# Functional derivative in Fourier space

I was wondering how the functional derivative $$\frac{ \delta _{ } F}{ \delta _{ } f(x)}$$ would read in Fourier/reciprocal space? To give some more detail, I'm thinking of the functional $$F=F[f(x)]$$ and suppose that $$f(x)$$ can be written as the Fourier series $$f(x) = \sum \limits_{k}^{} \tilde{f}_k e^{i kx}$$. Can $$\frac{ \delta _{ } F}{ \delta _{ } f(x)}$$ be written in terms of $$\frac{\partial F}{\partial \tilde{f}_k}$$ or something similar?

Let $$F$$ be a functional and $$f$$ a function in the domain of $$F$$. Furthermore assume that $$f(x)=\sum_k f_k e_k$$ where $$f_k$$ are real or complex numbers and $$e_k$$ are some basis functions (e.g. $$e_k=e^{ik2\pi x/L}$$). Then, $$\frac{\partial F[f]}{\partial f_k} = \frac{\partial F[\small\sum_k f_k e_k]}{\partial f_k} = \int \frac{\delta F[\small\sum_k f_k e_k]}{\delta \small\sum_k f_k e_k(y)} \frac{\partial \small\sum_k f_k e_k(y)}{\partial f_k} dy = \int \frac{\delta F[f]}{\delta f(y)} e_k(y) \, dy .$$ The rightmost integral is an inner product $$\langle \delta F[f], e_k \rangle,$$ where $$\delta F[f](y) = \frac{\delta F[f]}{\delta f(y)},$$ so by inversion we can write $$\frac{\delta F[f]}{\delta f(y)} = \sum_k \frac{\partial F[f]}{\partial f_k} e_k(y).$$