It is known that the system of functions $\{e^{inx}\}_{n \in \mathbb{Z}}$ is complete in $L^2[- \pi, \pi]$, and moreover, it is an orthonormal system.
What happens when you substitute one of the functions with a non integer exponent?
More exactly, what happens if you remove $e^{i0x}=1$ from the system, and add $e^{i0.1x}$? How can I check if the new system is complete? If $e^{i0x}=1$ can be approximated using the new system, then it is complete. How can I show this?
Thank you.
Suppose $e_1$, $e_2,e_3,\ldots$ is a complete orthonormal sequence in a Hilbert space. Let $f\in H$. When is $f$, $e_2,e_3,\ldots$ also a complete orthonormal sequence?
We can write $f=\sum_{n}a_n e_n$ for some square-summable sequence $(a_n)$. If $a_1=0$ then $f$, $e_2,e_3,\ldots$ isn't a complete orthonormal sequence, as its closed span avoids $e_1$. But if $a_1\ne0$ then it is a complete orthonormal sequence, as its closed span contains $a_1e_1=f-\sum_{n=2}^\infty a_ne_n$.