01:32 PMsubfallenBdd linear functionals on L-inf
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.) - Feb 14th 2011, 04:36 AMOpalg
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 .