An interplay between parameter \((p, q)\)-Boas transform and linear canonical transform (Q2050510)

Exhaustive references document the development of the theory of the linear canonical transform (LCT) and the Boas transform. A brief description of the linear canonical transform which contains four parameters given by Eqn. (1.1) is given. The Hilbert transform and its generalization such as fractional Hilbert transform (FRHT), which play a role for optical analysis and signal processing, are studied. The analytical signals associated with LCT and its applications are explained. The Boas transform (BT) was introduced by \textit{R.~P. Boas jun.} [Trans. Am. Math. Soc. 40, 287--308 (1936; Zbl 0015.21301)], and its relationship with the Hilbert transform is shown in Eqn. (1.7). The introductory part of the present paper also covers the concept of Boas transform and its generalization, the fractional Boas transform (FRBT) related to wavelets and signal processing. In Section~2, the authors employ the generalization of Boas transform known as the parameter \((p,q)\) Boas transform (PBT) which defines complex signals associated with PBT and LCT. The complex signals in the LCT domain are studied, and it is suggested that the resulting signals have no non-zero frequency components in comparison with the case of generalized analytic signals related to PHT. Further, the authors investigate the generalized Boas product transform (Theorem~2.4) which is an analog of Bedrosian's theorem in the LCT domain. Section~3 describes the generalization of BT by combining a rotation parameter and LCT with the classical BT, which is represented as \(\eta\)-LCBT (Definition~3.1) and some properties are studied. The authors point out that the \(\eta\)-LCBT does not satisfy the semigroup property. Then four versions of \(\eta\)-LCCSs are formulated, and their linear canonical transform spectrum and eigenfunction properties are analysed.