On a version of quaternionic function theory related to Chebyshev polynomials and modified Sturm-Liouville operators

In the last few years considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics and filter design played a key role in this development. It is pointed out in this paper that the operator $\mathcal {W}$ arising in the Helmholtz equation after the prolate spheroidal change of variables is the sum of three operators, $\mathcal {S}_{\xi ,\alpha }$, $\mathcal {S}_{\eta ,\beta }$ and $\mathcal {T}_{\phi }$, each of which acts on functions of one variable: two of them are modified Sturm-Liouville operators and the other one is, up to a variable coefficient, the Chebyshev operator. We believe that this fact reflects the essence of the separation of variables method in this case. We show that there exists a theory of functions with quaternionic values and of three real variables which is determined by the Moisil–Theodorescu-type operator with quaternionic variable coefficients, and that it is intimately related to the modified Sturm-Liouville operators and to the Chebyshev operator (we call it in this way, since its solutions are related to the classical Chebyshev polynomials). We address all the above and explore some basic facts of the arising quaternionic function theory. We further establish analogues of the basic integral formulae of complex analysis such as those of Borel-Pompeiu, Cauchy, and so on, for this version of quaternionic function theory. We conclude the paper by explaining the connections between the null-solutions of the modified Sturm-Liouville operators and of the Chebyshev operator, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other.