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Math Help - Orthonormal basis

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    Orthonormal basis

    Assume that  \{ P_n(x) \} ^{ \infty }_{n=0} is an orthonormal basis with respect to  <f,g> = \int _{-1} ^{1} f(t)g(t)dt

    Find an orthonormal basis  \{ Q_n (x) \} ^ { \infty } _{n=0} with respect to <f,g> = \int ^1 _0 f(t)g(t)dt
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    Quote Originally Posted by tttcomrader View Post
    Assume that  \{ P_n(x) \} ^{ \infty }_{n=0} is an orthonormal basis with respect to  <f,g> = \int _{-1} ^{1} f(t)g(t)dt

    Find an orthonormal basis  \{ Q_n (x) \} ^ { \infty } _{n=0} with respect to <f,g> = \int ^1 _0 f(t)g(t)dt
    The idea is to scrunch up the functions P_n so that they fit into the unit interval, then multiply them by a suitable constant so that they still have norm 1.

    In fact, let Q_n(x) = \frac1{\sqrt2}P_n(2x-1). Then you just have to check that \int_0^1Q_m(x)Q_n(x)\,dx is 1 if m=n and 0 otherwise.
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