Practical interpolation schemes for functions of more than one variable need not be finite dimensional as in the case of univariate functions. The so-called “blending function methods” are one example of such transfinite schemes. Another is the method of “pseudo-harmonic interpolation” described in this paper. For any bounded, convex planar domain $\mathcal{D}$, we show how to construct a bivariate function U which interpolates to arbitrary Dirichlet boundary conditions on the perimeter of $\mathcal{D}$. The interpolant $U(Q)$ satisfies the familar “maximum principle” and, in the case in which the domain $\mathcal{D}$ is circular, is actually a harmonic function in $\mathcal{D}$. Higher order schemes are described in which the function $U(Q)$ interpolates to specified normal derivatives on the boundary of $\mathcal{D}$ as well as to the Dirichlet boundary conditions. The interpolation schemes are also viewed as idempotent linear operators (i.e., projectors) on families of continuous functions on $\overlin...