Generalized harmonic functions

Harmonic analysis may be viewed as the decomposition and construction of functions from simpler components, in which harmonic functions have played no small part. That these functions can be represented or assembled from homogeneous polynomials that are themselves harmonic is one of its well-known features, and emerging theories that extend beyond classical harmonic functions lend further support to the importance of these expansions.
 

These later investigations have contributed to the notion of a generalized harmonic function, or a function that satisfies an equation which, in some sense or another, resembles that of Laplace. To account for the commonalities between these various generalizations and their harmonics in the case of the plane, certain symmetry constraints are imposed on the algebra ⟨z, z¯, ∂, ∂¯⟩ of Weyl, which lead to a natural notion for such functions. We illustrate this with a few examples and discuss some of the prominent cases that have led to these ideas.