The study of convolution powers of a finitely supported probability distribution [phi] on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power phi is the distribution of the nth step of the associated random walk. In the case that the random walk is aperiodic and irreducible, phi is well approximated by a single and appropriately scaled Gaussian density; this is the local (central) limit theorem. When such functions are allowed to be complex-valued, their convolution powers are seen to exhibit rich and disparate behavior, much of which never appears in the probabilistic setting. In the first half of this thesis, we study the asymptotic behavior of the convolution powers of complex-valued functions on Zd . This problem, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex-valued function [phi] on Zd , we ask and address four basic and fundamental questions about the convolution powers phi which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. In one dimension, we give a complete theory of sup-norm estimates and local limit theorems for the entire class of finitely supported complex-valued functions. This work extends results of I. J. Schoenberg, T. N. E. Greville, P. Diaconis and L. Saloff-Coste and, in the context of stability theory, results by V. Thom´ e and M. V. Fedoryuk. e In the second half of this thesis, we consider a class of "higher order" homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These socalled positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. We then introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has ¨ Holder continuous coefficients, we construct a fundamental solution to its heat equation by the method of E. E. Levi, adapted to parabolic systems by A. Friedman and S. D. Eidelman. Though our results in this direction are implied by the long-known results of S. D. Eidelman for 2b-parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an offdiagonal estimate, i.e., a heat kernel estimate, written in terms of the LegendreFenchel transform of the operator's principal symbol - an estimate which is seen to be sharp in many cases. We then turn to the study of such variablecoefficient operators whose coefficients are, at worst, bounded and measurable and we study their associated heat kernels. Following functional-analytic techniques of E. B. Davies and G. Barbatis, we prove heat kernel estimates in terms of the Legendre-Fenchel transform subject to a dimension-order restriction. Our work in this measurable-coefficient setting extends results of E. B. Davies and partially extends results of A. F. M. ter Elst and D. Robinson. All work in this thesis was done in collaboration with Laurent Saloff-Coste.