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Math Help - Least Squares with Weight Function

  1. #1
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    Least Squares with Weight Function

    Use the Laguerre polys
    L1=x-1, L2=x^2-4x+2, L3=x^3+9x^2+18x-6
    to compute least squares polys on (0, infinity) with respect to the weight fn w(x)=e^-x
    for the following functions:

    f(x)=x^2 , f(x)=e^-x, f(x)=x^3, f(x)=e^-2x
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  2. #2
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    Re: Least Squares with Weight Function

    Quote Originally Posted by monomoco View Post
    Use the Laguerre polys
    L1=x-1, L2=x^2-4x+2, L3=x^3+9x^2+18x-6
    to compute least squares polys on (0, infinity) with respect to the weight fn w(x)=e^-x
    for the following functions:

    f(x)=x^2 , f(x)=e^-x, f(x)=x^3, f(x)=e^-2x
    The Laguerre polynomials are orthogonal with respect to your weight on the given interval. So what do you know about expansion in terms of orthogonal functions?

    CB
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  3. #3
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    Re: Least Squares with Weight Function

    I am very confused about all this stuff, but I think when the functions are orthogonal we can use the following to calcualte coefficients in the polynomial:

    a_j = (1/a_j) *Integral from a to b of w(x)f(x)\Phi_j (x) dx

    Sorry, I don't really know Latex
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  4. #4
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    Re: Least Squares with Weight Function

    Quote Originally Posted by monomoco View Post
    I am very confused about all this stuff, but I think when the functions are orthogonal we can use the following to calcualte coefficients in the polynomial:

    a_j = (1/a_j) *Integral from a to b of w(x)f(x)\Phi_j (x) dx

    Sorry, I don't really know Latex
    Define the inner product:

    \langle f,g \rangle= \int_0^{\infty} f(x)g(x) w(x)dx

    on the space of real square integrable functions over (0,\infty) with respect to the weight w(x). Then if L_i,\ i=0, .. \ ,\infty is a complete basis of orthogonal functions for this space then:

    f(x)=\sum_{i=1}^{\infty} \frac{\langle f,L_i\rangle}{\|L_i\|}L_i(x)

    and the least squares (with respect to w ) approximation to f in the subspace spanned by L_n .. L_m is:

    f(x)=\sum_{i=n}^{m} \frac{\langle f,L_i\rangle}{\|L_i\|}L_i(x)

    CB
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  5. #5
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    Re: Least Squares with Weight Function

    Not so bad after all. Thanks very much!
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