On transfer operators on the circle with trigonometric weights

We study spectral properties of the transfer operators $L$ defined on the circle $\mathbb T=\mathbb R/\mathbb Z$ by $$(Lu)(t)=\frac{1}{d}\sum_{i=0}^{d-1} f\left(\frac{t+i}{d}\right)u\left(\frac{t+i}{d}\right),\ t\in\mathbb T$$ where $u$ is a function on $\mathbb T$. We focus in particular on the cases $f(t)=|\cos(\pi t)|^q$ and $f(t)=|\sin(\pi t)|^q$, which are closely related to some classical Fourier-analytic questions. We also obtain some explicit computations, particularly in the case $d=2$. Our study extends work of Strichartz \cite{Strichartz1990} and Fan and Lau \cite{FanLau1998}.