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Math Help - Polynomial Approximations

  1. #1
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    Polynomial Approximations

    The following is a worked example:




    In the following, for example

    \frac{(g,p_0)}{(p_0,p_0)}p_0 = \frac{(-1)}{2}.1

    What I don't understand is that why they got (g,p_0)=(-1).

    Since p_0 = 1

    And (g,p_0) = \int^1_{-1} 1.g(x)dx

    But what do we need to substitute for g? Because this function is piece-wise, how do I know which one of the two functions I have to put into the formula?
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  2. #2
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    Well, what are the definitions of (g, p_0), (g, p_1), etc.?
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Well, what are the definitions of (g, p_0), (g, p_1), etc.?
    Those are inner products. Since g and p_0 are continious functions on the given interval then their inner product is defined by

    (g,p_0)=\int^1_{-1}g(x)p_0 dx

    I belive that's the definition. The resulting polynomial we get in this problem is called "Legendre's polynomial", but unfortunently there is not much about it in my textbook.
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  4. #4
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    For this:

     (g,p_0) = \int^1_{-1} 1.g(x)dx

    What should I substitute for "g(x)"? The g(x) is defined as a piece wise function. Do I need to substitute "-1" or "2x-1"? (apparently neither of them give the correct answer)
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