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Math Help - Inner Product Orthogonal Question

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    Inner Product Orthogonal Question

    Find the polynomial p(x) \in P_3 which minimizes the distance \int\limits_{-1}^{1} (cos(\pi x) - p(x))^2 dx
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    Quote Originally Posted by flaming View Post
    Find the polynomial p(x) \in P_3 which minimizes the distance \int\limits_{-1}^{1} (cos(\pi x) - p(x))^2 dx
    Presumably P_3 means the set of cubic polynomials?

    Step 1 is to find an orthonormal basis \{f_0(x),f_1(x),f_2(x),f_3(x)\} for P_3, with respect to the inner product given by \langle f(x),g(x)\rangle = \int_{-1}^1f(x)g(x)\,dx. You do this by starting with the natural basis \{1,x,x^2,x^3\} and applying the Gram–Schmidt process.

    After you have done that, the rest is easy, because p(x) will be given by the formula p(x) = \sum_{j=0}^3\langle\cos(\pi x),f_j(x)\rangle f_j(x).
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