Hadamard transforms and analysis on Cayley–Dickson algebras

This article explores the innovative use of Hadamard transforms in Hermitian Clifford analysis within Cayley–Dickson algebras. The study focuses on the integration of the Hadamard matrix into these algebras, highlighting its role in establishing a crucial subgroup of the automorphism group. This involves treating each row vector of the matrix as a diagonal matrix. A key finding is the transformation of the Witt basis into the classical basis under the influence of the Hadamard transforms, illustrating a deep connection between these elements. The research also reveals how Hermitian Clifford vectors correspond to twisted vectors through these transforms. Moreover, it introduces a novel method for converting vectors into Cayley–Dickson numbers, expanding the scope of vector manipulation. This advancement allows for the expansion of unit-length vectors into orthogonal matrices, significantly enriching the field of Clifford analysis. Hadamard transforms also facilitate the definition of Hermitian Dirac operators, intricately connected to split complex numbers. The research establishes the Borel-Pompeiu integral formula and the Teodorescu transform for these operators, drawing on their relationship with twisted complex structures. This approach not only uncovers direct relationships between orthogonal and Hermitian Clifford analysis but also introduces the Almansi decomposition and Hermitian Clifford wavelet transforms.