The quaternion Fourier Transform (QFT) finds extensive applications across various domains, such as signal processing and optics. This paper strengthens two important uncertainty principles for QFT. First, we proposed two different sharper versions of generalized Heisenberg's uncertainty principle associated with quaternion Fourier trans-form in LP(R2, H) space for p? [1, 2]. Also a new version of Donoho Stark's uncertainty principle associated with QFT is discussed. Furthermore, in some particular case this new Donoho-Stark's uncertainty principle is the sharper version than the existing one in quaternion sense.

In recent years quaternion linear canonical transform (QLCT) has emerged due to its applications in various fields, including image and signal processing. This article discusses two spectrum-related theorems (real Paley–Wiener and Boas type). The real Paley–Wiener type theorem is formulated to describe the character of a compactly supported two-sided quaternion linear canonical transformed (QLCT) signal. The Boas-type theorem is also discussed to explain the property of right-sided QLCT of signals that vanish in the neighborhood of origin. Some potential applications of these theorems on some particular quaternion-type operators are also discussed.

The quaternion quadratic phase Fourier transform (QQPFT), an extension of the well-known quaternion Fourier transform (QFT), has emerged as a significant advancement in signal processing and optics. In this study, we aim to provide a direct proof of the Plancherel theorem within the context of the QQPFT. Specifically, we establish the theorem of the scalar inner product for the two-sided QQPFT and explore the quaternion inner product concept for the right-sided QQPFT. Additionally, we present a proof of the Plancherel theorem for quaternion values in the left-sided QQPFT. Also, we discuss the asymptotic behavior of the two-sided QQPFT and the right-sided QQPFT. Finally, as an application, we discuss the solution of some generalized quaternion differential equations

We explore various uncertainty principles within the framework of the quaternion quadratic phase Fourier transform (QQPFT), which is the quaternion extension of the quadratic phase Fourier transform (QPFT). First, using the orthogonal plane split (OPS) method of quaternions, we observe the relation between QPFT and QQPFT. Later, using the OPS method, we formulate sharp Young–Hausdorff inequality, Pitt’s inequality, logarithmic uncertainty principle, entropy uncertainty principle, Heisenberg uncertainty principle, and Nazarov’s uncertainty principle for QQPFT.

New type of convolution and correlation theorems associated with quadratic phase Fourier transform (QPFT) are studied. Applications of that in multiplicative filter design, which may be useful in optics and signal processing to recover the signals, are also discussed. Besides that, the real Paley–Wiener (PW) and Boas theorem for QPFT are proved, which analyses the characteristics of the signals associated with QPFT in the domain.

Quaternion signal processing is frequently used in color image processing. The quaternion windowed linear canonical transform (QWLCT), a generalization of the windowed linear canonical transform (WLCT), has a wide range of application domains, including signal processing and optics. In this paper, we study QWLCT-based characterization range, reproducing kernel, one-one map, Donoho-Stark inequality and Pitt’s inequality. Some useful uncertainty principles (UP) like Heisenberg UP, Lieb UP, and local UP are discussed. Moreover, some applications associated with QWLCT in linear time-varying (TV) systems are explained in detail.

This paper introduces the quaternion Schwarz type space, and quaternion linear canonical transform (QLCT) mapping properties are also discussed. Further, the quaternion pseudo-differential operator (QPDO) associated with QLCT is described. Some of its characteristics, including estimates, boundedness, and integral representation in quaternion Sobolev type space, are derived. Some applications of QLCT, quaternion differential equations, are also discussed.