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Problem Statement: We assume the following: $L^2[-\pi,\pi]$ is a real Hilbert space with the inner product

$$\langle f,g\rangle = \int_{-\pi}^\pi f(x)g(x)dx$$

and the set

$$\left\{\frac{1}{\sqrt{2\pi}},\frac{\sin{nx}}{\sqrt{\pi}},\frac{\cos{nx}}{\sqrt{\pi}} : n=1,2,3,\dotsc\right\}$$

is an orthogonal basis for $L^2[-\pi,\pi]$. Let $V$ be the vector space spaneed by the set

$$\left\{\frac{1}{\sqrt{2\pi}},\frac{\sin{x}}{\sqrt{\pi}},\frac{\cos{x}}{\sqrt{\pi}},\frac{\sin{2x}}{\sqrt{\pi}},\frac{\cos{2x}}{\sqrt{\pi}},\right\}$$

Find a function $g \in V$ such that $$ \int_{-\pi}^\pi xf(x) dx = \int_{-\pi}^\pi g(x)f(x)dx $$ for all $f \in V$, and show that $g$ is unique in $V$. Then, find all possible solutions $g \in L^2[-\pi,\pi]$ that satisfy the above condition.

My attempt: I have no idea what I'm doing with this one. I know that any $f\in V$ has a unique representation as a linear combination of elements of the basis given, so

$$f = \sum^5_{i=1}\langle f, e_i\rangle e_i$$

where $e_i$ is the $i$th element of the basis of $V$ given above.

Now, in Hilbert space language, we are trying to find $g \in V$ such that

$$\langle x,f\rangle = \langle g,f \rangle$$

for all $f\in V$.

But short of trying to write out a linear combination of the basis elements and evaluating the integrals directly, which I did try and it got messy, I'm unsure of where to go from here. A hint would be much appreciated.

poppy3345
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  • concerning the question: It is just a linear equations on a $5$ dimensional space and you have a orthonormal basis... – user251257 Sep 09 '15 at 23:30
  • How much Hilbert space theory are you familiar with? Do you know the projection theorem? Radon Nikodym theorem? – Jason Sep 09 '15 at 23:40
  • @user251257 I guess I'm not sure how to write out the linear equations you've mentioned for this problem. I had, at one point, tried to write $g(x) = a_1 e_1 + a_2 e_2 + ... + a_5 e_5$, as well as $f(x) = b_2 e_1 + b_2 e_2 + ... + b_5 e_5$ and calculate the integrals directly, but it got pretty messy, and I know there's a better way to do it... is that what you mean by linear equations, or is there something else with linear equations that I should try? – poppy3345 Sep 09 '15 at 23:42
  • @Jason I don't know either of those theorems by name, but I'll look them up and try to work with them on this problem. – poppy3345 Sep 09 '15 at 23:46
  • @Jason, why do you need Radon Nikodym? – user251257 Sep 09 '15 at 23:47
  • Not sure how helpful it would be, now that I'm looking at the problem properly. At first glance it looked like the construction of conditional expectation, but this is a concrete calculation rather than theory. See user251257's answer. – Jason Sep 09 '15 at 23:48

1 Answers1

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Proof sketch / Hints:

Since the equation is linear, $\langle f, g \rangle = \langle f, \operatorname{id} \rangle$ holds for every $f\in V$ if and only if it holds every $f$ in the orthonormal basis you gave above. Thus, we are looking at $$ \langle e_i, g \rangle = \langle e_i, \operatorname{id} \rangle $$ for $i = 1,\dotsc, 5$. Now, using $g = \sum_{j=1}^5 \gamma_j e_j$, we obtain $$ \sum_{j=1}^5 \gamma_j \langle e_i, e_j \rangle = \langle e_i, \operatorname{id} \rangle. $$ Due to orthonormality, we obtain $$ \gamma_i = \langle e_i, \operatorname{id} \rangle. $$

Now, $L^2[-\pi, \pi]$ also have an orthonormal basis...

Notes: About half of the coefficients $\gamma_i$ would be zero.

user251257
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