  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 $N\to \infty$. 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.
