Find the minimum of an integral

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December 18th 2011, 07:01 AM
cristi92

Find the minimum of an integral

Find $\min_{a,b \in \mathbb{R}}\int_{0}^{2\pi }(1+x-(a+b\cos x))^{2}dx$ without using mathematical analysis or calculus.

Do you have any ideea how can I do this?
December 18th 2011, 07:13 AM
FernandoRevilla

Re: Find the minimum of an integral

Quote:

Originally Posted by cristi92

Find $\min_{a,b \in \mathbb{R}}\int_{0}^{2\pi }(1+x-(a+b\cos x))^{2}dx$ without using mathematical analysis or calculus. Do you have any ideea how can I do this?

Hint On the space $\mathcal{C}[0,2\pi]$ consider the inner product $<f,g>=\int_0^{2\pi}f(x)g(x)dx$ . Then, the integral is the square of the norm of $1+x-(a+b\cos x)$ .
December 18th 2011, 07:21 AM
cristi92

Re: Find the minimum of an integral

Thank you!
December 18th 2011, 07:32 AM
ILikeSerena

Re: Find the minimum of an integral

Hmm, I don't get this. :confused:

I just verified that the result is ${2\pi^3 \over 3}$ .
But how do you get that from the norm?
December 18th 2011, 12:55 PM
FernandoRevilla

Re: Find the minimum of an integral

Quote:

Originally Posted by ILikeSerena

Hmm, I don't get this. :confused: I just verified that the result is ${2\pi^3 \over 3}$ . But how do you get that from the norm?

The given integral is $I(a,b)=\left\|{1+x-(a+b\cos x)}\right\|^2=d^2(1+x,a+b\cos x)$ , so the minimum of $I(a,b)$ is the square root of the distance of $1+x$ to its orthogonal projection onto $\textrm{Span}[1,\cos x]$ .
December 18th 2011, 10:55 PM
ILikeSerena

Re: Find the minimum of an integral

Nice! :)

I'm still wondering though how to arrive at the result.... :confused:

The only way I can think of, is calculating the integrals of (1+x) with 1 and with cosx (after norming them) to find the projection, which is (1+pi).
And then calculate the integral of ((1+x)-(1+pi))^2, which is indeed $2\pi^3 \over 3$ .

But isn't this analysis and calculus?
Is there an easier way?
December 18th 2011, 11:24 PM
FernandoRevilla

Re: Find the minimum of an integral

Quote:

Originally Posted by ILikeSerena

But isn't this analysis and calculus? Is there an easier way?

Well, there is no strict boundary between Calculus and Algebra, but if you use the inner product and the concept of orthogonal projection we can consider more an algebraic than analytic method. Besides, if we compute the Gram matrix $G$ with respect to the basis $B=\{1,x\cos x\}$ of $V=\textrm{span}[1,x,\cos x]$ we can express $<f(x),g(x)>=\sqrt{[f]_B^t\;G\;[g]_B}$ and from this point, the problem is purely algebraic.
December 19th 2011, 10:30 AM
ILikeSerena

Re: Find the minimum of an integral

I see.
Thanks!