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In this paper we construct two distinct generalized holomorphic orthogonal function systems over infinite cylinders in [ABSTRACT FROM AUTHOR]

Morais has recently introduced certain complete orthogonal sets of monogenic polynomials over 3D prolate spheroids with remarkable properties. The underlying functions take on either values in the reduced and full quaternions (identified, respectively, with ℝ3 and ℝ4), and are generally assumed to be nullsolutions of the well known Riesz and Moisil-Théodoresco systems in ℝ3. In continuation of these studies, we recall some fundamental properties of the polynomials, and prove some recursive formulae between them. As a consequence, we obtain a two-term type recurrence relation satisfied by those basis polynomials. These results are then employed to investigate a rather wide class of approximation properties for monogenic functions over 3D prolate spheroids in terms of spheroidal monogenics. [ABSTRACT FROM AUTHOR]

Presents information on a study which generalized the preconditioner proposed by Potts and Steidl to solve ill-conditioned block Toeplitz systems. Construction of preconditioners; Clustering of eigenvalues; Convergence rate and operation cost of solving the block-Toeplitz-Toeplitz-block matrix; Mathematical model of the linear image restoration problem.

S. G. Georgiev, Complete 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 ) and are generally assumed to be null-solutions of the well-known Riesz system in . 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 . 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. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Mehler formula for the Hermite polynomials allows for an integral representation of the one–dimensional Fractional Fourier transform. In this paper, we introduce a multi–dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials of Clifford analysis. [ABSTRACT FROM AUTHOR]