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The AGT relations reduce S-duality to the modular transformations ofconformal blocks. It was recently conjectured that for the four-point conformalblock the modular transform up to the non-perturbative contributions can bewritten in form of the ordinary Fourier transform when$\beta\equiv-\epsilon_1/\epsilon_2=1$. Here we extend this conjecture togeneral values of $\epsilon_1,\epsilon_2$. Namely, we argue that for a properlynormalized four-point conformal block the S-duality is perturbatively given bythe Fourier transform for arbitrary values of the deformation parameters$\epsilon_1,\epsilon_2$. The conjecture is based on explicit perturbativecomputations in the first few orders of the string coupling constant$g^2\equiv-\epsilon_1\epsilon_2$ and hypermultiplet masses.

S-duality as Fourier transform for arbitrary $\epsilon_1,\epsilon_2$