  In this paper we are concerned with the construction of a general principle
that will allow us to produce regular spectral triples with finite and simple
dimension spectrum. We introduce the notion of weak heat kernel asymptotic
expansion (WHKAE) property of a spectral triple and show that the weak heat
kernel asymptotic expansion allows one to conclude that the spectral triple is
regular with finite simple dimension spectrum. The usual heat kernel expansion
implies this property. Finally we show that WHKAE is stable under quantum
double suspension, a notion introduced by Hong and Szymanski. Therefore quantum
double suspending compact Riemannian spin manifolds iteratively we get many
examples of regular spectral triples with finite simple dimension spectrum.
This covers all the odd dimensional quantum spheres. Our methods also apply to
the case of noncommutative torus.
