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Homework Help: Find the constant polynomial g closest to f

  1. Oct 7, 2014 #1
    1. The problem statement, all variables and given/known data
    In the real linear space C(1, 3) with inner product (f,g) = integral (1 to 3) f(x)g(x)dx, let f(x) = 1/x and show that the constant polynomial g nearest to f is g = (1/2)log3.

    2. Relevant equations


    3. The attempt at a solution
    I seem to be able to get g = log 3 but I do not know where the 1/2 comes from. Here is what I did:
    Let fn = summation (k=0 to n) (f, gk) gk
    Therefore, (f, gk) = integral (1 to 3) (1/t)(gk)dt and (f, g0) = 0
    Hence, (f, g1) = integral (1 to 3) (1/t)dt = log(t) evaluated from 1 to 3 = log (3) - log(1) = log(3) - 0 = log(3).
    I don't see where the 1/2 comes from or where my mistake is?
     
  2. jcsd
  3. Oct 7, 2014 #2

    pasmith

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    Homework Helper

    You need to find the constant [itex]k[/itex] which minimizes the distance between [itex]f = 1/t[/itex] and [itex]g = k[/itex]. The distance is defined in terms of the inner product by [itex]\| f - g \| = \sqrt{(f-g,f-g)}[/itex]. Since squaring is strictly increasing on the positive reals it suffices instead to minimize
    [tex]\| f - g \|^2 = \int_1^3 (f(t) - g(t))^2\,dt.[/tex]
     
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