Home Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform
Article Open Access

Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform

  • Mohammad Younus Bhat EMAIL logo , Ibrahim M. Almanjahie , Aamir H. Dar and Javid G. Dar
Published/Copyright: November 11, 2022
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

Wigner-Ville transform or Wigner-Ville distribution (WVD) associated with quaternion offset linear canonical transform (QOLCT) was proposed by Bhat and Dar. This work is devoted to the development of the theory proposed by them, which is an emerging tool in the scenario of signal processing. The main contribution of this work is to introduce WVD and ambiguity function (AF) associated with the QOLCT (WVD-QOLCT/AF-QOLCT). First, the definition of the WVD-QOLCT is proposed, and then several important properties such as dilation, nonlinearity, and boundedness are derived. Second, we derived the AF for the proposed transform. A bunch of important properties, including the reconstruction formula associated with the AF, are studied.

Keywords: quaternion algebra; quaternion offset linear canonical transform; Wigner-Ville distribution; ambiguity function; dilation
MSC 2010: 11R52; 42C40; 42C30; 43A30

1 Introduction

In the modern era of information theory, Wigner-Ville distribution (WVD), or more commonly Wigner-Ville transform (WVT), is an important and effective tool. It was the great Eugene Wigner who introduced the concept of WVD during the calculation of quantum corrections. Moving on the same road in 1948, Ville derived it independently by using the quadratic representation of a signal in the local time-frequency domain. The survey on this novel transform continued. This transform can be generalized to a linear canonical transform (LCT) by changing the kernel of the Fourier transform (FT) to that of an LCT in the WVT domain. For more about this transform, we refer to [1,2,3, 4,5,6, 7,8,9, 10,11].

On the one hand, the quaternion Fourier transform (QFT) is worthwhile to explore in the era of communications. A bunch of needed properties like energy conservation, uncertainty principle, shift, modulation, differentiation, convolution, and correlation of QFT have been explored. Various number of transforms generalized from QFTs are closely related, for instance, the fractional QFT, quaternion wavelet transform, quaternion linear canonical transform (QLCT), and many more. Like the case of QFTs, one can also extend the WVD to the quaternion algebra while having similar properties to that in the classical sense. Many researchers generalized the WVD to the novel quaternion algebra, termed the quaternion WVD. One can see [12,13,14, 15,16,17, 18,19].

On the other hand, the ambiguity function (AF) that was introduced by Woodward in 1953 plays an important role in the Information theory. It has been applied in various fields, such as radar signal processing, sonar technology, and optical information theory (see [20,21, 22,23,24, 25,26]). In recent times, a lot of research work has been carried out by coupling the classical AF with other transforms. Other notable works are found in [27,28, 29,30]. The WVD and AF associated with the LCT have great advantages and flexibility over the classical WVD and AF, respectively. The generalized WVD and AF have achieved better detection performance. Therefore, it is worthwhile to study a new generalized WVD and AF.

Recently, Bhat and Dar [31] have studied convolution and correlation theorems for WVD for the quaternion offset linear canonical transform (QOLCT). However, due to the noncommutativity of the multiplications in quaternion signals, only a special condition for the convolution and correlation theorems can be established. To be precise, we are bound to take real-valued signals in the function space over the Hamiltonian group. We continued their studies and established some more properties like dilation, nonlinearity, and boundedness. Nevertheless, to extend the scope of study, we introduced an ambiguity function for the QOLCT and were at ease to obtain the classical properties associated with it.

The article is organized as follows. In Section 2, we look at some basics about the quaternion algebra and the QOLCT. Section 3 is devoted to the definition of WVD associated with the QOLCT (WVD-QOLCT). Various properties of the WVD-QOLCT are investigated. In Section 4, we first introduce the AF for the QOLCT. We continued the study and obtained various properties like nonlinearity, boundedness, and a reconstruction formula for the proposed transform.

2 Preliminary

Here, we look at some basics needed for the rest of the article.

2.1 Quaternion algebra

The quaternion algebra is an extended version of complex numbers. Hamilton invented it in 1843, and in his honour, it is denoted by H . The elements of H have the Cartesian form as

H = { q ∣ q ≔ [ q ] 0 + i [ q ] 1 + j [ q ] 2 + k [ q ] 3 , [ q ] i ∈ R , i = 0 , 1 , 2 , 3 } ,

where i , j , and k are imaginary units obeying the Hamilton’s multiplication rules (see [18]),

(1) i j = − j i = k , j k = − k j = i , k i = − i k = j ,

(2) i 2 = j 2 = k 2 = i j k = − 1 .

Looking at (1), H is obviously noncommutative. So, we cannot directly extend a number of results on complex numbers to the quaternion. For the sake of convenience, we represent a quaternion q to be the sum of scalar q 1 and purely 3 D quaternion q . Thus, we can write explicitly every quaternion as follows:

q = q 1 + i q 2 + j q 3 + k q 4 ∈ H , q 1 , q 2 , q 3 , q 4 ∈ R .

The conjugation is obtained by changing the sign of the pure part, i.e.,

q ¯ = q 1 − i q 2 − j q 3 − k q 4 .

It is evident that quaternion conjugation is linear but anti-involution,

p ¯ ¯ = p , p + q ¯ = p ¯ + q ¯ , p q ¯ = q ¯ p ¯ , ∀ p , q ∈ H .

Modulus of a quaternion q can be defined as follows:

∣ q ∣ = q q ¯ = q 1 2 + q 2 2 + q 3 2 + q 4 2 , ∣ p q ∣ = ∣ p ∣ ∣ q ∣ .

It is evident that

∣ p q ∣ = ∣ p ∣ ∣ q ∣ , ∣ q ∣ = ∣ q ¯ ∣ , p , q ∈ H .

Any function f : R 2 → H can be viewed as

f ( x , y ) ≔ f 1 ( x , y ) + i f 2 ( x , y ) + j f 3 ( x , y ) + k f 4 ( x , y ) ,

where ( x , y ) ∈ R × R .

The inner product of the functions f , g in H can be defined as follows:

(3) ⟨ f , g ⟩ L 2 ( R 2 , H ) = ∫ R 2 f ( x ) g ( x ) ¯ d 2 x .

If we let f = g , then

(4) ‖ f ‖ 2 2 = ⟨ f , f ⟩ L 2 ( R 2 , H ) = ∫ R 2 ∣ f ( x ) ∣ 2 d 2 x .

The space L 2 ( R 2 , H ) can be defined like

(5) L 2 ( R 2 , H ) = { f ∣ f : R 2 → H , ‖ f ‖ 2 < ∞ } .

Lemma 1

If f , g ∈ L 2 ( R 2 , H ) , then the Cauchy-Schwarz inequality [15] holds

(6) ∣ ⟨ f , g ⟩ L 2 ( R 2 , H ) ∣ 2 ≤ ‖ f ‖ L 2 ( R 2 , H ) 2 ‖ g ‖ L 2 ( R 2 , H ) 2

if and only if f = λ g for some quaternionic parameter λ ∈ H , the equality holds.

2.2 QLCT

QLCT was first of all defined by Morais et al. [19]. They have considered two matrixes with real values as

A 1 = a 1 b 1 c 1 d 1 , A 1 = a 2 b 2 c 2 d 2 ∈ R 2 × 2 ,

with a 1 d 1 − b 1 c 1 = 1 and a 2 d 2 − b 2 c 2 = 1 . Hitzer in [16] generalized the definitions of [19] into the case of two-sided QLCT of signals f ∈ L 1 ( R 2 , H ) like

(7) ℒ A 1 , A 2 i , j [ f ] ( u ) = ∫ R 2 K A 1 i ( t 1 , u 1 ) f ( t ) K A 2 j ( t 2 , u 2 ) d 2 t , b 1 , b 2 ≠ 0 ; d 1 e i c 1 d 1 2 u 1 2 f ( d 1 u 1 , t 2 ) K A 2 j ( t 2 , u 2 ) , b 1 = 0 , b 2 ≠ 0 ; d 2 K A 1 i ( t 1 , u 1 ) f ( t 1 , d 2 u 2 ) e j c 2 d 2 2 u 2 2 , b 1 ≠ 0 , b 2 = 0 ; d 1 d 2 e i c 1 d 1 2 u 1 2 f ( d 1 u 1 , d 2 u 2 ) e j c 2 d 2 2 u 2 2 , b 1 = b 2 = 0 ,

where t = ( t 1 , t 2 ) and u = ( u 1 , u 2 ) .

Theorem 1

(Inversion formula for QLCT) Suppose f ∈ L ( R 2 , H ) , then the inversion of the two-sided QLCT of f is given as follows:

(8) f ( x ) = { ℒ A 1 , A 2 − 1 } { ℒ A 1 , A 2 H [ f ] } ( x ) = ∫ R 2 K A 1 − i ( x 1 , u 1 ) ℒ A 1 , A 2 H [ f ] ( u ) K A 2 − j ( x 2 , u 2 ) d 2 u .

Theorem 2

(Plancherel for QLCT) Every 2D quaternion-valued signal f ∈ L 2 ( R 2 , H ) , and its QLCT are related to the Placherel identity in the following way:

(9) ‖ f ‖ L 2 ( R 2 , H ) = ‖ ℒ A 1 , A 2 H [ f ] ‖ Q , 2 .

2.3 QOLCT

Amid the noncommutative nature in the multiplication of quaternions, we have three types of the QOLCT. The first one is left-sided QOLCT, the second one is right-sided QOLCT, and the third one is two-sided QOLCT.

Definition 1

Let A i = a i b i ∣ r i c i d i ∣ s i be a matrix parameter such that a i , b i , c i , d i , r i , s i ∈ R , and a i d i − b i c i = 1 for i = 1 , 2 . The two-sided QOLCT of any quaternion-valued function f ∈ L 2 ( R 2 , H ) is given by

(10) O A 1 , A 2 i , j [ f ( t ) ] ( u ) = ∫ R 2 K A 1 i ( t 1 , u 1 ) f ( t ) K A 2 j ( t 2 , u 2 ) d 2 t , b 1 , b 2 ≠ 0 , d 1 e i c 1 d 1 2 ( u 1 − r 1 ) 2 + u 1 r 1 f ( d 1 u 1 − d 1 r 1 , t 2 ) K A 2 j ( t 2 , u 2 ) , b 1 = 0 , b 2 ≠ 0 , d 2 K A 1 i ( t 1 , u 1 ) f ( t 1 , d 2 u 2 − d 2 r 2 ) e j c 2 d 2 2 ( u 2 − r 2 ) 2 + u 2 r 2 , b 1 ≠ 0 , b 2 = 0 , d 1 d 2 f ( d 1 u 1 − d 1 r 1 , d 2 u 2 − d 2 r 2 ) e i c 1 d 1 2 ( u 1 − r 1 ) 2 + u 1 r 1 e j c 2 d 2 2 ( u 2 − r 2 ) 2 + u 2 r 2 , b 1 = 0 , b 2 = 0 ,

where t = ( t 1 , t 2 ) and u = ( u 1 , u 2 ) , and the quaternion kernels K A 1 i ( t 1 , u 1 ) and K A 2 j ( t 2 , u 2 ) , are respectively, given as follows:

(11) K A 1 i ( t 1 , u 1 ) = 1 2 π b 1 i e i 2 b 1 [ a 1 t 1 2 − 2 t 1 ( u 1 − r 1 ) − 2 u 1 ( d 1 r 1 − b 1 s 1 ) + d 1 ( u 1 2 + r 1 2 ) ] , b 1 ≠ 0

(12) K A 2 j ( t 2 , u 2 ) = 1 2 π b 2 j e j 2 b 2 [ a 2 t 2 2 − 2 t 2 ( u 2 − r 2 ) − 2 u 2 ( d 2 r 2 − b 2 s 2 ) + d 2 ( u 2 2 + r 2 2 ) ] , b 2 ≠ 0 .

Remark 1

One can define the left- and right-sided QOLCT by simply placing the two above kernels both on the left or on the right.

Note that, when r 1 = r 2 = s 1 = s 2 = 0 , the two-sided QOLCT reduces to the QLCT.

Also, when A 1 = A 2 = 0 1 − 1 0 0 0 , the conventional two-sided QFT is recovered.

Theorem 3

Suppose f ∈ L 2 ( R 2 , H ) , then the inversion of two-sided QOLCT is given by

(13) f ( t ) = ∫ R 2 K A 1 − i ( t 1 , u 1 ) O A 1 , A 2 i , j [ f ] ( u ) K A 2 − j ( t 2 , u 2 ) d 2 u .

Theorem 4

(Plancherel for QOLCT) Every two-dimensional quaternion-valued function f ∈ L 2 ( R 2 , H ) and its two-sided QOLCT are related to the Plancherel identity in the following way:

(14) ∥ O A 1 , A 2 i , j [ f ] ∥ L 2 ( R 2 , H ) = ∥ f ∥ L 2 ( R 2 , H ) .

3 WVD-QOLCT

Recently, authors in [31] defined WVD-QOLCT and studied some properties associated with it and also established a special condition for the convolution and correlation theorems for the WVD-QQPFT, which are important for signal processing. In this section, we continue to study the properties of WVD-QOLCT, and prior to that, we first define the WVD-QOLCT.

Definition 2

Let A i = a i b i ∣ r i c s d i ∣ s i be a matrix parameter such that a s , b i , c i , d i , r i , s i ∈ R , and a i d i − b i c i = 1 , for i = 1 , 2 . The WVD associated with the two-sided QOLCT (WVD-QOLCT) of signals f , g ∈ L 2 ( R 2 , H ) is given as follows:

(15) W f , g A 1 , A 2 ( t , w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z , b i ≠ 0 ,

where t = ( t 1 , t 2 ) , w = ( w 1 , w 2 ) , z = ( z 1 , z 2 ) , and the quaternion kernels K A 1 i ( z 1 , w 1 ) and K A 2 j ( z 2 , w 2 ) are given by (11) and (12), respectively.

Note: If f = g , then W f , f A 1 , A 2 ( t , w ) we call it the Auto WVD-QOLCT. Otherwise, it is labelled as the cross WVD-QOLCT.

In this article, we always assume that b i ≠ 0 , i = 1 , 2 , otherwise the proposed transform reduces to a chirp multiplication. Hence, for any f , g ∈ L 2 ( R 2 , H ) , we have,

(16) W f , g A 1 , A 2 ( t , w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 w = O A 1 , A 2 i , j f t + z 2 g t − z 2 ¯ = O A 1 , A 2 i , j { h f , g ( t , w ) } ,

where h f , g ( t , w ) = f t + z 2 g t − z 2 ¯ is known as quaternion correlation product.

Applying the inverse QOLCT to (16), we obtain

(17) h f , g ( t , z ) = { O A 1 , A 2 i , j } − 1 { W f , g A 1 , A 2 ( t , w ) } ,

which implies

(18) f t + z 2 g t − z 2 ¯ = { O A 1 , A 2 i , j } − 1 { W f , g A 1 , A 2 ( t , w ) } = ∫ R 2 K A 1 − i ( t 1 , w 1 ) W f , g A 1 , A 2 ( t , w ) K A 2 − j ( t 2 , w 2 ) d 2 w .

We now study several important properties of the WVD-QOLCT defined by (2).

Theorem 5

(Dilation) Let f , g ∈ L 2 ( R 2 , H ) , then

(19) W D m f , D m g A 1 , A 2 ( t , w ) = W f , g B 1 , B 2 t m , w m ,

where A i = a i b i ∣ r i c s d i ∣ s i , B i = a i b i m 2 ∣ r i m m 2 c s d i ∣ m s i , i = 1 , 2 , and D m f ( t ) = 1 m f t m , m ≠ 0 .

Proof

We have from Definition 2

W D m f , D m g A 1 , A 2 ( t , w ) = 1 m 2 ∫ R 2 K A 1 i ( z 1 , w 1 ) f t m + z 2 m g t m − z 2 m ¯ K A 2 j ( z 2 , w 2 ) d 2 z = ∫ R 2 K A 1 i ( m x 1 , w 1 ) f t m + x 2 g t m − x 2 ¯ K A 2 j ( m x 2 , w 2 ) d 2 x = ∫ R 2 1 2 π b 1 i e i 2 b 1 [ a 1 m 2 x 1 2 − 2 m x 1 ( w 1 − r 1 ) − 2 w 1 ( d 1 r 1 − b 1 s 1 ) + d 1 ( w 1 2 + r 1 2 ) ] f t m + z 2 m g t m − z 2 m ¯ × 1 2 π b 2 j e j 2 b 2 [ a 2 m 2 x 2 2 − 2 m x 2 ( w 2 − r 2 ) − 2 w 2 ( d 2 r 2 − b 2 s 2 ) + d 2 ( w 2 2 + r 2 2 ) ] d 2 x = ∫ R 2 1 2 π b 1 i e i 2 b 1 a 1 1 m 2 x 1 2 − 2 1 1 m 2 x 1 w 1 m − r 1 m − 2 1 m 2 w 1 m d 1 r 1 m − b 1 m 2 m s 1 + d 1 1 m 2 w 1 m 2 + r 1 m 2 f t m + z 2 m × g t m − z 2 m ¯ 1 2 π b 2 j e j 2 b 2 a 2 1 m 2 x 2 2 − 2 1 1 m 2 x 2 w 2 m − r 2 m − 2 1 m 2 w 2 m d 2 r 2 m − b 2 m 2 m s 2 + d 2 1 m 2 w 2 m 2 + r 2 m 2 d 2 x = ∫ R 2 1 2 π b 1 i e i 2 b 1 m 2 a 1 x 1 2 − 2 x 1 w 1 m − r 1 m − 2 w 1 m d 1 r 1 m − b 1 m 2 m s 1 + d 1 w 1 m 2 + r 1 m 2 f t m + z 2 m × g t m − z 2 m ¯ 1 2 π b 2 j e j 2 b 2 m 2 a 2 x 2 2 − 2 x 2 w 2 m − r 2 m − 2 w 2 m d 2 r 2 m − b 2 m 2 m s 2 + d 2 w 2 m 2 + r 2 m 2 d 2 x = ∫ R 2 K B 1 i x 1 , w 1 m f t m + z 2 m g t m − z 2 m ¯ K B 2 j x 2 , w 2 m d 2 x = W f , g B 1 , B 2 t m , w m , where B i = a i b i m 2 ∣ r i m m 2 c s d i ∣ m s i , i = 1 , 2 ,

which completes the proof.□

Theorem 6

(Shift) Let f , g ∈ L 2 ( R 2 , H ) , then

(20) W T m f , T m g A 1 , A 2 ( t , w ) = W f , g A 1 , A 2 ( t − m , w ) ,

where T m f ( t ) = f ( t − m ) , t = ( t 1 , t 2 ) .

Proof

From Definition 2, we have

W T m f , T m g A 1 , A 2 ( t,w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 − m g t − z 2 − m ¯ K A 2 j ( z 2 , w 2 ) d 2 z = ∫ R 2 K A 1 i ( z 1 , w 1 ) f ( t − m ) + z 2 g ( t − m ) − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z = W f , g A 1 , A 2 ( t - m,w ) ,

which completes the theorem.□

Theorem 7

(Boundedness) Let f , g ∈ L 2 ( R 2 , H ) . Then,

(21) ∣ W f , g A 1 , A 2 ( t , w ) ∣ ≤ 2 π b 1 b 2 ‖ f ‖ L 2 ( R 2 , H ) ‖ g ‖ L 2 ( R 2 , H ) .

Proof

With the help of Cauchy-Schwarz inequality for quaternion domain (6), we obtain

∣ W f , g A 1 , A 2 ( t,w ) ∣ 2 = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d n 2 ≤ ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z 2 = 1 4 π 2 ∣ b 1 b 2 ∣ ∫ R 2 f t + z 2 g t − z 2 ¯ d 2 z 2 ≤ 1 4 π 2 ∣ b 1 b 2 ∣ ∫ R 2 f t + z 2 2 d 2 z ∫ R 2 g t − z 2 ¯ 2 d 2 z = 1 4 π 2 ∣ b 1 b 2 ∣ 4 ∫ R 2 ∣ f ( w ) ∣ 2 d 2 w 4 ∫ R 2 ∣ g ( y ) ¯ ∣ 2 d 2 y = 4 π 2 ∣ b 1 b 2 ∣ ‖ f ‖ L 2 ( R 2 , H ) 2 ‖ g ‖ L 2 ( R 2 , H ) 2 ,

with the changing of variables w = t + z 2 and y = t − z 2 in last second step, we obtain

∣ W f , g A 1 , A 2 ( t , w ) ∣ ≤ 2 π ∣ b 1 b 2 ∣ ‖ f ‖ L 2 ( R 2 , H ) ‖ g ‖ L 2 ( R 2 , H ) ,

which completes the proof.□

Theorem 8

(Nonlinearity) Let f and g be two quaternion functions in L 2 ( R 2 , H ) . Then,

(22) W f + g A 1 , A 2 = W f , f A 1 , A 2 + W f , g A 1 , A 2 + W g , f A 1 , A 2 + W g , g A 1 , A 2 .

Proof

By Definition 2, we have

W f + g A 1 , A 2 ( t , w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 + g t + z 2 f t − z 2 + g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 f t − z 2 ¯ + f t + z 2 g t − z 2 ¯ + g t + z 2 f t − z 2 ¯ + g t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z = ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 f t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z + ∫ R 2 K A 1 i ( z 1 , w 1 ) f t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z + ∫ R 2 K A 1 i ( z 1 , w 1 ) g t + z 2 f t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z + ∫ R 2 K A 1 i ( z 1 , w 1 ) g t + z 2 g t − z 2 ¯ K A 2 j ( z 2 , w 2 ) d 2 z = W f , f A 1 , A 2 + W f , g A 1 , A 2 + W g , f A 1 , A 2 + W g , g A 1 , A 2 ,

which completes the proof.□

Theorem 9

If ‖ f ‖ ( R 2 , H ) = ‖ g ‖ ( R 2 , H ) = 1 , then we have,

(23) ∫ R 2 ∫ R 2 ‖ W f , g A 1 , A 2 ( t , w ) ‖ 2 d 2 t d 2 w = 4 .

Proof

As W f , g A 1 , A 2 ( t , w ) is the QOLCT of h f , g ( t , z ) , then by Plancherel for QOLCT (14), we obtain

∫ R 2 ∣ W f , g A 1 , A 2 ( t , w ) ∣ 2 d 2 w = ∫ R 2 ∣ h f , g A 1 , A 2 ( t , z ) ∣ 2 d 2 z = ∫ R 2 f t + z 2 g t − z 2 ¯ 2 d 2 z .

Setting x = t + z 2 and y = t − z 2 and integrating the above equation with respect to d t , we have

∫ R 2 ∫ R 2 ∣ W f , g A 1 , A 2 ( t , w ) ∣ 2 d 2 w = ∫ R 2 ∫ R 2 f t + z 2 g t − z 2 ¯ 2 d 2 t d z = 4 ∫ R 2 ∫ R 2 ∣ f ( x ) ∣ 2 ∣ g ( y ) ∣ 2 ¯ d 2 y d t ,

which completes the proof.□

Theorem 10

(Reconstruction formula) For f , g ∈ L 2 ( R 2 , H ) , where g does not vanish at 0, we obtain the following inversion formula of the WVD-QOLCT:

(24) f ( t ) = 1 g ( 0 ) ¯ ∫ R 2 K A 1 − i ( w 1 , z 1 ) W f , g A 1 , A 2 t 2 , u K A 2 − j ( w 2 , z 2 ) d 2 w .

Proof

See [31].□

Theorem 11

(Orthogonality relation) If f 1 , f 2 , g 1 , g 2 ∈ L 2 ( R 2 , H ) are quaternion-valued signals. Then,

(25) ⟨ W f 1 , g 1 A 1 , A 2 ( t , w ) , W f 2 , g 2 A 1 , A 2 ( t , w ) ⟩ = [ ⟨ f 1 , f 2 ⟩ ⟨ g 2 , g 1 ⟩ ] H .

Proof

See [31].□

Consequences of Theorem 11:

  1. If g 1 = g 2 = g , then

    (26) ⟨ W f 1 , g A 1 , A 2 ( t , w ) , W f 2 , g A 1 , A 2 ( t , w ) ⟩ = ‖ g ‖ L 2 ( R 2 ) 2 ⟨ f 1 , f 2 ⟩ .

  2. If f 1 = f 2 = f , then

    (27) ⟨ W f , g 1 A 1 , A 2 ( t , w ) , W f , g 2 A 1 , A 2 ( t , w ) ⟩ = ‖ f ‖ L 2 ( R 2 ) 2 ⟨ g 1 , g 2 ⟩ .

  3. If f 1 = f 2 = f and g 1 = g 2 = g , then

    (28) ⟨ W f , g A 1 , A 2 ( t , w ) , W f , g A 1 , A 2 ( t , w ) ⟩ = ∫ R 2 ∫ R 2 ∣ W f , g A 1 , A 2 ( t , w ) ∣ 2 d 2 w d 2 t = ‖ f ‖ L 2 ( R 2 ) 2 ‖ g ‖ L 2 ( R 2 ) 2 .

Equation (28) represents the Plancherel theorem for WVD-QOLCT.

Now, we move forward towards next section that is AF associated with QOLCT and its properties.

4 AF associated with QOLCT

Wood ward in 1953 introduced the classical AF for mathematical analysis of sonar and radar signals. Recently, authors in [32] introduced the AF associated with QLCT (AF-QLCT). In this section, we generalize the AF-QLCT to the offset linear canonical domain.

Definition 3

Let A i = a i b i ∣ r i c s d i ∣ s i be a matrix parameter such that a s , b i , c i , d i , r i , s i ∈ R , and a i d i − b i c i = 1 , for i = 1 , 2 . The AF associated with the two-sided QOLCT (AF-QOLCT) of signals f , g ∈ L 2 ( R 2 , H ) , is given by

(29) A f , g A 1 , A 2 ( t , w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f z + t 2 g z − t 2 ¯ K A 1 j ( z 2 , w 2 ) d 2 z .

Theorem 12

Let f , g ∈ L 2 ( R 2 , H ) , then

(30) A f ( − t ) A 1 , A 2 ( t , w ) = A f ( t ) B 1 , B 2 ( t , − w ) ,

where B i = a i b i ∣ − r i c s d i ∣ − s i for i = 1 , 2 .

Proof

From (29), we have

A f ( − t ) A 1 , A 2 ( t , w ) = ∫ R 2 K A 1 i ( z 1 , w 1 ) f − z + t 2 g − z − t 2 ¯ K A 1 j ( z 2 , w 2 ) d 2 z = ∫ R 2 1 2 π i e i 2 b 1 [ a 1 z 1 2 − 2 z 1 ( w 1 − r 1 ) − 2 w 1 ( d 1 r 1 − b 1 s 1 ) + d 1 ( w 1 2 + r 1 2 ) ] f × − z + t 2 g − z − t 2 ¯ 1 2 π j e j 2 b 2 [ a 2 z 2 2 − 2 z 2 ( w 2 − r 2 ) − 2 w 2 ( d 2 r 2 − b 2 s 2 ) + d 2 ( w 2 2 + r 2 2 ) ] d 2 z .

On setting − z = z ′ , we have from the above equation

A f ( − t ) A 1 , A 2 ( t , w ) = ∫ R 2 1 2 π i e i 2 b 1 [ a 1 ( − z 1 ′ ) 2 − 2 ( − z 1 ′ ) ( w 1 − r 1 ) − 2 w 1 ( d 1 r 1 − b 1 s 1 ) + d 1 ( w 1 2 + r 1 2 ) ] f z ′ + t 2 g z ′ − t 2 ¯ × 1 2 π j e j 2 b 2 [ a 2 ( − z 2 ′ ) 2 − 2 ( − z 2 ′ ) ( w 2 − r 2 ) − 2 w 2 ( d 2 r 2 − b 2 s 2 ) + d 2 ( w 2 2 + r 2 2 ) ] d 2 z ′ = ∫ R 2 1 2 π i e i 2 b 1 [ a 1 z 1 ′ 2 − 2 z 1 ′ ( − w 1 − ( − r 1 ) ) − 2 ( − w 1 ) ( d 1 ( − r 1 ) − b 1 ( − s 1 ) ) + d 1 ( ( − w 1 ) 2 + ( − r 1 ) 2 ) ] f z ′ + t 2 g z ′ − t 2 ¯ × 1 2 π j e j 2 b 2 [ a 2 z 2 ′ 2 − 2 z 2 ′ ( − w 2 − ( − r 2 ) ) − 2 ( − w 2 ) ( d 2 ( − r 2 ) − b 2 ( − s 2 ) ) + d 2 ( ( − w 2 ) 2 + ( − r 2 ) 2 ) ] d 2 z ′ = ∫ R 2 K B 1 i ( z 1 ′ , − w 1 ) f z ′ + t 2 g z ′ − t 2 ¯ K B 1 j ( z 2 ′ , − w 2 ) d 2 z ′ = A f ( t ) B 1 , B 2 ( t , − w ) , where B i = a i b i ∣ − r i c s d i ∣ − s i for i = 1 , 2 ,

which completes the proof.□

Theorem 13

(Dilation) Let us consider the matrix parameters A i = a i b i ∣ r i c s d i ∣ s i and B i = a i b i m 2 ∣ r i m m 2 c s d i ∣ m s i , for i = 1 , 2 . Then, for f , g ∈ L 2 ( R 2 , H ) , we have

(31) A D m f , D m g A 1 , A 2 ( t , w ) = A f , g B 1 , B 2 t m , w m ,

where D m f ( t ) = 1 m f t m , m ≠ 0 .

Proof

The proof follows by using the procedure of Theorem 5.□

Theorem 14

(Nonlinearity) Let f and g be two quaternion functions in L 2 ( R 2 , H ) . Then,

(32) A f + g A 1 , A 2 = A f , f A 1 , A 2 + A f , g A 1 , A 2 + A g , f A 1 , A 2 + A g , g A 1 , A 2 .

Proof

The proof follows by using the procedure of Theorem 8.□

Next theorem assures that every original quaternion signal can be uniquely reconstructed by the AF-QOLCT with a quaternion constant.

Theorem 15

(Reconstruction formula for AF-QOLCT). For f , g ∈ L 2 ( R 2 , H ) , where g does not vanish at 0, we obtain the following inversion formula of the AF-QOLCT:

(33) f ( t ) = 1 g ( 0 ) ¯ ∫ R 2 K A 1 − i w 1 , t 1 2 A f , g A 1 , A 2 ( t , w ) K A 2 − j w 2 , t 2 2 d 2 w .

Proof

It is clear from (29) and (18) that

f z + t 2 g z − t 2 ¯ = ∫ R 2 K A 1 − i ( w 1 , z 1 ) A f , g A 1 , A 2 ( t , w ) K A 2 − j ( w 2 , z 2 ) d 2 w .

On setting z = t 2 , we have from the above equation

f ( t ) g ( 0 ) ¯ = ∫ R 2 K A 1 − i w 1 , t 1 2 A f , g A 1 , A 2 ( t , w ) K A 2 − j w 2 , t 2 2 d 2 w ,

which implies

□ f ( t ) = 1 g ( 0 ) ¯ ∫ R 2 K A 1 − i w 1 , t 1 2 A f , g A 1 , A 2 ( t , w ) K A 2 − j w 2 , t 2 2 d 2 w .

5 Conclusion

Based on the QOLCT and the classical WVD theory, this article first proposes the definition of WVD in the QOLCT domain, namely WVD-QOLCT. Various properties of the WVD-QOLCT, including dilation, shift, and boundedness, are derived in detail. Next, we propose the definition of ambiguity function associated with QOLCT (AF-QOLCT) and establish various properties associated with it viz nonlinearity, dilation and reconstruction formula. The obtained results are new and meaningful. This article is useful for studying the areas of numerical simulations and empirical analysis like [27,28, 29,30].

Acknowledgments

The authors thank and extend their appreciation to the Deanship of Scientific research at King Khalid University for funding this work through the research group programme under the grant number R.G.P.2/132/43.

  1. Funding information: Funding for this work was received under the grant number R.G.P.2/132/43.

  2. Author contributions: All authors have accepted responsibility for the entire contents of this manuscript and approved its submission. All the authors equally contributed towards this work.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during this study.

References

[1] M. Bahri, Correlation theorem for Wigner-Ville distribution, Far East J. Math. Sci. 80 (2013), no. 1, 123–133. Search in Google Scholar

[2] R. F. Bai, B. Z. Li, and Q. Y. Cheng, Wigner-Ville distribution associated with the linear canonical transform, J. Appl. Maths. (2012), Article ID 740161. 10.1155/2012/740161Search in Google Scholar

[3] M. Y. Bhat and A. H. Dar, Scaled Wigner distribution in the offset linear canonical domain, Optik - Int. J. Light Electron Opt. 262 (2022), 169286. 10.1016/j.ijleo.2022.169286Search in Google Scholar

[4] L. Debnath, B. V. Shankara, and N. Rao, On new two- dimensional Wigner-Ville nonlinear integral transforms and their basic properties, Int. Trans. Sp. Funct. 21 (2010), no. 3, 165–174. 10.1080/10652460903069231Search in Google Scholar

[5] W. B. Gao and B. Z. Li, Convolution and correlation theorems for the windowed offset linear canonical transform, 2019, arxiv: 1905.01835v2 [math.GM]. 10.1109/ICSIDP47821.2019.9173354Search in Google Scholar

[6] Y. G. Li, B. Z Li, and H. F. Sun, Uncertainty principle for Wigner-Ville distribution associated with the linear canonical transform, Abstr. Appl. Anal. (2014), Article ID 470459. 10.1155/2014/470459Search in Google Scholar

[7] Y. E. Song, X. Y. Zhang, C. H. Shang, H. X. Bu, and X. Y. Wang, The Wigner-Ville distribution based on the linear canonical transform and its applications for QFM signal parameters estimation, J. App. Math. (2014), 8 pages. 10.1155/2014/516457Search in Google Scholar

[8] D. Urynbassarova, B. Z. Li, and R. Tao, The Wigner-Ville distribution in the linear canonical transform domain, IAENG Int. J. Appl. Math. 46 (2016), no. 4, 559–563. Search in Google Scholar

[9] D. Urynbassarova, B. Zhao, and R. Tao, Convolution and correlation theorems for Wigner-Ville distribution associated with the offset linear canonical transform, Optik - Int. J. Light Electron Opt. 157 (2018), 544–466, http://dx.doi.org/10.1016/j.ijleo.2017.08.099 Search in Google Scholar

[10] D. Wei, Q. Ran, and Y. Li, A convolution and correlation theorem for the linear canonical transform and its application, Circuits Syst. Signal Process. 31 (2012), no. 1, 301–312. 10.1007/s00034-011-9319-4Search in Google Scholar

[11] D. Wei, Q. Ran, and Y. Li, New convolution theorem for the linear canonical transform and its translation invariance property, Optik - Int. J. Light Electron Opt. 123 (2012), no. 16, 1478–1481. 10.1016/j.ijleo.2011.08.054Search in Google Scholar

[12] Z. C. Zhang, Unified Wigner-Ville distribution and ambiguity function in the linear canonical transform domain. Optik - Int. J. Light Electron Opt. 114 (2015), 45–60. 10.1016/j.sigpro.2015.02.016Search in Google Scholar

[13] M. Bahri, E. S. M. Hitzer, A. Hayashi, and R. Ashino, An uncertainty principle for quaternion Fourier transform, Comps. Maths Appl. 56 (2008), no. 9, 2398–2410. 10.1016/j.camwa.2008.05.032Search in Google Scholar

[14] M. Y. Bhat and A. H. Dar, The algebra of 2D Gabor quaternionic offset linear canonical transform and uncertainty principles, J. Anal. 30 (2022), 637–649. 10.1007/s41478-021-00364-zSearch in Google Scholar

[15] Y. El Haoui and S. Fahlaoui, Generalized uncertainty principles associated with the quaternionic offset linear canonical transform, https://arxiv.org/abs/1807.04068v1. 10.1080/17476933.2021.1916919Search in Google Scholar

[16] E. M. S. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algeb. 17 (2007), no. 3, 497–517. 10.1007/s00006-007-0037-8Search in Google Scholar

[17] H. Y. Huo, W. C. Sun, and L. Xiao, Uncertainty principles associated with the offset linear canonical transform. Math. Methods Appl. Sci. 42 (2019), no. 2, 466–474. 10.1002/mma.5353Search in Google Scholar

[18] K. I. Kou, Jian-YuOu, and J. Morais, On uncertainty principle for quaternionic linear canonical transform, Abstr. Appl. Anal. (2013), Article ID 725952. 10.1155/2013/725952Search in Google Scholar

[19] K. I. Kou, J. Morais, and Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis, Math. Meth. Appl. Sci. 36 (2013), no. 9, 1028–1041, https://doi.org/10.1002/mma.2657. Search in Google Scholar

[20] A. H. Dar and M. Y. Bhat, Scaled ambiguity function and scaled Wigner distribution for LCT signals, Optik - Int. J. Light Electron Opt. 267 (2022), 169678. 10.1016/j.ijleo.2022.169678Search in Google Scholar

[21] G. Kutyniok, Ambiguity functions, Wigner distributions and Cohen’s class for LCA groups, J. Math. Anal. Appl. 277 (2003), 589–608. 10.1016/S0022-247X(02)00630-3Search in Google Scholar

[22] Z. Zhang, New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain, Signal Process. 118 (2016)51–61. 10.1016/j.sigpro.2015.06.010Search in Google Scholar

[23] H. Zhao, Q. Ran, J. Ma, and L. Tan, Linear canonical ambiguity function and linear canonical transform moments, Optik - Int. J. Light Electron Opt. 122 (2011), no. 6, 540–543. 10.1016/j.ijleo.2010.04.006Search in Google Scholar

[24] Z. C. Zhang, Sampling theorem for the short-time linear canonical transform and its applications. Signal Proces. 113 (2015), 138–146. 10.1016/j.sigpro.2015.01.020Search in Google Scholar

[25] Z. C. Zhang, Novel Wigner distribution and ambiguity function associated with the linear canonical transform. Optik - Int. J. Light Electron Opt. 127 (2015), no. 12, 4995–5012. 10.1016/j.ijleo.2016.02.028Search in Google Scholar

[26] J. A. Johnston, Wigner distribution and FM radar signal design, Proc. Inst. Electr. Eng. F Radar Signal Process. 136 (1989), 81–88. 10.1049/ip-f-2.1989.0014Search in Google Scholar

[27] O. Bazighifan, An approach for studying asymptotic properties of solutions of neutral differential equations, Symmetry 12 (2020), no. 4, 555. 10.3390/sym12040555Search in Google Scholar

[28] O. Moaaz, E. M. Elabbsy, and E. Shaaban, Oscillation criteria for a class of third order damped differential equations, Arab. J. Math. Sci. 24 (2018), no. 1, 16–30. 10.1016/j.ajmsc.2017.07.001Search in Google Scholar

[29] O. Moaaz, D. Chalishajar, and O. Bazighifan, Some qualitative behaviour of general class of difference equations, Mathematics 7 (2019), 585. 10.3390/math7070585Search in Google Scholar

[30] S. S. Santra, T. Ghosh and O. Bazighifan, Explicit criteria for the oscillation of second order differential equations with several sub-linear neutral coefficients, Adv. Cont. Disc. Models 643 (2020). 10.1186/s13662-020-03101-1Search in Google Scholar

[31] M. Y. Bhat and A. H. Dar, Convolution and correlation theorems for Wigner-Ville distribution associated with the quaternion offset linear canonical transform, Sig. Imag Vid. Processing 16 (2022), 1235–1242, https://doi.org/10.1007/s11760-021-02074-2. Search in Google Scholar

[32] X. Fan, K. I. Kou and M. S. Liu, Quaternion Wigner-Ville distribution associated with linear canonical transforms, Signal Proces. 130 (2017), 129–141. 10.1016/j.sigpro.2016.06.018Search in Google Scholar
Received: 2022-05-12
Revised: 2022-08-25
Accepted: 2022-10-11
Published Online: 2022-11-11

© 2022 Mohammad Younus Bhat et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On some summation formulas
  3. A study of a meromorphic perturbation of the sine family
  4. Asymptotic behavior of even-order noncanonical neutral differential equations
  5. Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation
  6. Starlike and convexity properties of q-Bessel-Struve functions
  7. Mathematical modeling and optimal control of the impact of rumors on the banking crisis
  8. On linear chaos in function spaces
  9. Convergence of generalized sampling series in weighted spaces
  10. Persistence landscapes of affine fractals
  11. Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces
  12. Various notions of module amenability on weighted semigroup algebras
  13. Regularity and normality in hereditary bi m-spaces
  14. On a first-order differential system with initial and nonlocal boundary conditions
  15. On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints
  16. Local linear approach: Conditional density estimate for functional and censored data
  17. Some properties of graded generalized 2-absorbing submodules
  18. Eigenvalue inclusion sets for linear response eigenvalue problems
  19. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation
  20. More properties of generalized open sets in generalized topological spaces
  21. An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
  22. An accurate and efficient local one-dimensional method for the 3D acoustic wave equation
  23. On a weighted elliptic equation of N-Kirchhoff type with double exponential growth
  24. On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces
  25. Entire and meromorphic solutions for systems of the differential difference equations
  26. Multiplication operators on the Banach algebra of bounded Φ-variation functions on compact subsets of ℂ
  27. Mannheim curves and their partner curves in Minkowski 3-space E13
  28. Characterizations of the group invertibility of a matrix revisited
  29. Iterates of q-Bernstein operators on triangular domain with all curved sides
  30. Data analysis-based time series forecast for managing household electricity consumption
  31. A robust study of the transmission dynamics of zoonotic infection through non-integer derivative
  32. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing
  33. Review Article
  34. Remarks on some variants of minimal point theorem and Ekeland variational principle with applications
  35. Special Issue on Recent Methods in Approximation Theory - Part I
  36. Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces
  37. Approximation of integrable functions by general linear matrix operators of their Fourier series
  38. Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation
  39. Approximation of conic sections by weighted Lupaş post-quantum Bézier curves
  40. On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation
  41. Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings
  42. Linear barycentric rational collocation method for solving biharmonic equation
  43. A note on the convergence of Phillips operators by the sequence of functions via q-calculus
  44. Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
  45. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part I
  46. Positive solutions for fractional differential equation at resonance under integral boundary conditions
  47. Source term model for elasticity system with nonlinear dissipative term in a thin domain
  48. A numerical study of anomalous electro-diffusion cells in cable sense with a non-singular kernel
  49. On Opial-type inequality for a generalized fractional integral operator
  50. Special Issue on Advances in Integral Transforms and Analysis of Differential Equations with Applications
  51. Mathematical analysis of a MERS-Cov coronavirus model
  52. Rapid exponential stabilization of nonlinear continuous systems via event-triggered impulsive control
  53. Novel soliton solutions for the fractional three-wave resonant interaction equations
  54. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach
  55. Special Issue on Problems, Methods and Applications of Nonlinear Analysis
  56. Some recent results on singular p-Laplacian equations
  57. Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity
  58. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems
  59. Existence of solutions for a nonlinear problem at resonance
  60. Asymptotic stability of solutions for a diffusive epidemic model
  61. Special Issue on Computational and Numerical Methods for Special Functions - Part I
  62. Fully degenerate Bernoulli numbers and polynomials
  63. Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform
  64. Some identities related to degenerate Stirling numbers of the second kind
  65. Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind
  66. λ-q-Sheffer sequence and its applications
  67. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part I
  68. General decay for a nonlinear pseudo-parabolic equation with viscoelastic term
  69. Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces
  70. Computation of solution of integral equations via fixed point results
  71. Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points
  72. Notes on continuity result for conformable diffusion equation on the sphere: The linear case
https://doi.org/10.1515/dema-2022-0175
Keywords for this article
quaternion algebra; quaternion offset linear canonical transform; Wigner-Ville distribution; ambiguity function; dilation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On some summation formulas
  3. A study of a meromorphic perturbation of the sine family
  4. Asymptotic behavior of even-order noncanonical neutral differential equations
  5. Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation
  6. Starlike and convexity properties of q-Bessel-Struve functions
  7. Mathematical modeling and optimal control of the impact of rumors on the banking crisis
  8. On linear chaos in function spaces
  9. Convergence of generalized sampling series in weighted spaces
  10. Persistence landscapes of affine fractals
  11. Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces
  12. Various notions of module amenability on weighted semigroup algebras
  13. Regularity and normality in hereditary bi m-spaces
  14. On a first-order differential system with initial and nonlocal boundary conditions
  15. On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints
  16. Local linear approach: Conditional density estimate for functional and censored data
  17. Some properties of graded generalized 2-absorbing submodules
  18. Eigenvalue inclusion sets for linear response eigenvalue problems
  19. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation
  20. More properties of generalized open sets in generalized topological spaces
  21. An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
  22. An accurate and efficient local one-dimensional method for the 3D acoustic wave equation
  23. On a weighted elliptic equation of N-Kirchhoff type with double exponential growth
  24. On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces
  25. Entire and meromorphic solutions for systems of the differential difference equations
  26. Multiplication operators on the Banach algebra of bounded Φ-variation functions on compact subsets of ℂ
  27. Mannheim curves and their partner curves in Minkowski 3-space E13
  28. Characterizations of the group invertibility of a matrix revisited
  29. Iterates of q-Bernstein operators on triangular domain with all curved sides
  30. Data analysis-based time series forecast for managing household electricity consumption
  31. A robust study of the transmission dynamics of zoonotic infection through non-integer derivative
  32. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing
  33. Review Article
  34. Remarks on some variants of minimal point theorem and Ekeland variational principle with applications
  35. Special Issue on Recent Methods in Approximation Theory - Part I
  36. Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces
  37. Approximation of integrable functions by general linear matrix operators of their Fourier series
  38. Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation
  39. Approximation of conic sections by weighted Lupaş post-quantum Bézier curves
  40. On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation
  41. Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings
  42. Linear barycentric rational collocation method for solving biharmonic equation
  43. A note on the convergence of Phillips operators by the sequence of functions via q-calculus
  44. Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
  45. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part I
  46. Positive solutions for fractional differential equation at resonance under integral boundary conditions
  47. Source term model for elasticity system with nonlinear dissipative term in a thin domain
  48. A numerical study of anomalous electro-diffusion cells in cable sense with a non-singular kernel
  49. On Opial-type inequality for a generalized fractional integral operator
  50. Special Issue on Advances in Integral Transforms and Analysis of Differential Equations with Applications
  51. Mathematical analysis of a MERS-Cov coronavirus model
  52. Rapid exponential stabilization of nonlinear continuous systems via event-triggered impulsive control
  53. Novel soliton solutions for the fractional three-wave resonant interaction equations
  54. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach
  55. Special Issue on Problems, Methods and Applications of Nonlinear Analysis
  56. Some recent results on singular p-Laplacian equations
  57. Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity
  58. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems
  59. Existence of solutions for a nonlinear problem at resonance
  60. Asymptotic stability of solutions for a diffusive epidemic model
  61. Special Issue on Computational and Numerical Methods for Special Functions - Part I
  62. Fully degenerate Bernoulli numbers and polynomials
  63. Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform
  64. Some identities related to degenerate Stirling numbers of the second kind
  65. Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind
  66. λ-q-Sheffer sequence and its applications
  67. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part I
  68. General decay for a nonlinear pseudo-parabolic equation with viscoelastic term
  69. Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces
  70. Computation of solution of integral equations via fixed point results
  71. Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points
  72. Notes on continuity result for conformable diffusion equation on the sphere: The linear case
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 25.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2022-0175/html
Scroll to top button