Abstract
AbstractWe 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 ℂp be the complex plane generated by 1 and p and let Jp be the corresponding complex structure on ℍ. We identify totally regular variables with real‐affine holomorphic functions from (ℍ, Jp) to (ℂp, Lp), where Lp is the complex structure defined by left multiplication by p. We show that every Jp‐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 that minimizes the Dirichlet energy on a domain containing the points. Copyright © 2009 John Wiley & Sons, Ltd.
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