The mathematical theory of multipole translation operators for the three-dimensional Laplace and Helmholtz equations is summarized and extended. New results for the Laplace equation include an elementary proof of the inner-to-inner translation theorem, from which follows the definition of a far-field signature function analogous to that of the Helmholtz equation. The theory for the Helmholtz equation is developed in terms of a new convolutional form of the translation operator, which is connected to Rokhlin's diagonal form by means of Wigner 3-j symbols.
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