Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

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SUMMARY

To find a polynomial orthogonal to f(x) = x² - 1/2 on the interval L2[0,1], one can use a linear polynomial of the form Ax + B. By integrating the product of Ax + B and x² - 1/2 over the interval from 0 to 1, the coefficients A and B can be determined such that the integral equals zero. The simplest solution identified in the discussion is f(x) = x, which satisfies the orthogonality condition.

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mandygirl22
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Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1].

I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do not know anything about it and the book does not make any sense out of it. Thanks for your help!
 
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Well, just DO it! The simplest kind of polynomial is linear. Integrate Ax+ B times [math]x^2- 1/2[/math] from 0 to 1 and choose A and B so the integral is 0.
 
*hits forehead* I knew there had to be some simple way of doing it that I was ignoring! Thanks!

(and the REALLY HARD solution ended up being f(x)=x) :smile:
 

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