# Confusion related to density estimation

I have this confusion related to how this density was estimate. I have attached the screenshot of the paper

Any references related to basis expansion and all. I didn't get how this was derived actually.

The space of square integrable functions (in particular densities) can be expressed as a series defined in terms of an orthonormal basis using a Fourier series expansion

$$f(x)=\sum_{j=1}^{\infty} c_ne^{inx},$$

where $c_n = \dfrac{1}{2n}\int_{-\pi}^{\pi} f(x) e^{inx}dx$.

A similar representation can be obtained using wavelets.

All of this is related to intrinsic properties of Hilbert spaces:

An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable.

The result of interest might be the following

Theorem. Every Hilbert space $H\neq\{0\}$ has an orthonormal basis.

In my opinion, this question is very wide but hopefully this answer will give you some guidelines to continue your study in the direction of interest.