TITLE: Completeness of the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ in $L^2[0,1]$
QUESTION [5 upvotes]: In the book R. M. Young, An introduction to non-harmonic Fourier series, I came across the following problem (page 18):
Problem. Show that the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ is complete in $L^2[0,1]$.
I tried to apply the Müntz-Szasz theorem but it didn't work out. Any ideas or approaches how to show completeness here? Thanks!

REPLY [3 votes]: This is just a comment on the present responses which I am adding in the hope that it will be of interest--it will be too long for that format.  Firstly, it treats  completeness for more general functions of the form $$\frac1{x+\lambda_n}$$ where $(\lambda_n)$ is a sequence of positive numbers (even complex ones with positive real parts) and displays the connection with the Müntz-Szasz theorem.
It also uses the Hahn Banach theorem to connect  with uniqueness theorems for spaces of analytic functions as is done  above.
We begin by replacing the $L^2$-space with $C([0,1)$--anything complete in the latter remains so in the former.  We now consider for any measure $\mu$  the function $$f(z)=\int_0^1 \frac{d\mu(x)}{x+z}.$$  This is bounded and analytic in the right half plane.  We now use the fact that a sequence $(\lambda_n)$ there is the zero set of such a function if and only if $$\sum\left(1-\left|\frac{\lambda_n-1}{\lambda_n+1}\right|\right)<\infty.$$
This suffices to prove your result, together with a wide-ranging set of generalisations.
The same proof, with the definition
$$f(z)=\int_0^1 x^zd\mu(x),$$
can be used to prove the classic Müntz-Szasz theorem (which assumes that the sequence increases to infinity along the real line), but also variants which allow it to converge in more exotic manners to the boundary of the half-plane.