Least energy quaternionic regular Lagrange interpolation

✍ Scribed by Alessandro Perotti

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 193 KB
• Volume: 33
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1208

✦ Synopsis

Abstract

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 that minimizes the Dirichlet energy on a domain containing the points. Copyright © 2009 John Wiley & Sons, Ltd.