Least energy quaternionic regular Lagrange interpolation

We revisit the concept of a totally regular variable of functions in one quaternionic variable and its application to Lagrange interpolation. We consider left--regular functions in the kernel of a modified Cauchy--Fueter operator. For every imaginary unit p, let C_p be the complex plane generated by 1 and p and let J_p be the corresponding complex structure on H. We identify totally regular variables with real--affine holomorphic functions from (H,J_p) to (C_p,L_p), where L_p is the complex structure defined by left multiplication by p. We show that every J_p--biholomorphic map gives rise to a family of Lagrange interpolation formulas for any set of N distinct points in H. In the case of quaternionic regular polynomials of degree at most N, there exists a unique regular interpolating polynomial which minimizes the Dirichlet energy on a domain containing the points.