Polynomial interpolation over quaternions (Q401383)

The author studies the Lagrange interpolation problem for polynomials over the skew field of real quaternions. A two-sided Lagrange problem is formulated: NEWLINE\[NEWLINE\begin{aligned} &\sum_{k=0}^n \alpha^k_{i} f_k =c_i\quad \text{ for} \quad i=1,\ldots,n\quad \\ &\sum_{k=0}^n \beta^k_{j} f_k =d_j\quad \text{for} \quad j=1,\ldots,m\quad \end{aligned} NEWLINE\]NEWLINE with data sets \(\{\alpha_1,\ldots,\alpha_n\}\) and \(\{\beta_1,\ldots,\beta_m\}\). For its unique solution, quaternionic polynomials (left,right), their zeros, minimal polynomials, conjugacy classes are introduced. Necessary conditions are formulated, which can be assumed without loss of generality. The solvability is given in terms of certain Sylvester equations with relations to some backward-shift operators in the ring of real quaternionic polynomials. More difficult is the problem with both ``left'' and ``right'' interpolation conditions. It is shown that there can exist infinitely many low-degree solutions. In the homogeneous case, the solution set forms a quasi-ideal in the ring of real quaternionic polynomials. One should note that in [\textit{K. Gürlebeck} and \textit{W. Sprößig}, Quaternionic analysis and elliptic boundary value problems. Berlin: Akademie-Verlag (1989; Zbl 0699.35007)] a quaternionic holomorphic (one-sided) Lagrange polynomial was constructed.