Polynomial L^2 approximations

Apr 2010
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Here is a study problem for my final exam tomorrow (its in 9 hours... help!):

Find the quadratic or lower-order polynomial that is the best approximation to cos(pie X) (using the L^2 norm) on the interval [-1, 1].

Then repeat for a quartic or lower-order polynomial. Hints: (1) You need an orthonormal set of polynomials.

Thanks!
 

dwsmith

MHF Hall of Honor
Mar 2010
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Here is a study problem for my final exam tomorrow (its in 9 hours... help!):

Find the quadratic or lower-order polynomial that is the best approximation to cos(pie X) (using the L^2 norm) on the interval [-1, 1].

Then repeat for a quartic or lower-order polynomial. Hints: (1) You need an orthonormal set of polynomials.

Thanks!

I don't understand your notation of \(\displaystyle L^2\) norm. What do you mean?
 
Apr 2010
26
0
i think the L^2 inner product on the interval [-1, 1]
 

dwsmith

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Mar 2010
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i think the L^2 inner product on the interval [-1, 1]
Is that this: \(\displaystyle <f,g>=\frac{1}{\pi}\int_{-1}^{1}f(x)g(x)dx\) in your book?
 
Apr 2010
26
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i have it as just the (f, g) = the integral of f(x) conjugate times g(x) dx... i.e. f(x) bar g(x) dx
 

CaptainBlack

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Here is a study problem for my final exam tomorrow (its in 9 hours... help!):

Find the quadratic or lower-order polynomial that is the best approximation to cos(pie X) (using the L^2 norm) on the interval [-1, 1].

Then repeat for a quartic or lower-order polynomial. Hints: (1) You need an orthonormal set of polynomials.

Thanks!
You either know what the first few polynomials of the orthonormal basis polynomials on the interval are (They are multiples of the Legendre polynomials, see here) or you should generate them using the Gram-Schmidt process.

let them be \(\displaystyle P_0\), \(\displaystyle P_1\), \(\displaystyle P_2\), ...

Then the best quadratic polynomial approximation to \(\displaystyle f(x)\in L^2_{[-1,1]}\) in the required sense is:

\(\displaystyle P(x)=\langle f,P_0 \rangle P_0(x) + \langle f,P_1 \rangle P_1(x) +\langle f,P_2 \rangle P_2(x)\)

where \(\displaystyle \langle .,. \rangle\) denotes the usual inner-product on \(\displaystyle L^2_{[-1,1]}\)

CB
 
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