I am aware of the typical Fourier coefficients of a $2 \pi$ periodic function $f$ given by it's $L^2$ inner product with the basis $e^{i n \theta} $ $n \in \mathbb{Z}$. However, an article I'm reading proposes the following setting. Consider the following circle action on $\mathbb{C}^2$ given by $$e^{i \theta}z = \big( e^{i \theta}z_1, e^{i \theta} z_2 \big)$$ where $z = (z_1, z_2) \in \mathbb{C}^2$. Suppose $V \in C^\infty (\mathbb{C}^2)$ is a smooth function. Let $\hat{V}(z)$ be defined by $$\hat{V}(z) = \frac{1}{2\pi}\int_0^ {2\pi}V(e^{i \theta}z) d\theta$$ What is the meaning of the following phrase?
The decomoposition of $\hat{V}(z)$ into it's Fourier coefficients with respect to the circle action $$e^{i \theta }z = \big( e^{i \theta}z, e^{- i \theta}z\big)$$
Note that the second and first circle action are different. A google search has found me no nice definition so a reference or explanation might be helpful. I suspect there's a general definition of fourier coefficients with respect to a circle action.