  This article presents a reformulation of the Theory of Functional
Connections: a general methodology for functional interpolation that can embed
a set of user-specified linear constraints. The reformulation presented in this
paper exploits the underlying functional structure presented in the seminal
paper on the Theory of Functional Connections to ease the derivation of these
interpolating functionals--called constrained expressions--and provides
rigorous terminology that lends itself to straightforward derivations of
mathematical proofs regarding the properties of these constrained expressions.
Furthermore, the extension of the technique to and proofs in n-dimensions is
immediate through a recursive application of the univariate formulation. In
all, the results of this reformulation are compared to prior work to highlight
the novelty and mathematical convenience of using this approach. Finally, the
methodology presented in this paper is applied to two partial differential
equations with different boundary conditions, and, when data is available, the
results are compared to state-of-the-art methods.
