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
Read more about this topic: Zeta Function Regularization
Famous quotes containing the words heat and/or kernel:
“Even if you find yourself in a heated exchange with your toddler, it is better for your child to feel the heat rather than for him to feel you withdraw emotionally.... Active and emotional involvement between parent and child helps the child make the limits a part of himself.”
—Stanley I. Greenspan (20th century)
“All true histories contain instruction; though, in some, the treasure may be hard to find, and when found, so trivial in quantity that the dry, shrivelled kernel scarcely compensates for the trouble of cracking the nut.”
—Anne Brontë (1820–1849)