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Compute llf-gll^2

  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) = intergal (1 to 3) (f(x)g(x))dx, let f(x) = 1/x. Knowing that g = (1/2)log3 is the constant polynomial g that is nearest to f. Compute llg-fll2 for this g.

    2. Relevant equations


    3. The attempt at a solution
    I devised llg-fll = llgll - llfll = sqrt(g, g) - sqrt(f,f). Therefore, llg-fll2 = (g, g) - (f, f)
    Using the equations, this equals integral(1 to 3)(1/2log3)2dx - integral (1 to 3)(1/x)2dx
    Simplifying I get (1/2log3)2x evaluated from 1 to 3 +1/x evaluated from 1 to 3 = log23-2/3

    My book says the answer is supposed to be 2/3 - 1/2log23 but I do not get this. Where am I going wrong?
     
  2. jcsd
  3. Oct 7, 2014 #2

    pasmith

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    Homework Helper

    This is wrong. [tex]\|g - f\|^2 = (g,g) - 2(f,g) + (f,f).[/tex] Hence [tex]\|g - f\| = \sqrt{(g,g) - 2(f,g) + (f,f)}.[/tex] However you should in finding [itex]g[/itex] have needed to calculate [tex]\|g - f\|^2 = \|f - g\|^2 = \int_1^3 (f(t) - g(t))^2\,dt[/tex] anyway, so you can simply substitute the found [itex]g[/itex].
     
    Last edited: Oct 7, 2014
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