Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall p<2$.
I thought taking a function $f=\sum_{n=1}^\infty \alpha_n f_n$ with $\sum\alpha_n<\infty$ and $\Vert f_n\Vert_\infty \le 1$ which will produce a convergent series on the right side (maybe a power series).
How can I choose my $f=\sum \alpha_n f_n$?
EDIT: After Math1000 answered the case $p\in[0,1]$ I still need to figure out the case when $p\in(1,2)$. Any help will be appreciated.
Consider the sawtooth wave $f(x)=\frac1\pi x$, $x\in(-\pi,\pi]$. The Fourier coefficients are \begin{align} a_0 &= 0\\ a_n &= \frac1\pi \int_{-\pi}^\pi \frac1\pi x\sin nx\ \mathsf dx = \frac2{n^2}\sin n\pi -\frac2n\cos n\pi = (-1)^{n+1}\frac2n,n\geqslant 1\\ b_n &= \frac1\pi \int_{-\pi}^\pi \frac1\pi x\cos nx\ \mathsf dx =\frac 2n\sin n\pi = 0,\ n\geqslant 1. \end{align} So the Fourier series converges, i.e. for $x\in (-\pi,\pi]$, $$\sum_{n=1}^\infty (-1)^{n+1}\frac2n\sin nx = \frac1\pi x,$$ but the Fourier coefficients $a_n$ are not in $\ell^p$, $0<p\leqslant 1$ as $$\sum_{n=1}^\infty |a_n|^p = 2^p\sum_{n=1}^\infty\frac1{|n|^p}=\infty.$$ To find an example where the coefficients $\hat f(n)$ are not in $l^p$ for a given $1<p<2$ we would need $\hat f(n)\in\Theta((2-p)^n)$. I am not sure if such there exists a such sequence of coefficients that satisfies this condition for all $p\in(0,2)$.