= Solution
For period two in each direction, use <Fourier coefficients>
$$
f_k=\frac14\int_{-1}^1\int_{-1}^1f(x,y)e^{-i\pi(k_1x+k_2y)}\,dx\,dy,\qquad k\in\mathbb Z^2.
$$
Integration of the periodic <Poisson equation> gives the necessary compatibility condition $f_0=0$. With it, the normalization fixes $u_0=0$ and every other <Fourier mode> gives $-\pi^2|k|^2u_k=f_k$. The Fourier-Galerkin approximation is therefore
$$
\boxed{u_N(x,y)=-\sum_{\substack{|k_1|,|k_2|\le N\\k\ne0}}\frac{f_k}{\pi^2|k|^2}e^{i\pi(k_1x+k_2y)}.}
$$
One may evaluate the coefficients by a <discrete Fourier transform> for a pseudospectral implementation, with the corresponding quadrature/aliasing error distinguished from truncation.
Convergence with <spectral accuracy> means faster decay than every fixed algebraic power of $N$. A smooth periodic extension gives this rate; an analytic periodic extension to a complex strip gives the stronger exponential rate $O(e^{-cN})$ on a slightly narrower strip. Division by $|k|^2$ preserves that Fourier decay, so the normalized solution has the same type of spectral convergence.
\b[Two qualifications are required by the literal source assumptions.] The mean-zero condition on $f$ was not stated: $f\equiv1$ meets the printed analyticity and endpoint equalities but has no periodic solution. Also endpoint equality of $f$ alone does not make its periodic extension analytic. For example $f(x,y)=x^2-1/3$ is analytic, has matching endpoint values and zero mean, but its nonzero <Fourier coefficients> are $f_{(k,0)}=2(-1)^k/(\pi^2k^2)$. Then $u_{(k,0)}=-2(-1)^k/(\pi^4k^4)$, with only algebraic decay. Thus the standard spectral-speed conclusion is valid under the intended analytic-periodic torus assumption and compatibility, not for every function satisfying only the displayed endpoint values.
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