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Functional optimization problem

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Maximize the functional [tex]\int_{-1}^1 x^3 g(x)[/tex], where g is subject to the following conditions:

    [tex]\int^1_{-1} g(x)dx = \int^1_{-1} x g(x)dx = \int^1_{-1} x^2 g(x)dx = 0[/tex] and [tex]\int^1_{-1} |g(x)|^2 dx = 1[/tex].

    2. Relevant equations

    In the previous part of the problem, I computed [tex]\min_{a,b,c} \int^1_{-1} |x^3 - a - bx - cx^2|^2 dx[/tex]. I'm not sure how this is related, or if it is at all.

    3. The attempt at a solution

    Thus far, I have only tried to look for patterns. In particular, I've tried simply looking for functions g satisfying the conditions, without trying to maximize. I've found a few, and they seem to be closely related to the exponential function. I will continue to look, but I think I may need a boost to get started. I'll be very grateful for any hints anyone can give me.

    EDIT: Hours of trying to solve this, then finally posting it to PF, then trying the Cauchy-Schwartz inequality with a bit of tricky algebra and finding the solution is really frustrating.
     
    Last edited: Oct 27, 2009
  2. jcsd
  3. Oct 28, 2009 #2

    lanedance

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    hi phreak, so i think this is a calculus of variations problem, have you tried using the Euler Lagrange equation with Lagrange multipliers?
     
  4. Oct 28, 2009 #3

    lanedance

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    to expand on that, when you have the problem of optimising an integral for an unknown function g, where the integrand is given by f
    [tex] \int dx f(g, g', x) [/tex]

    subject to constraints h_i
    [tex] \int dx h_i(g, g', x) = c_i[/tex]

    then write the total equation as
    [tex] L(g, g', x) = f(g, g', x) + \lambda_i h_i(g, g', x)[/tex]
    where the lambda's are as yet undetermined lagrange multipliers

    then the the optimising function must satisfy the Euler-Lagrange equation
    [tex] \frac{\partial L(g, g', x)}{\partial g} - \frac{d}{dx} \frac{\partial L(g, g', x)}{\partial g'} = 0[/tex]
     
  5. Oct 28, 2009 #4

    lanedance

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    note in your case a lot of thing simplify as there is no g' term in your equations
     
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