Zeta Function Regularization - Heat Kernel Regularization

Heat Kernel Regularization

The sum

is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the can sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that

where is a step function, with steps of at . A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if

then the series for converges in the half-plane and is uniformly convergent on every compact subset of the half-plane . In almost all applications to physics, one has