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Gaussian bounds on noise correlation of functions play an important role inhardness of approximation, in quantitative social choice theory and in testing.The author (2008) obtained sharp gaussian bounds for the expected correlationof $\ell$ low influence functions $f^{(1)},\ldots, f^{(\ell)} : \Omega^n \to[0,1]$, where the inputs to the functions are correlated via the $n$-foldtensor of distribution $\mathcal{P}$ on $\Omega^{\ell}$.  It is natural to ask if the condition of low influences can be relaxed to thecondition that the function has vanishing Fourier coefficients. Here we answerthis question affirmatively. For the case of two functions $f$ and $g$, wefurther show that if $f,g$ have a noisy inner product that exceeds the gaussianbound, then the Fourier supports of their large coefficients intersect.

Gaussian Bounds for Noise Correlation of Resilient Functions