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Change of variables and use of dirac delta function

  1. Aug 24, 2012 #1
    Dear community,

    I have a scalar function in a Hilbert space of the form f(x) = F(x)* · h o g(x) where F and h are vectors and I need to write < f, f >_L^2 in the shape ∫h(x) ∫H(x,y) h(y) dy dx.

    My derivation is the following using delta function:

    < f, f >_L^2 = ∫F(x)* · h o g(x) ∫δ(x-y) F(y)* h o g(x) dy dx

    applying change of variables on both spatial integrals

    < f, f >_L^2 = ∫|Dg-1(x)| F(g-1(x))* · h(x) ∫δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))* h(y) dy dx =

    ∫ h(x) ∫|Dg-1(x)| F(g-1(x))* δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))* h(y) dy dx

    yielding

    H(x,y) = |Dg-1(x)| F(g-1(x)) δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))*

    ¿Is the use of delta function somehow misleading?
    ¿In which situations (continuity, differentiability) may I
    use this derivation?
    ¿Is my derivation correct?

    Thank you so much.
     
    Last edited: Aug 24, 2012
  2. jcsd
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