Minimize integral using orthogonal basis

  • #1
psie
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Homework Statement
Determine the polynomial of degree at most that minimizes . (Hint: first find an orthogonal basis for a suitably chosen space of polynomials of degree .)
Relevant Equations
norm, inner product, least squares approximation.
I'm posting to inquire about a possible typo in the given answer in the back of the book, or if maybe I did something wrong, because my answer does not agree with the one stated.

So the exercise is about finding the least squares approximation. The norm is the norm and the corresponding inner product is I choose the polynomials and as a basis and make them orthogonal according to Gram-Schmidt, i.e. I find that and are orthogonal to each other under the inner product above. Put , then the orthogonal projection of on the subspace spanned by is given by Now, and
Plugging these into the equation for , gives but the answer given is . Is there a way to check one's answer to know if you got the right one?
 
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  • #2
Sorry, I just realized now I forgot to use instead of .:doh:
 
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  • #3
psie said:
Sorry, I just realized now I forgot to use instead of .:doh:
... and you tell me this now that I have literally checked every single integral and step and was right about to answer where your typo was ... :devil: :biggrin:
 
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