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Main goal of this paper is to generalize Hadamard's real part theorem and invariant forms of Borel-Carathéodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in the framework of quaternionic analysis.

We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.

Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.

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Communicated by S. G. GeorgievComplete orthogonal systems of monogenic polynomials over 3D prolate spheroids have recently experienced an upsurge of interest because of their many remarkable properties. These generalized polynomials and their applications to the theory of quasi-conformal mappings and approximation theory have played a major role in this development. In particular, the underlying functions of three real variables take on values in the reduced quaternions (identified with R 3 ) and are generally assumed to be null-solutions of the well-known Riesz system in R 3 . The present paper introduces and explores a new complete orthogonal system of monogenic functions as solutions to this system for the space exterior of a 3D prolate spheroid. This will be made in the linear spaces of square integrable functions over R. The representations of these functions are explicitly given. Some important properties of the system are briefly discussed, from which several recurrence formulae for fast computer implementations can be derived.

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