Orthogonal system in Sobolev Space $H^1(-\pi,\pi)$

Question

I am having trouble with this exercise:

Prove that $\{e^{int}\}_{n\in\mathbb{Z}}\bigcup \{sinh(t)\}$ constitute a complete orthogonal system in Sobolev space $H^1(-\pi,\pi)$, defined as:

$$H^1(-\pi,\pi):=\{f \in L^2(-\pi,\pi): f'\in L^2(-\pi,\pi) \}$$

which forms a Hilbert space, provided the inner product:

$$\langle f(t),g(t) \rangle_1 = \int_{-\pi}^{\pi}{f(t)\overline{g(t)}dt}+\int_{-\pi}^{\pi}{f'(t)\overline{g'(t)}dt}$$

I have so far established that it would be sufficient to prove that $\{e^{int}\}_{n\in\mathbb{Z}}^\bot$ has dimension 1 and $sinh(t)\in\{e^{int}\}_{n\in\mathbb{Z}}^\bot$

But I don't know how to prove that $\{e^{int}\}_{n\in\mathbb{Z}}^\bot$ has dimension 1.

There is also a hint: the standard Fourier coefficients of a function orthogonal to $\{e^{int}\}_{n\in\mathbb{Z}}$ are like:

$$g_n = \frac{c (-1)^n n}{n^2 + 1}$$ with $n \in \mathbb{Z}$ and $c \in \mathbb{C}$.

Any help would be very appreciated.

Answer 1

OK, I think I just overthought this. Once I get that

$$g_n = c\frac{(-1)^n n}{n^2+1}$$

It's clear that any such $g(t)\in \{e^{int}\}_{n \in \mathbb{Z}}^{\bot}$ could written as:

$$g(t)= \sum_{n\in \mathbb{Z} } g_n e^{int} = c \sum_{n\in \mathbb{Z}} \frac{(-1)^n n}{n^2 +1} e^{int}$$

which obviously lives in a one-dimensional space.

Now, if I prove that $sinh(t)\in \{e^{int}\}_{n \in \mathbb{Z}}^{\bot}$ the proof is done.

Please tell me if I am right or not. I am new on this topic.

Answer 2

The main tricky thing here is that you cannot really differentiate a given function in $H^1$ term-by-term since the resulting sum may diverge. Instead you want to integrate term-by-term, which is more likely to be safe. So if you consider arbitrary $g' \in L^2,g'=\sum_n g_n e^{int}$, then you can write $g=f_0 + g_0 t + \sum_{n \neq 0} \frac{g_n}{in} e^{int}$.

Then $\langle g,e^{int} \rangle_{H^1}=\langle g,e^{int} \rangle_{L^2} + \langle g',ine^{int} \rangle_{L^2}$. But now these two inner products can both be evaluated by interchanging integration and summation. In particular $\langle g',ine^{int} \rangle_{L^2}=g_n \langle e^{int},ine^{int} \rangle$ and $\langle g,e^{int} \rangle_{L^2}=f_0 \langle 1,e^{int} \rangle_{L^2} + g_0 \langle t,e^{int} \rangle_{L^2} + \frac{g_n}{in} \| e^{int} \|_{L^2}^2$. Sum those up and set them all equal to zero.