I am studying Fourier series and I am wanting to understand weighted sums of the form
$$ \sum_{n=-\infty}^{\infty}|c_n|^{r} (|n|+1)^{r-2}, $$
where, conceivably, $c_n$ denotes the Fourier coefficients of some function. As I understand, even if $\sum_{n=-\infty}^{\infty} |c_n|^r < \infty$ for $r \neq 2$, it need not be the case that $c_n$ are Fourier coefficients of a $L^r$ function (in contradistinction with the $r=2$ case). However, such weighted sums were used by Hardy and Littlewood when studying rearrangements of Fourier coefficients and when certain $c_n$ correspond to Fourier coefficients of $L^r$ functions. I understand the weighted sums in this context, but I would really like to know why, if it is the case, the weights $(|n|+1)^{r-2}$ are natural in the context of $L^r$ functions. Why would Hardy and Littlewood consider this weight in the first place?
The relevant result is that (for $r>2$), if $c_n$ comprise a set of numbers and $c_n^{\ast}$ denotes the symmetric decreasing rearrangement of their moduli $|c_n|$, then $$ \sum_{n=-\infty}^{\infty}|c_n^*|^{r} (|n|+1)^{r-2}<\infty, $$ is necessary and sufficient for the $c_n$ to correspond to the Fourier coefficients of a $L^r$ function.
note: $L^r = L^r(\mathbb{T})$, where $\mathbb{T}$ is the 1-dimensional torus.
Hakkeyoi!