We prove that the spectral structure on the N -dimensional standard sphere of radius (N − 1) 1/2 compatible with a projection onto the first n-coordinates converges to the spectral structure on the n-dimensional Gaussian space with variance 1 as N → ∞. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space. 2 2α 2 dx.