Real valued Fourier series in two dimensions with no assumption on boundary conditions

by Indrasis Mitra   Last Updated June 30, 2020 04:20 AM

For a two-dimensional real valued function with Dirichlet condition on the boundaries, the Fourier series representation is: $$ f(x,y)=\sum_{n=1}^\infty \sum_{m=1}^\infty c_{nm}\sin\left({n\pi\, x\over a}\right)\sin\left({m\pi\, y\over b}\right), \quad 0<x<a,\ 0<y<b \tag 1 $$

I would like to know the general representation of a two-dimensional Fourier series i.e. for a real-valued function of two variables in its trigonometric form. From general I mean that the representation should not assume Dirichlet conditions on the boundaries of the rectangular domain as Eq. $1$ does.

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