= Double Fourier sine series
{title2=$f(x,y)=\sum_{m,n\geq1}c_{mn}\sin(m\pi x/L_x)\sin(n\pi y/L_y)$}
On a rectangle $(0,L_x)\times(0,L_y)$, products of the one-dimensional <Fourier sine basis> give an orthogonal basis of <L2 space>. <Fourier orthogonality> in each variable gives
$$
c_{mn}=\frac4{L_xL_y}\int_0^{L_x}\int_0^{L_y}f(x,y)\sin\frac{m\pi x}{L_x}\sin\frac{n\pi y}{L_y}\,dy\,dx.
$$
For a constant function one, these coefficients are $4[1-(-1)^m][1-(-1)^n]/(\pi^2mn)$: only odd indices occur. The product modes vanish on the rectangle's boundary, making them natural <Dirichlet Laplacian eigenfunctions>. In the <heat equation>, their amplitudes decay as $e^{-\kappa\pi^2(m^2/L_x^2+n^2/L_y^2)t}$. Initial data not vanishing at the boundary are still represented in the interior and in the square-integrable sense; the positive-time boundary values need not agree continuously with the initially assigned boundary values.
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