On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

Abstract

This paper shows a hypercomplex function theory emerging in the representation of paravector-valued monogenic functions over the (m + 1)-dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well-known Whittaker-Cannon theorem to special monogenic functions defined in an open hyperball in R m+1 . More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three-hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non-Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non-Cannon basis.

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

“…This exploration established links with Bernoulli special monogenic polynomials, Euler special monogenic polynomials, and Bessel special monogenic polynomials. An intriguing research discovery is highlighted in [45], where the authors extended the well-known Whittaker-Cannon theorem in open hyperballs in by employing Hadamard's three-hyperballs theorem [7]. Specifically, the hypercomplex Cannon functions were proven to preserve the effectiveness properties of both Cannon and non-Cannon bases.…”