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Math Help - Bdd linear functionals on L-inf

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    Bdd 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.)
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    Quote Originally Posted by subfallen View Post
    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 x^n\ (0\leqslant n\leqslant N) form a basis for the (N+1)-dimensional space of polynomials with degree at most N. The subspace spanned by 1,\,x^2,\,x^3,\ldots,x^N has dimension N. Let f(x) be a nonzero element in its one-dimensional orthogonal complement with respect to the inner product \langle u(x),v(x)\rangle = \int_0^1u(x)v(x)\,dx and let g(x) = f(x)/\langle x,f(x)\rangle.
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