Adaptive Fourier Decomposition of Slice Regular Functions

A Cauchy kernel for slice regular functions

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results. AMS Classific...

FFT formulations of adaptive Fourier decomposition

Extension results for slice regular functions of a quaternionic variable

In this paper we prove a new representation formula for slice regular functions, which shows that the value of a slice regular function f at a point q = x + yI can be recovered by the values of f at the points q + yJ and q + yK for any choice of imaginary units I, J,K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a...

Adaptive Fourier Decomposition of Functions in the Orthogonal Rational System of Quaternionic Values

We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems {Bn(x)} which can be obtained in the respective contexts through Gram-Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic-valued signals of finite energy. This study is a generalization of the m...

A Phragmén - Lindelöf principle for slice regular functions

The celebrated 100-year old Phragmén-Lindelöf theorem, [15, 16], is a far reaching extension of the maximum modulus theorem for holomorphic functions that in its simplest form can be stated as follows: Theorem 1.1. Let Ω ⊂ C be a simply connected domain whose boundary contains the point at infinity. If f is a bounded holomorphic function on Ω and lim supz→z0 |f(z)| ≤ M at each finite boundary p...