Heat and Noise on Cubes and Spheres: The Sensitivity of Randomly Rotated Polynomial Threshold Functions

We establish a precise relationship between spherical harmonics and Fourier basis functions over a hypercube randomly embedded in the sphere. In particular, we give a bound on the expected Boolean noise sensitivity of a randomly rotated function in terms of its spherical sensitivity, which we define according to its evolution under the spherical heat equation. As an application, we prove an average case of the Gotsman-Linial conjecture, bounding the sensitivity of polynomial threshold functions subjected to a random rotation.