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Fourier-Walsh transform

{title2=$T_\rho$}
{wiki=Noise_stability}

The noise operator averages $f(y)$ over a random $y$ correlated with $x$ by $\rho$. It acts diagonally on the <Fourier-Walsh transform>:
$$
T_\rho f=\sum_S\rho^{|S|}\widehat f(S)\chi_S.
$$

Noise operator on the Boolean hypercube

Noise stability

Hypercontractive inequality on the Boolean hypercube

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 168 / 1 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 168 / 2 / i / Solution

Noise operator on the Boolean hypercube ( $T_{ρ}$ )

Noise stability ( $Stab_{ρ} (f)$ )

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Noise operator on the Boolean hypercube (Tρ​)

Noise stability (Stabρ​(f))