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 transform 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 this article, 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 (Formula presented.) domain.

In this paper, 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.

The offset linear canonical transform (OLCT) is an important tool in signal processing and optics. Recently, the quaternion offset linear canonical transform (QOLCT) has been introduced which is the quaternion extension of the OLCT and the generalized form of quaternion Fourier transform(QFT). In this article, the Plancherel's theorem of the scalar inner product for the two-sided QOLCT is introduced. Also, the quaternion inner product theorems for the right sided and left sided QOLCT have been discussed. Further, as an application of the Plancherel's theorem, the real Paley-Wiener theorem and Donoho-Stark uncertainty principle have been explored as well as the solution of particular type of quaternion differential equations are discussed using QOLCT. Additionally, the advantages of QOLCT over QLCT and QFT is illustrated graphically using example and the use of Plancherel's theorem in filter analysis is demonstrated.

In this paper, we define new type of convolution and correlation theorems associated with the offset linear canonical transform (OLCT). Additionally, we discuss their applications in multiplicative filter design, which may prove useful in optics and signal processing for signal recovery. Furthermore, we explore the real Paley-Wiener (PW) and Boas theorems for the OLCT, analyzing signal characteristics for OLCT within the L2(R) domain.

The quadratic phase Fourier transform (QPFT) is a generalization of several well known integral transforms, including the linear canonical transform (LCT), fractional Fourier transform (FrFT), and Fourier transform (FT). This paper introduces the multidimensional QPFT and investigates its theoretical properties, including Parseval’s identity and inversion theorems. Also, multidimensional convolution is proposed for QPFT in the multidimensional setting. Additionally, Boas type theorem for the multidimensional QPFT is established. As applications, multidimensional multiplicative filter design and the solution of multidimensional integral and differential equations using the proposed convolution operation are explored.

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.

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.

In this paper, the relation between linear canonical transform (LCT) and quaternion linear canonical transform (QLCT) by using orthogonal plan split (OPS) method is studied. Further, several inequalities like sharp Young–Hausdorff inequality and Pitt's inequality are proved. Moreover, entropy uncertainty principle, sharp local uncertainty principle, and Heisenberg's uncertainty principle for the two-sided QLCT are also discussed.

The linear canonical transform (LCT), a generalization of the Fourier transform (FT), has a wide range of application domains including signal processing and optics. In this article, LCT based new convolution and correlation theorems are derived. Also discussed the real Paley–Wiener and Boas theorems, which give a characterization of two classes of functions in terms of the behaviour of their linear canonical transforms.

In this article, we studied the inverse theorems and Parseval's identity for higher dimensional linear canonical transform. The existing convolution for the single and multi-variable functions is also discussed. We introduced a generalized convolution of multiple variables and derived their properties for the multidimensional linear canonical transform including convolution theorem and product theorem.

In this paper, the idea of wave packet transform has been generalized. We have obtained reconstruction formula, characterization range, orthogonality, some important estimates and convolution of linear canonical wave packet transform. We have also discussed uncertainty principles and some properties on periodic functions for this transform.