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

by Javi Arizu   Last Updated May 15, 2018 20:20 PM

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.

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