# How to change variables if you only know the invariant measure

I want to find an equation relating the coefficients of two different harmonic expansions of the same function (and a relation between their respective basis functions).

$$f_1\left(x\right)=\sum_{}^{} a_n \Phi_n\left(x\right)$$

$$f_2\left(y\right)=\sum_{}^{} b_k \Psi_k\left(y\right)$$

The functions should be equal, but they are expressed in terms of different variables that exist in 1D spaces with different invariant measures. For the first function I have $f_1\left(x\right)$ with invariant measure $\text{d}x$, while for function 2 I have $f_2\left(y\right)$ with invariant measure $\sin\left(y\right)\text{d}y$. Both functions are periodic with period $2\pi$, and $x\in\left[0,2\pi\right)$, $y\in\left[0,2\pi\right)$, and I know that:

$$\int_{0}^{2\pi}f_1\left(x\right)\text{d}x=1$$

and

$$\int_{0}^{2\pi}f_2\left(y\right)\sin\left(y\right)\text{d}y=1$$

both functions are probability density functions whose argument is an angle. In a sense, $f_1\left(x\right)$ is a pdf on the unit circle, while $f_2\left(y\right)$ is a pdf on a deformed unit circle. So the functions should be equal point wise, i.e. $f_1\left(x=a\right)=f_2\left(y=a\right)$. Is there a way to find the relationship between $x$ and $y$?

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Equate the percentiles of the two distributions, so that $\int_0^x f_1(t) dt = \int_0^{Y(x)} f_2(t) \sin(t) dt$. (Or maybe you want the second without the $\sin t$, but the same idea.)