In recent years, the construction of numerous generalized transformations within quaternion algebra has received more attention by some researchers like the quaternion linear canonical wavelet transform [1], the quaternion Wigner-Ville distribution [2, 3, 4, 5], the quaternion shearlet transform [6], and the quaternion windowed Fourier transform [7]. Further, the authors of [8] have studied the linear canonical S-transform (LCST), which is the extension of the traditional S-transform in the framework of the linear canonical transform (LCT). It is known that several results of the traditional S-transform are modified in the LCST domain, such as modulation, translation, uncertainty inequalities and so on (see, e.g., [9, 10, 11].

Recently, the authors in [12][13] have studied the quaternion linear canonical S-transform and obtained the uncertainty relations. However, Lieb and Nazarov inequalities for this transformation have not been published to date. Therefore, in the present work, we will establish Lieb’s inequality for the quaternion linear canonical S-transform. To achieve the result, we recall several results of the quaternion linear canonical S-transform and make a relation between the quaternion linear canonical transform and the quaternion linear canonical S-transform.

The structure of this work is organized as follows. In Section 2, some basic facts regarding quaternion algebra are collected. The definition of the quaternion Fourier transform (QFT) and useful properties are provided in Section 3. Section 4 briefly reviews the definition of the quaternion linear canonical transform (QLCT) and its connection with the quaternion Fourier transform (QFT). Lastly, Section 5 contains the introduction of the quaternion linear canonical S-transform (QLCST) and the derivation of Lieb and Nazarov inequalities related to the proposed transformation.

Let be the associated algebra of real quaternion. The quaternion is represented as [14]

where the three diferent imaginary quaternion units i, j and k follow the multiplication rules:

Relation (2) shows that quaternion multiplication is not commutative. For every ,

where and denote the scalar component and vector component, respectively.

The conjugate of quaternion ¯p is defined as

It satisfies the property

The scalar and vector parts of a quaternion p are

The modulus of a quaternion p is defined by

It is straightforward to verify that, for every , the followings hold:

The inner product for two quaternion functions is defined by

with symmetric scalar product

For in relation (9), we get norm, i.e.,

This part recalls the definition of two-sided quaternion Fourier transform . We list properties which will be needed in the next section. For more information about the QFT and its properties, we refer the reader to [15, 16, 17, 18, 19].

Definition 3.1. Let , the two-sided quaternion Fourier transform of f is defined as

The following shows that the function in (11) can be recovered using its QFT.

Definition 3.2.Let f inand in . The inversion formula of the quaternion Fourier transform is calculated by

The next, from relation (12), one obtains

In the sequel, we introduce the two-sided quaternion linear canonical transform (shortly QLCT) and its relationship to the quaternion Fourier transform (QFT). For a detailed information on this transformation, the reader may consult to [20], [21].

Definition 4.1.Let and belong to the special linear group . The QLCT of a suitable function is defined as

and

where

In this research, we always assume the QLCT with . It is obvious that when with , the QLCT (14) changes to QFT

where is defined by (11).

Definition 4.2.For any quaternion signalwith , the inversion of the QLCT is described by

According to the QLCT definition (14), we obtain

Hence,

where

By equation (19), we can easily derive for every

which is known as Parseval’s formula for the QLCT.

In the sequel, we start with a definition of the quaternion linear canonical S-transform (QLCST). We collect its essential properties, which will be useful for deriving the main results in this research.