# Faster Optimization in S-Graphs Exploiting Hierarchy

Hriday Bavle1, Jose Luis Sanchez-Lopez1, Javier Civera2, Holger Voos1 *This
work was partially funded by the Fonds National de la Recherche of Luxembourg
(FNR), under the project C22/IS/17387634/DEUS, and by the Spanish Government
under Grant PID2021-127685NB-I00 and by the Aragón Government under Grant DGA
T45 17R/FSE. For the purpose of Open Access, the author has applied a CC BY
public copyright license to any Author Accepted Manuscript version arising
from this submission.1Authors are with the Automation and Robotics Research
Group, Interdisciplinary Centre for Security, Reliability and Trust,
University of Luxembourg. Holger Voos is also associated with the Faculty of
Science, Technology and Medicine, University of Luxembourg, Luxembourg.
{hriday.bavle, joseluis.sanchezlopez<EMAIL_ADDRESS>is with I3A,
Universidad de Zaragoza, Spain<EMAIL_ADDRESS>
###### Abstract

3D scene graphs hierarchically represent the environment appropriately
organizing different environmental entities in various layers. Our previous
work on situational graphs [1], [2] extends the concept of 3D scene graph to
SLAM by tightly coupling the robot poses with the scene graph entities,
achieving state-of-the-art results. Though, one of the limitations of S-Graphs
is scalability in really large environments due to the increased graph size
over time, increasing the computational complexity.

To overcome this limitation in this work we present an initial research of an
improved version of S-Graphs exploiting the hierarchy to reduce the graph size
by marginalizing redundant robot poses and their connections to the
observations of the same structural entities. Firstly, we propose the
generation and optimization of room-local graphs encompassing all graph
entities within a room-like structure. These room-local graphs are used to
compress the S-Graphs marginalizing the redundant robot keyframes within the
given room. We then perform windowed local optimization of the compressed
graph at regular time-distance intervals. A global optimization of the
compressed graph is performed every time a loop closure is detected. We show
similar accuracy compared to the baseline while showing a 39.81% reduction in
the computation time with respect to the baseline.

##  I Introduction

Figure 1: Generation of S-Graph utilizing the improved optimization technique.
The redundant keyframes within a given room are marginalized (red spheres)
after performing room-local optimization. The remaining keyframes (black
spheres) along with their connected walls-rooms-floors, undergo local and
global optimization steps. The orange boxes over the keyframes highlight the
S-Graph undergoing the global optimization step where all the keyframes are
utilized except the marginalized ones. Figure 2: System architecture of the
presented approach, divided into front-end and back-end. The front-end
comprises of modules for generating the four-layered S-Graph. In the back-end,
the generated S-Graph undergoes different optimization steps. In this example
keyframe $X_{t+1}$ belonging to the room is marginalized out to compress the
graph.

S-Graphs [1], [2] enable mobile robots with a deep understanding of their
situation through the real-time generation of a 3D scene graph tightly coupled
to the robot poses in the form of a four-layered optimizable graph. S-Graphs
consists of a keyframes layer comprising the robot poses, a walls layer
encompassing the wall-plane entities, a rooms layer constraining the wall-
planes of appropriate rooms, and a floors layer constraining all the rooms
within a given floor level. As the robot explores large-scale environments,
this optimizable graph increases in the number of keyframes and semantic
elements thus increasing the computational complexity and time.

To overcome the mentioned limitation, in this preliminary work, we present an
approach for exploiting the inherent hierarchy extracted by the baseline
S-Graphs to reduce the computational complexity while maintaining the robot
pose and its map accuracy. Every time a room is detected all the enclosed
robot keyframes, along with its attached wall-planes, and the room node form a
room-local graph which is locally optimized and then utilized to marginalize
the redundant keyframe nodes. In this current work, each room is kept with
only one robot pose, the rest are set to marginalized. This marginalization is
propagated to the entire S-Graph to generate a compressed version of the
graph. Furthermore, we perform optimization of the compressed graph in two
steps, local and global optimization. The local optimization is performed over
a window of keyframes as the robot explores the environment while the global
optimization over the complete compressed graph is performed only on loop
closures. Initial experimentation performed over simulated and real data show
reduced computational complexity of our proposed optimization framework while
maintaining the robot pose and map accuracy. Fig. 1 highlights the presented
approach where the underlying S-Graph undergoes a global optimization step
that excludes the redundant keyframes (colored in red) within a given room
structure.

(a) Proposed

(b) S-Graphs+ [2]

(c) Ground Truth

Figure 3: Qualitative results of a real-world experiment showing the map
accuracy by our proposed work and baseline S-Graphs+ with respect to the
ground truth map. The black spheres are the keyframes while the red spheres in
our proposed work are the marginalized keyframes belonging to the rooms.

##  II Related Work

Several methods exist in the literature for performing factor graph
compression maintaining the map accuracy while bonding the computational
complexity. [3] presents an incremental soothing and mapping approach
considering the information gain of measurement to the state estimates, thus
instead of solving the entire graph only solving parts of the graph affected
by the new measurements. [4] presents a hierarchical optimization technique
grouping nodes into different subgraphs at a lower level based on simple
distance-based criteria. These subgraphs form a reduced higher-level graph
which is then optimized quickly and efficiently. Authors [5] present an
information-theoretic approach for factor graph compression where laser scans
measurements and their corresponding robot poses are removed such that the
expected loss of information with respect to the current map is minimized.

In recent years, 3D scene graphs have emerged as an efficient way to represent
the environment, [2], [6], [7]. Although these works efficiently represent the
environment in different hierarchical levels, there can be redundant
information stored at different levels which is neither removed nor
compressed, which can lead to increased computational complexity while
exploring very large-scale environments. Inspired by factor graph compression
techniques, in this work, we present an ongoing research work of marginalizing
information from 3D scene graphs exploiting its underlying hierarchy.

##  III Proposed Approach

Fig. 2 present an overview of the proposed work. The robot odometry and 3D
LiDAR information are utilized for generating the four-layered S-Graph. The
point cloud information mapped at each robot keyframe is utilized to extract
wall planes, creating the walls layer. The extracted walls are utilized by the
room detector to extract rooms, and by the floor detector to extract floors.
During the creation of the S-Graph the underlying optimization step can be
divided into three different stages namely 1. Room-Local Optimization 2. Local
Optimization 3. Global Optimization, all explained in the following sections.

###  III-A Room-Local Optimization

Room-Local optimization generates a local S-Graph lying within the given
structure of a room, whenever a room is detected. To generate a room-local
S-Graph we first look for a subset keyframes $K_{s}\subset K$ which are
bounded by the walls of a four-wall room ${R}_{j}$ which satisfy the following
condition:

$\displaystyle\forall\boldsymbol{n}_{i}\in{R}_{j},\quad\boldsymbol{n_{i}}\cdot(\boldsymbol{k}_{i}-\boldsymbol{\leftidx{{}^{M}}{\Pi}}_{i})<$

where $\boldsymbol{\leftidx{{}^{M}}{\Pi}}_{i}=\boldsymbol{n}_{i}\cdot d_{i}$
and $\boldsymbol{n}$, $d$ are the normal and the distance of wall-plane $i$.
$\boldsymbol{k}_{i}$ is the translation of the keyframe $K_{i}$. Furthermore,
we add in the room-local S-Graph all the wall-planes $\boldsymbol{\pi}_{i}$
connected keyframe subset $K_{s}$. Lastly, we incorporate the room vertex
$R_{j}$ and the floor $F_{l}$. Inspired from [8], all the keyframes outside
the room but connected to the wall-planes $\boldsymbol{\pi}_{i}$ are
incorporated in room-local graph but kept constant during optimization.

The room-local S-Graph is then optimized to perform a room-local optimization
step which optimizes only a subset of the entire graph. Since all the robot
poses within a given room observe the same structure, only the first robot
pose within is room maintained. Other robot poses can be safely marginalized
from the graph along with their observations. This optimized and marginalized
information is used to create a compressed version of the S-Graph.

###  III-B Local Optimization

The marginalized information from the room-local optimization step is included
to generate a compressed S-Graph excluding the marginalized keyframes and
their edges. Marginalization leads to a generation of a disconnected graph.
These disconnections are between the marginalized keyframes and their non-
marginalized neighbors. To obtain a connected graph, we check for the $k$
edges $e_{k}$, $k=\\{1,\dots,k\\}$ of the $i\textsuperscript{th}$ marginalized
keyframe $K_{i}$ with $n$ non-marginalized neighbors $K_{n}$,
$n=\\{1,\dots,n\\}$ and connect the closest non-marginalized neighbors $K_{1}$
and $K_{2}$ with a new edge $e_{f}$. The information matrix of the new edge is
the summation of the information matrices of the eliminated edges in $e_{k}$.

After the generation of the compressed S-Graph, the local optimization
performs a windowed optimization over the last $N$ subset of the keyframes
$K_{N}$ and their connected walls, rooms, and floors within the compressed
graph. The additional keyframes not in $K_{N}$ that observe the walls and
rooms are included in the optimization but kept constant.

###  III-C Global Optimization

Global optimization considers the optimization of the entire compressed
S-Graph without the marginalized keyframes and their edges, if a loop closure
is detected between the robot keyframes. In global optimization, only the
first keyframe is fixed for optimization stability.

##  IV Experimental Validation

We validate our approach on different simulated and real datasets captured
using a velodyne 3D LiDAR and a legged robotic platform. Currently, we compare
our approach with the baseline S-Graphs+ [2]. In both the simulated and real
experiments, the robot performs several random and repeated trajectories in
indoor environments with different room configurations. For the simulated
experiments, we report the ATE of the robot pose along with the computation
time required during the optimization steps. For real experiments, we provide
qualitative results in terms of map quality generated by our approach and the
baseline while comparing its computation time.

Table. I shows the results obtained on two simulated datasets when comparing
them against the baseline S-Graphs+ [2]. As can be seen from the table, our
proposed method is able to provide better or similar accuracy in terms of the
ATE when compared with the baseline, but given our proposed method is able to
improve the computation time of the global optimization with respect to the
baseline by $33\%$.

Figure. 3 shows qualitative results of the obtained map during the execution
of the real experiment. As can be appreciated from the figure our proposed
approach is able to provide the same map quality when comparing it with the
baseline and the ground truth map. But in this experiment as well, the average
computation of S-Graphs+ is $129$ ms while our proposed approach has an
average computation time of $60$ ms.

TABLE I: ATE (cm) and the computation time [ms] of S-Graphs+ and our proposed method on simulated datasets. Best results are boldfaced. |  ATE (cm) $\downarrow$ |  Computation Time (mean) [ms] $\downarrow$
---|---|---
| Dataset |  |
Method | C1F1 | C1F2 | C1F1 | C1F2
S-Graphs+ [2] | 4.20 | 1.59 | 71.0 | 69.0
Proposed (ours) | 4.09 | 1.73 | 53.0 | 41.0

##  V Conclusion

In this paper, we present an initial research strategy for compression and
faster optimization for the real-time generation of situational graphs. We
exploit the inherent hierarchical representations generated by S-Graph to
generate room-local optimizable graphs, which are used to generate a
compressed version of the underlying S-Graph. We also propose to break down
the optimization of the compressed graph into two steps namely, local
(windowed) optimization and global optimization. We perform preliminary
experiments to compare our approach to the baseline S-Graphs+ achieving
similar results in terms of accuracy but improved computation time. As for
future works, we would further validate our presented approach over large-
scale indoor environments with multiple floors incorporating the floor-local
optimization step in the graph in addition to the room-local optimization
step.

## References

  * [1] H. Bavle, J. L. Sanchez-Lopez, M. Shaheer, J. Civera, and H. Voos, “Situational graphs for robot navigation in structured indoor environments,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 4, pp. 9107–9114, 2022.
  * [2] ——, “S-graphs+: Real-time localization and mapping leveraging hierarchical representations,” _IEEE Robotics and Automation Letters_ , vol. 8, no. 8, pp. 4927–4934, 2023.
  * [3] M. Kaess, A. Ranganathan, and F. Dellaert, “isam: Incremental smoothing and mapping,” _IEEE Transactions on Robotics_ , vol. 24, no. 6, pp. 1365–1378, 2008.
  * [4] G. Grisetti, R. Kümmerle, C. Stachniss, U. Frese, and C. Hertzberg, “Hierarchical optimization on manifolds for online 2d and 3d mapping,” in _2010 IEEE International Conference on Robotics and Automation_ , 2010, pp. 273–278.
  * [5] H. Kretzschmar and C. Stachniss, “Information-theoretic compression of pose graphs for laser-based slam,” _The International Journal of Robotics Research_ , vol. 31, no. 11, pp. 1219–1230, 2012. [Online]. Available: https://doi.org/10.1177/0278364912455072
  * [6] U.-H. Kim, J.-M. Park, T. jin Song, and J.-H. Kim, “3-D Scene Graph: A Sparse and Semantic Representation of Physical Environments for Intelligent Agents,” _IEEE Transactions on Cybernetics_ , vol. 50, no. 12, pp. 4921–4933, dec 2020.
  * [7] N. Hughes, Y. Chang, and L. Carlone, “Hydra: A real-time spatial perception system for 3d scene graph construction and optimization,” in _Robotics: Science and Systems_ , 2022.
  * [8] R. Mur-Artal and J. D. Tardós, “ORB-SLAM2: an Open-Source SLAM System for Monocular, Stereo and RGB-D Cameras,” _IEEE transactions on robotics_ , vol. 33, no. 5, pp. 1255–1262, 2017.


Uncertainty Inequalities for 3D Octonionic-valued Signals Associated with
Octonion Offset Linear Canonical Transform

M. Younus Bhat${}^{1},$ and Aamir H. Dar2

1 Department of Mathematical Sciences, Islamic University of Science and
Technology Awantipora, Pulwama, Jammu and Kashmir 192122, India.E-mail: g
<EMAIL_ADDRESS>
2 Department of Mathematical Sciences, Islamic University of Science and
Technology Awantipora, Pulwama, Jammu and Kashmir 192122, India.E-mail:
<EMAIL_ADDRESS>
> Abstract. The octonion offset linear canonical transform ($\mathbb{O}-$OLCT)
> can be defined as a time-shifted and frequency-modulated version of the
> octonion linear canonical transform ($\mathbb{O}-$LCT), a more general
> framework of most existing signal processing tools. In this paper, we first
> define the ($\mathbb{O}-$OLCT) and provide its closed-form representation.
> Based on this fact, we study some fundamental properties of proposed
> transform including inversion formula, norm split and energy conservation.
> The crux of the paper lies in the generalization of several well known
> uncertainty relations for the ($\mathbb{O}-$OLCT) that include Pitt’s
> inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality
> and local uncertainty inequalities.
>
>
> Keywords: Quaternion offset linear canonical transform(QOLCT); Octonion;
> Octonion Fourier transform$(\mathbb{O}-$OFT); Octonion offset linear
> canonical transform($\mathbb{O}-$OLCT); Uncertainty principle.
>
>
> 2000 Mathematics subject classification:  42B10; 43A32; 94A12; 42A38; 30G30.

##  1\.  Introduction

The hyper-complex Fourier transform(FT) is of the interest in the present era.
It treats multi-channel signals as an algebraic whole without losing the
spectral relations. Presently, many hyper-complex FTs exists in literature
which are defined by different approaches, see [1, 2]. The developing interest
in hyper-complex FTs including applications in watermarking, color image
processing, image filtering, pattern recognition and edge detection [3]-[8].
Among the various hyper-complex FTs, the most basic ones are the quaternion
Fourier transforms(QFTs). QFTs are most widely studied in recent years because
of its wide applications in optis and signal processing. QFT[9] is very useful
in Cayley-Dickson algebra of order 4(Quaternions) as it is a substitute to the
two-dimensional complex Fourier transform (CFT). Various properties and
applications of the QFT were established in [10]-[13]. The generalization of
quaternion Fourier transform (QFT)is quaternion linear canonical transform
(QLCT), which is more effective signal processing tool than QFT due to its
extra parameters, see[14, 15, 16, 17, 18, 19, 20].
Later, the quaternion linear canonical transform (QLCT) with four parameters
has been generalized to a six parameter transform known as quaternion offset
linear canonical transform (QOLCT). Due to the time shifting and frequency
modulation parameters, the QOLCT has gained more flexibility over classical
QLCT. Hence has found wide applications in image and signal processing, see
[21, 22, 23, 24].
On the other hand the Cayley-Dickson algebra of order 8 is known as octonion
algebra which deserve special attention in the hyper-complex signal
processing. The octonion Fourier transform ($\mathbb{O}-$FT) was proposed by
Hahn and Snopek in 2011[25]. From then $\mathbb{O}-$FT is becoming the hot
area of research in modern signal processing community. Some properties and
uncertainty relations and applications associated with $\mathbb{O}-$FT have
been studied, see[27, 28, 29, 30].In 2021 Gao and Li [31] proposed octonion
linear canonical transform($\mathbb{O}-$LCT) as a generalization of
$\mathbb{O}-$FT by substituting the Fourier kernel with the LCT kernel. They
established some vital properties like inversion formula, isometry, Riemann-
Lebesgue lemma and proved Heisenberg’s and Donoho-Stark’s uncertainty
principles. The generalization of $\mathbb{O}-$FT to other transforms viz
linear canonical transform, offset linear canonical transform, Stockwell
transform, Quadratic phase transform etc. is still in its infancy.

So motivated and inspired by this, we shall propose the novel octonion offset
linear canonical transform ($\mathbb{O}-$OLCT) which have never been proposed
up to date, therefore it is worthwhile to rigorously study the Octonion offset
linear canonical transform $\mathbb{O}-$OLCT which can be productive for
signal processing theory and applications.
The highlights of the paper are pointed out below:

  * •

To introduce a novel integral transform coined as the octonion offset linear
canonical transform ($\mathbb{O}-$OLCT).

  * •

To study the fundamental properties of the proposed transform, including the
closed-form representation, norm split into quaternions, inversion formula and
energy conservation.

  * •

To formulate several classes of uncertainty inequalities, such as the Pitt’s
inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality and
local uncertainty inequality associated with the octonion offset linear
canonical transform ($\mathbb{O}-$OLCT).

The rest of the paper is organized as follows: In Section 2, some general
definitions and basic properties of octonions are summarized. The definition
and the properties of the $\mathbb{O}-$OLCT are studied in Section 3. In
Section 3, we develop a series of uncertainty inequalities such as the Pitt’s
inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality and
local uncertainty inequality associated with the $\mathbb{O}-$OLCT. Finally, a
conclusion is extracted in Section 5.

##  2\. Preliminaries

In this section, we collect some basic facts on the octonion algebra and the
offset linear canonical transform(OLCT), which will be needed throughout the
paper.

###  2.1. Octonion algebra

The octonion algebra denoted by $\mathbb{O},$ [32] is generated by the eighth-
order Cayley-Dickson construction. According to His construction, a
hypercomplex number $o\in\mathbb{O}$ is an ordered pair of quaternions
$q_{0},q_{1}\in\mathbb{H}$

$\displaystyle o$ $\displaystyle=$ $\displaystyle(q_{0},q_{1})$
$\displaystyle=$ $\displaystyle((z_{0},z_{1}),(z_{2},z_{3}))$ $\displaystyle=$
$\displaystyle q_{0}+q_{1}.e_{4}$ $\displaystyle=$
$\displaystyle(z_{0}+z_{1}.e_{2})+(z_{2}+z_{3}.e_{2}).e_{4}$

which has equivalent form

$o=s_{o}+\sum_{i=1}^{7}s_{i}e_{i}=s_{0}+s_{1}e_{1}+s_{2}e_{2}+s_{3}e_{3}+s_{4}e_{4}+s_{5}e_{5}+s_{6}e_{6}+s_{7}e_{7}$
(2.2)

that is $o$ is a hypercomplex number defined by eight real numbers
$s_{i},i=0,1,\dots,7$ and seven imaginary units $e_{i}$ where $i=1,2,\dots,7.$
The octonion algebra is non-commutative and non-associative algebra. The
multiplication of imaginary units in the Cayley-Dickson algebra of octonions
are presented in Table I or in diagram called Fano scheme, shown in Figure 1.

Table I

Multiplication Rules in Octonion Algebra.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\cdot&1&e_{1}&e_{2}&e_{3}&e_{4}&e_{5}&e_{6}&e_{7}\\\
\hline\cr 1&1&e_{1}&e_{2}&e_{3}&e_{4}&e_{5}&e_{6}&e_{7}\\\ \hline\cr
e_{1}&e_{1}&-1&e_{3}&-e_{2}&e_{5}&-e_{4}&-e_{7}&e_{6}\\\ \hline\cr
e_{2}&e_{2}&-e_{3}&-1&e_{1}&e_{6}&e_{7}&-e_{4}&-e_{5}\\\ \hline\cr
e_{3}&e_{3}&e_{2}&-e_{1}&-1&e_{7}&-e_{6}&e_{5}&-e_{4}\\\ \hline\cr
e_{4}&e_{4}&-e_{5}&-e_{6}&-e_{7}&-1&e_{1}&e_{2}&e_{3}\\\ \hline\cr
e_{5}&e_{5}&e_{4}&-e_{7}&e_{6}&-e_{1}&-1&-e_{3}&e_{2}\\\ \hline\cr
e_{6}&e_{6}&e_{7}&e_{4}&-e_{5}&-e_{2}&e_{3}&-1&-e_{1}\\\ \hline\cr
e_{7}&e_{7}&-e_{6}&e_{5}&e_{4}&-e_{3}&-e_{2}&e_{1}&-1\\\ \hline\cr\end{array}$

The conjugate of an octonion is defined as

$\overline{o}=s_{0}-s_{1}e_{1}-s_{2}e_{2}-s_{3}e_{3}-s_{4}e_{4}-s_{5}e_{5}-s_{6}e_{6}-s_{7}e_{7}$
(2.3)

Therefore norm is defined by $|o|=\sqrt{o\overline{o}}$ and
$|o|^{2}=\sum_{i=o}^{7}s_{i}.$ Also $|o_{1}o_{2}|=|o_{1}||o_{2}|,\forall
o_{1},o_{2}\in\mathbb{O}.$

From (2.1) it is evident that every $o\in\mathbb{O}$ can be represented in
quaternion form as

$o=a+be_{4}$ (2.4)

where $a=s_{0}+s_{1}e_{1}+s_{2}e_{2}+s_{3}e_{3}$ and
$b=s_{4}+s_{5}e_{1}+s_{6}e_{2}+s_{7}e_{3}$ are both quaternions. By direct
verification we have following lemma.

######  Lemma 2.1.

Let $a,b\in\mathbb{H},$ then
(1) $e_{4}a=\overline{a}e_{4};$ (2) $e_{4}(ae_{4})=-\overline{a};$
$(3)\quad(ae_{4})e_{4}=-a;$
(4) $a(be_{4})=(ba)e_{4}$; (5) $(ae_{4})b=(a\overline{b})e_{4};$ (6)
$(ae_{4})(be_{4})=-\overline{b}a.$

It is clear from above Lemma that, for an octonion
$a+be_{4},a,b\in\mathbb{H},$ we have

$\overline{a+be_{4}}=\overline{a}-be_{4}$ (2.5)

and

$|a+be_{4}|^{2}=|a|^{2}+|b|^{2}.$ (2.6)

An octonion-valued function $f:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ has
following explicit form

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle
f_{0}+f_{1}(x)e_{1}+f_{2}(x)e_{2}+f_{3}(x)e_{3}+f_{4}(x)e_{4}+f_{5}(x)e_{5}+f_{6}(x)e_{6}+f_{7}(x)e_{7}$
$\displaystyle=$ $\displaystyle g(x)+h(x)e_{4}$

where each $f_{i}(x)$ is a real vaiued functions, $g,h\in\mathbb{H}$ and
$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}.$ For each octonion-valued function
$f(x)$ over $\mathbb{R}^{3}$ and $1\leq p<\infty,$ the $L^{p}-$norm of $f$ is
defined by

$\|f\|^{p}_{p}=\int_{\mathbb{R}^{3}}|f(x)|^{p}dx.$ (2.8)

And for $p=\infty$, then the $L^{\infty}$-norm is defined by

$\|f\|_{\infty}=esssup_{x\in\mathbb{R}^{3}}|f(x)|.$ (2.9)

###  2.2. Offset linear canonical transform

The offset linear canonical transform(OLCT)[33] of any function
$f:\mathbb{R}\longrightarrow\mathbb{O}$ with respect to the matrix parameter
$A=(a,b,c,d,e,\tau,\eta)$ is defined as

$\mathcal{O}_{A}\big{[}f(x)\big{]}w)=\int_{\mathbb{R}}f(x)K_{A}^{i}(x,w)dx.$
(2.10)

with

$K_{A}(x,w)=\dfrac{1}{\sqrt{2\pi|b|}}e^{\frac{{i}}{2b}\big{[}ax^{2}-2x(w-\tau)-2w(d\tau-b\eta)+d(w^{2}+\tau^{2})-\frac{\pi}{2}\big{]}},b\neq
0\,$ (2.11)

where $e^{\frac{-i\pi}{4}}$ is the polar form of $\frac{1}{\sqrt{i}}.$
And the Offset linear canonical transform in quaternion[34] setting is given
as:

Let $A_{s}=\begin{bmatrix}a_{s}&b_{s}|&\tau_{s}\\\
c_{s}&d_{s}|&\eta_{s}\end{bmatrix}\in\mathbb{R}^{2\times 2}$ be a matrix
parameter satisfying ${\rm det}(A_{s})=1$, for $s=1,2$. Then QLCT of signal
$f\in L^{2}\left(\mathbb{R}^{2},\mathbb{H}\right)$ is defined by

$\mathcal{O}_{i,j}^{A_{1},A_{2}}[f]({w})=\int_{\mathbb{R}^{2}}f({x})K_{A_{1}}^{{i}}(x_{1},w_{1})K_{A_{2}}^{{j}}(x_{2},w_{2})\rm{d}{x},$
(2.12)

where ${w}=(w_{1},w_{2}),x=(x_{1},x_{2})\in\mathbb{R}^{2}$ , and the kernel
signals $K_{A_{1}}^{{i}}(x_{1},w_{1})$, $K_{A_{2}}^{{j}}(x_{2},w_{2})$ are
respectively given by

$K_{A_{1}}^{{i}}(x_{1},w_{1})=\dfrac{1}{\sqrt{2\pi
b_{1}{i}}}\,e^{\frac{{i}}{2b_{1}}\big{[}a_{1}x_{1}^{2}-2x_{1}(w_{1}-\tau_{1})-2w_{1}(d_{1}\tau_{1}-b_{1}\eta_{1})+d_{1}(w_{1}^{2}+\tau_{1}^{2})\big{]}},b_{1}\neq
0\,$ (2.13)

$K_{A_{2}}^{{j}}(x_{2},w_{2})=\dfrac{1}{\sqrt{2\pi
b_{2}{j}}}\,e^{\frac{{j}}{2b_{2}}\big{[}a_{2}x_{2}^{2}-2x_{2}(w_{2}-\tau_{2})-2w_{2}(d_{2}\tau_{2}-b_{2}\eta_{2})+d_{2}(w_{2}^{2}+\tau_{2}^{2})\big{]}},\,b_{2}\neq
0\,$ (2.14)

##  3\. Octonionic offset linear canonical transform

In this section we shall formally introduce the notion of proposed transform
”The Octonion Offset Linear Canonical Transform($\mathbb{O}-$OLCT) and study
its important properties like closed form representation, inversion formula,
split of norm and energy conservation. Prior to establishing the fundamental
properties for the proposed transform, we shall revisit the definitions of the
octonion Fourier transform($\mathbb{O}-$FT)[35] and the octonion linear
canonical transform($\mathbb{O}-$LCT)[31]. Lets begin with definition of
$\mathbb{O}-$FT.

###  3.1. Octonion Fourier transform

Let $\mu_{i},i=1,2\dots,7$ denote the imaginary units in Cayley-Dickson
algebra of octonions, then for an octonion-valued function $f\in
L^{1}(\mathbb{R}^{3},\mathbb{O})$ the one dimensional $\mathbb{O}-$FT [35] is
given by

$\mathcal{F}_{\mu_{4}}\\{f\\}(w)=\int_{\mathbb{R}}f(x)e^{-\mu_{4}2\pi xw}dx,$
(3.1)

with inversion

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\mu_{4}}^{-1}\\{\mathcal{F}_{\mu_{4}}\\{f\\}\\}(x)$
(3.2) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\mathcal{F}_{\mu_{4}}\\{f\\}(w)e^{\mu_{4}2\pi
xw}dx,$

And for octonion valued function $f\in L^{1}(\mathbb{R}^{3},\mathbb{O})\cap
L^{2}(\mathbb{R}^{3},\mathbb{O})$, the three dimensional $\mathbb{O}-$FT [20,
24] is defined as

$\mathcal{F}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)=\int_{\mathbb{R}^{3}}f(x)e^{-\mu_{1}2\pi
x_{1}w_{1}}e^{-\mu_{2}2\pi x_{2}w_{2}}e^{-\mu_{4}2\pi x_{3}w_{3}}dx,$ (3.3)

with inversion

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\mathcal{F}^{-1}_{\mu_{1},\mu_{2},\mu_{4}}\\{\mathcal{F}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}\\}(x)$
(3.4) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\mathcal{F}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)e^{-\mu_{1}2\pi
x_{1}w_{1}}e^{-\mu_{2}2\pi x_{2}w_{2}}e^{-\mu_{4}2\pi x_{3}w_{3}}dw,$

where $w=(w_{1},w_{2},w_{3}),\quad x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}.$
The multiplication in the above integrals is done from left to right as the
octonion is non- associative. And the order of imaginary units in (3.3) is not
accidental, see[36]. The $\mathbb{O}-$FT of 3D octonion-valud signals follows
the multiplication rules of Table-I(octonion algebra) because the octonion-
valued 3D signals has octonion structure.

###  3.2. Octonion linear canonical transform

In 2021 Gao,W.B and Li,B.Z [31] introduced linear canonical transform in
octonion setting they called it the octonion linear canonical transform
($\mathbb{O}-$LCT) and defined it as
Let $f\in L^{1}(\mathbb{R}^{3},\mathbb{O}),$ then the one dimensional
$\mathbb{O}-$LCT with respect to the uni-modular matrix $A=(a,b,c,d)$ is given
by

$\mathcal{L}^{A}_{\mu_{4}}\\{f(x)\\}w)=\int_{\mathbb{R}}f(x)K_{A}^{\mu_{4}}(x,w)dx,$
(3.5)

where

$K^{\mu_{4}}_{A}(x,w)=\dfrac{1}{\sqrt{2\pi|b|}}e^{\frac{{\mu_{4}}}{2b}\big{[}ax^{2}-2xw-+dw^{2}-\frac{\pi}{2}\big{]}},\quad
b\neq 0\,$

with inversion

$\displaystyle
f(x)=\int_{\mathbb{R}}\mathcal{L}^{A}_{\mu_{4}}K_{A}^{-\mu_{4}}(x,w)dx,$ (3.6)

where $K_{A}^{-\mu_{4}}(x,w)=K_{A^{-1}}^{\mu_{4}}(w,x)$ and
$A^{-1}=(d,-b,-c,a).$

And for octonion valued function $f\in L^{1}(\mathbb{R}^{3},\mathbb{O})\cap
L^{2}(\mathbb{R}^{3},\mathbb{O})$, the three dimensional $\mathbb{O}-$LCT with
respect to the matrix parameter $A_{k}=(a_{k},b_{k},c_{k},d_{k}),$ satisfying
$det(A_{k})=,\quad k=1,2,3$ is defined as

$\mathcal{L}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)=\int_{\mathbb{R}^{3}}f(x)K^{\mu_{1}}_{A_{1}}(x_{1},w_{1})K^{\mu_{2}}_{A_{1}}(x_{2},w_{2})K^{\mu_{4}}_{A_{3}}(x_{3},w_{3})dx$
(3.7)

where $x=(x_{1},x_{2},x_{3}),\,w=(w_{1},w_{2},w_{3}),$ and
$K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})$ , $K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})$ and
$K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})$ are kernel signals given by

$K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})=\dfrac{1}{\sqrt{2\pi|b_{1}|}}e^{\frac{{\mu_{1}}}{2b_{1}}\big{[}a_{1}x^{2}_{1}-2x_{1}w_{1}+d_{1}w^{2}_{1}-\frac{\pi}{2}\big{]}},\quad
b_{1}\neq 0\,$
$K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})==\dfrac{1}{\sqrt{2\pi|b_{2}|}}e^{\frac{{\mu_{2}}}{2b_{2}}\big{[}a_{2}x^{2}_{2}-2x_{2}w_{2}+d_{2}w^{2}_{2}-\frac{\pi}{2}\big{]}},\quad
b_{2}\neq 0\,$

and

$K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})=\dfrac{1}{\sqrt{2\pi|b_{3}|}}e^{\frac{{\mu_{4}}}{2b_{3}}\big{[}a_{3}x^{2}_{3}-2x_{3}w_{3}+d_{3}w^{2}_{3}-\frac{\pi}{2}\big{]}},\quad
b_{3}\neq 0.$

Now we are in a position to define octonion offset linear canonical transform
($\mathbb{O}-$OLCT).

According to the one dimensional octonion Fourier transform
($\mathbb{O}-$FT)(3.1) and the one dimensional octonion linear canonical
transform ($\mathbb{O}-$OLCT) (3.5) , we can obtain the definition of one
dimensional octonion offset linear canonical transform($\mathbb{O}-$OLCT).

######  Definition 3.1 (One dimensional $\mathbb{O}-$OLCT).

Let $f\in L^{1}(\mathbb{R},\mathbb{O}),$ then one dimensional
$\mathbb{O}-$OLCT with respect a uni-modular matrix parameter
$A=(a,b,c,d,e,\tau,\eta)$ is defined as follows:

$\mathcal{O}^{A}_{\mu_{4}}\\{f(x)\\}w)=\int_{\mathbb{R}}f(x)K_{A}^{\mu_{4}}(x,w)dx.$
(3.8)

where

$K^{\mu_{4}}_{A}(x,w)=\dfrac{1}{\sqrt{2\pi|b|}}e^{\frac{{\mu_{4}}}{2b}\big{[}ax^{2}-2x(w-\tau)-2w(d\tau-b\eta)+d(w^{2}+\tau^{2})-\frac{\pi}{2}\big{]}},\quad
b\neq 0\,$ (3.9)

The following lemma gives the relationship of one dimensional
$\mathbb{O}-$OLCT and one dimensional $\mathbb{O}-$FT of an octonion-valued
signals.

######  Lemma 3.1.

The one dimensional $\mathbb{O}-$OLCT of a signal $f\in
L^{1}(\mathbb{R},\mathbb{O})$ can be reduced to one dimensional
$\mathbb{O}-$FT (3.1) as

$\mathcal{O}^{A}_{\mu_{4}}\\{f(x)\\}w)=\frac{1}{\sqrt{2\pi|b|}}\mathcal{F}_{\mu_{4}}\left\\{f(x)e^{[\frac{a}{2b}x^{2}+\frac{1}{b}x\tau]}\right\\}\left(\frac{w}{2\pi|b|}\right)e^{\mu_{4}[\frac{d}{2b}(w^{2}+\tau^{2})+\frac{w}{b}(b\eta-d\tau)-\frac{\pi}{4}]}$
(3.10)

where $b\neq 0$ and

$\mathcal{F}_{\mu_{4}}\\{f(x)\\}(w)=\int_{\mathbb{R}}f(x)e^{-\mu_{4}2\pi
xw}dx$ (3.11)

represents the one dimensional $\mathbb{O}-$FT of an octonion-valued signal
$f(x)$.

By applying Lemma 3.1 and (3.2), we get the formula for the inversion of one
dimensional $\mathbb{O}-$OLCT, which is given below as a theorem.

######  Theorem 3.1.

Let $f$ be an octonion-valued signal $\in L^{1}(\mathbb{R},\mathbb{O}).$ Then
the inversion formula of the one dimensional $\mathbb{O}-$OLCT is

$f(x)=\int_{\mathbb{R}}\mathcal{O}^{A}_{\mu_{4}}[f](w)\overline{K^{\mu_{4}}_{A}(x,w)}dw,$
(3.12)

where $\overline{K^{\mu_{4}}_{A}(x,w)}=K^{-\mu_{4}}_{A}(x,w)$ and $b\neq 0.$

On replacing the complex unit $i$ in the ordinary offset linear canonical
transform by the imaginary units in the octonions, the three dimensional
octonion offset linear canonical transform($\mathbb{O}-$OLCT) could be defined
as

######  Definition 3.2.

(Three dimensional $\mathbb{O}-$OLCT)
Let $A_{k}=\left[\begin{array}[]{cccc}a_{k}&b_{k}&|&\tau_{k}\\\
c_{k}&d_{k}&|&\eta_{k}\\\ \end{array}\right]$, be a matrix parameter such that
$a_{k}$, $b_{k}$, $c_{k}$, $d_{k}$, $p_{k}$, $q_{k}\in\mathbb{R}$ and
$a_{k}d_{k}-b_{k}c_{k}=1,$ for $k=1,2,3.$ The three dimensional
$\mathbb{O}-$OLCT of an octonion-valued signal $f$ over $\mathbb{R}^{3},$ is
given by

$\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)=\int_{\mathbb{R}^{3}}f(x)K^{\mu_{1}}_{A_{1}}(x_{1},w_{1})K^{\mu_{2}}_{A_{1}}(x_{2},w_{2})K^{\mu_{4}}_{A_{3}}(x_{3},w_{3})dx$
(3.13)

where $x=(x_{1},x_{2},x_{3}),\,w=(w_{1},w_{2},w_{3}),$ and
$K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})$ , $K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})$ and
$K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})$are kernel signals given by

$K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})==\dfrac{1}{\sqrt{2\pi|b_{1}|}}e^{\frac{{\mu_{1}}}{2b_{1}}\big{[}a_{1}x^{2}_{1}-2x_{1}(w_{1}-\tau_{1})-2w_{1}(d_{1}\tau_{1}-b_{1}\eta_{1})+d_{1}(w^{2}_{1}+\tau^{2}_{1})-\frac{\pi}{2}\big{]}},\quad
b\neq 0\,$ (3.14)
$K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})=\dfrac{1}{\sqrt{2\pi|b_{2}|}}e^{\frac{{\mu_{2}}}{2b_{2}}\big{[}a_{2}x^{2}_{2}-2x_{2}(w_{2}-\tau_{2})-2w_{2}(d_{2}\tau_{2}-b_{2}\eta_{2})+d_{2}(w^{2}_{2}+\tau^{2}_{2})-\frac{\pi}{2}\big{]}},\quad
b\neq 0\,$ (3.15)

and

$K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})==\dfrac{1}{\sqrt{2\pi|b_{3}|}}e^{\frac{{\mu_{4}}}{2b_{3}}\big{[}a_{3}x^{2}_{3}-2x_{3}(w_{3}-\tau_{3})-2w_{3}(d_{3}\tau_{3}-b_{3}\eta_{3})+d_{3}(w^{2}_{3}+\tau^{2}_{3})-\frac{\pi}{2}\big{]}},\quad
b\neq 0\,.$ (3.16)

Note that, for $b_{k}=0,k=1,2,3$, the $\mathbb{O}-$OLCT boils down to chirp
multiplication operator and it is of no particular intrest for our objective
in this work. Hence for the sake of braveity, we always set $b_{k}\neq 0$ in
the paper unless stated otherwise.

Also note that,the kernels with imaginary units $\mu_{1},\mu_{2},\mu_{4}$ are
octonion-valued and does not reduce to the quaternion cases, thus the present
integral transform is more interesting and complicated.
Now we will obtain the closed-form representation of $\mathbb{O}-$OLCT defined
in (3.13), let us begin by setting

$\xi_{k}={\frac{{1}}{2b_{k}}\big{[}a_{k}x^{2}_{k}-2x_{k}(w_{k}-\tau_{k})-2w_{k}(d_{k}\tau_{k}-b_{k}\eta_{k})+d_{k}(w^{2}_{k}+\tau^{2}_{k})-\frac{\pi}{2}\big{]}},\quad
k=1,2,3.$ (3.17)

Thus,

$\displaystyle
K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}e^{\mu_{1}\xi_{1}}e^{\mu_{2}\xi_{2}}e^{\mu_{4}\xi_{3}}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}(c_{1}+\mu_{1}s_{1})(c_{2}+\mu_{2}s_{2})(c_{3}+\mu_{4}s_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}(c_{1}c_{2}c_{3}+s_{1}c_{2}c_{3}\mu_{1}+c_{1}s_{2}c_{3}\mu_{2}$
$\displaystyle\quad+$ $\displaystyle
s_{1}s_{2}c_{3}\mu_{3}+c_{1}c_{2}s_{3}\mu_{4}+s_{1}c_{2}s_{3}\mu_{5}+c_{1}s_{2}s_{3}\mu_{6}+s_{1}s_{2}s_{3}\mu_{7}),$

where $c_{k}=\cos\xi_{k}$ and $s_{k}=\sin\xi_{k},\quad k=1,2,3.$
Using (3.2) in (3.13) we get closed-form representation of $\mathbb{O}-$OLCT
given by following lemma.

######  Lemma 3.2 (Closed-form representation).

The $\mathbb{O}-$OLCT of a three dimensional signal
$f:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ has the closed-form
representation:

$\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\\{f\\}(w)={\mathbf{\Phi}}_{0}(w)+{\mathbf{\Phi}}_{1}(w)+{\mathbf{\Phi}}_{2}(w)+{\mathbf{\Phi}}_{3}(w)+{\mathbf{\Phi}}_{4}(w)+{\mathbf{\Phi}}_{5}(w)+{\mathbf{\Phi}}_{6}(w)+{\mathbf{\Phi}}_{7}(w)$
(3.19)

where we put the integrals

$\displaystyle{\mathbf{\Phi}}_{0}(w)=\int_{\mathbb{R}^{3}}f_{eee}(x)\frac{1}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}c_{2}c_{3}dx,$

and,

$\displaystyle{\mathbf{\Phi}}_{1}(w)=\int_{\mathbb{R}^{3}}f_{oee}(x)\frac{\mu_{1}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}c_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{2}(w)=\int_{\mathbb{R}^{3}}f_{eoe}(x)\frac{\mu_{2}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}s_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{3}(w)=\int_{\mathbb{R}^{3}}f_{ooe}(x)\frac{\mu_{3}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}s_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{4}(w)=\int_{\mathbb{R}^{3}}\frac{\mu_{4}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}f_{eeo}(x)c_{1}c_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{5}(w)=\int_{\mathbb{R}^{3}}f_{oeo}(x)\frac{\mu_{5}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}c_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{6}(w)=\int_{\mathbb{R}^{3}}f_{eoo}(x)\frac{\mu_{6}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}s_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Phi}}_{7}(w)=\int_{\mathbb{R}^{3}}f_{ooo}(x)\frac{\mu_{7}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}s_{2}s_{3}dx,$

where the functions $f_{xyz},x,y,z\in\\{e,o\\},$ are eight components of $f$
of different parity with respect appropriate variable, for example
$f_{oeo}(x)$ is odd with respect to $x_{1}$, even with respect $x_{2}$ and odd
with respect to $x_{3}.$

Under suitable conditions, the original octonion-valued signal $f$ can be
reconstructed from $\mathbb{O}-$OLCT by its inverse transform.

######  Definition 3.3.

The inverse $\mathbb{O}-$OLCT of signal
$g:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ is defined by

$\left\\{O^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\right\\}^{-1}\\{g\\}(x)=\int_{\mathbb{R}^{3}}f(w)\overline{K_{A_{3}}^{\mu_{4}}(x_{1},w_{1})}\overline{K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})}\overline{K_{A_{1}}^{\mu_{1}}(x_{3},w_{3})}dw$
(3.20)

It has the closed-form representation:

$\left\\{O^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\right\\}^{-1}\\{g\\}(x)={\mathbf{\Psi}}_{0}(w)+{\mathbf{\Psi}}_{1}(w)+{\mathbf{\Psi}}_{2}(w)+{\mathbf{\Psi}}_{3}(w)+{\mathbf{\Psi}}_{4}(w)+{\mathbf{\Psi}}_{5}(w)+{\mathbf{\Psi}}_{6}(w)+{\mathbf{\Psi}}_{7}(w)$
(3.21)

where we put the integrals

$\displaystyle{\mathbf{\Psi}}_{0}(w)=\int_{\mathbb{R}^{3}}f_{eee}(x)\frac{1}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}c_{2}c_{3}dx,$

and,

$\displaystyle{\mathbf{\Psi}}_{1}(w)=\int_{\mathbb{R}^{3}}f_{oee}(x)\frac{-\mu_{1}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}c_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{2}(w)=\int_{\mathbb{R}^{3}}f_{eoe}(x)\frac{-\mu_{2}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}s_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{3}(w)=\int_{\mathbb{R}^{3}}f_{ooe}(x)\frac{-\mu_{3}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}s_{2}c_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{4}(w)=\int_{\mathbb{R}^{3}}f_{eeo}(x)\frac{-\mu_{4}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}c_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{5}(w)=\int_{\mathbb{R}^{3}}f_{oeo}(x)\frac{-\mu_{5}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}c_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{6}(w)=\int_{\mathbb{R}^{3}}f_{eoo}(x)\frac{-\mu_{6}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}c_{1}s_{2}s_{3}dx,$
$\displaystyle{\mathbf{\Psi}}_{7}(w)=\int_{\mathbb{R}^{3}}f_{ooo}(x)\frac{-\mu_{7}}{2\pi\sqrt{2\pi|b_{1}b_{2}b_{3}|}}s_{1}s_{2}s_{3}dx,$

The simple way to define inverse of $\mathbb{O}-$OLCT is to introduce
Inversion theorem.

######  Theorem 3.2 ( Inversion for three dimensional $\mathbb{O}-$OLCT ).

Every octonion-valued signal $f:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ can
be reconstructed by the formula

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left\\{\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\right\\}^{-1}[\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}](x)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)K_{A_{3}}^{-\mu_{4}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})K_{A_{1}}^{-\mu_{1}}(x_{3},w_{3})dw,$

###### Proof.

By using the definition of QOLCT[OWN1], the one dimensional $\mathbb{O}-$OLCT
and three dimensional $\mathbb{O}-$OLCT the proof of theorem 3.2 follows. ∎

For the clarity of the formulas we denote $x_{l,m,n}=(lx_{1},mx_{2},nx_{3}),$
$l,m,n\in\\{+,-\\},$ i.e. $x_{+-+}=(x_{1},-x_{2},x_{3})$ and denoting the even
and odd part of a function $f(x)$ by $f_{e}(x)$ and $f_{o}(x)$ where
$f_{e}=(f(x_{+++})+f(x_{++-}))/2$ which is only even in the third variable
$x_{3}.$ Similarly, $f_{o}=(f(x_{+++})-f(x_{++-}))/2.$

######  Lemma 3.3 (Norm split).

Let $f:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ beoctonion-valued signal and
$\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)$ be the
$\mathbb{O}-$OLCT of $f,$ then

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|^{2}\right.$
$\displaystyle+\left.\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|^{2}\right)$

###### Proof.

From (2.1) every octonion-valued signal $f$ has explicit form $f=g+h\mu_{4}$
where $g,h\in\mathbb{H}.$ And by the even and odd part we further express the
$\mathbb{O}-$OLCT in (3.13) as follows,

$\displaystyle\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}g(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})dx$
$\displaystyle+\int_{\mathbb{R}^{3}}h(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})\mu_{4}K_{A_{3}}^{\mu_{4}}(x_{3},w_{3})dx$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}g_{e}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})c_{3}dx$
$\displaystyle+\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}h_{o}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})s_{3}dx$
$\displaystyle+\left(\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}h_{e}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})c_{3}dx\right.$
$\displaystyle-\left.\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}g_{o}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})s_{3}dx\right)\mu_{4}.$

With $c_{k}=\cos\xi_{k}$ and $s_{k}=\sin\xi_{k}$ where $\xi_{k}$ is given in
(3.17). From (3.3) it is clear that $\mathbb{O}-$OLCT can be divided into four
QOLCTs. Thus the norm of $\mathbb{O}-$OLCT splits into four norms of
quaternion functions as:

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{{2\pi|b_{3}|}}\left\|\int_{\mathbb{R}^{3}}g_{e}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})c_{3}dx\right\|^{2}_{2}$
$\displaystyle+\frac{1}{{2\pi|b_{3}|}}\left\|\int_{\mathbb{R}^{3}}h_{o}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})s_{3}dx\right\|^{2}_{2}$
$\displaystyle+\frac{1}{{2\pi|b_{3}|}}\left\|\int_{\mathbb{R}^{3}}h_{e}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})c_{3}dx\right\|^{2}_{2}$
$\displaystyle+\frac{1}{{2\pi|b_{3}|}}\left\|\int_{\mathbb{R}^{3}}g_{o}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})s_{3}dx\right\|^{2}_{2}$

where the equality is by the fact that $f_{e}$ and $f_{o}$ are orthogonal in
$L^{2}$ inner product.

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}_{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|^{2}_{2}\right.$
$\displaystyle+\left.\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|^{2}_{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|^{2}_{2}\right)$

Which completes the proof. ∎

Again denoting the quaternion form of $f$ by $f=g+h\mu_{4}$, we have

$\displaystyle\|f\|^{2}_{2}$ $\displaystyle=$
$\displaystyle\|g+h\mu_{4}\|^{2}_{2}$ $\displaystyle=$
$\displaystyle\|g\|^{2}+\|h\|^{2}$ $\displaystyle=$
$\displaystyle\|g_{e}\|^{2}+\|g_{o}\|^{2}+\|h_{e}\|^{2}+\|h_{o}\|^{2}$

Now by the Plancherel theorem for the QOLCT [OWN1], we have

$\left\|\int_{\mathbb{R}^{3}}g_{e}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})c_{3}dx\right\|^{2}_{2}=\left\|g_{e}(x)\right\|^{2}_{2}$
(3.25)

Similarly results hold for the remaining three norms.

On applying (3.2) and (3.25) in lemma 3.3 we have proved the following Energy
conservation relation for $\mathbb{O}-$OLCT.

######  Theorem 3.3 (Energy conservation).

Let $f:\mathbb{R}^{3}\longrightarrow\mathbb{O}$ be a continuous and square
integrable octonion-valued signal function. Then we have

$\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}=\frac{1}{2\pi|b_{3}|}\|f\|^{2}_{2}$
(3.26)

Now we move towards our main section in which we present some uncertainty
principles associated with $\mathbb{O}-$OLCT.

##  4\. Uncertainty principles for $\mathbb{O}-$OLCT

We know that in signal processing there are different types of uncertainty
principles in the QFT, QLCT and QOLCT domains. Recently in [31] authors
investigate Heisenberg’s uncertainty principle and Donoho-Stark’s uncertainty
principle for $\mathbb{O}-$LCT. Considering that the $\mathbb{O}-$OLCT is a
generalized version of the $\mathbb{O}-$FT or $\mathbb{O}-$LCT, it is natural
and interesting to study uncertainty principles of a octonion-valued function
and its $\mathbb{O}-$OLCT. So in this section we shall investigate some
uncertainty principles for $\mathbb{O}-$OLCT.

###  4.1. Pitt’s inequality and the logarithmic uncertainty principle

Here we prove the sharp Pitt’s inequality for the $\mathbb{O}-$OLCT and derive
the associated logarithmic uncertainty inequality. The proof of the Pitt’s
inequality heavily depends on the QOLCT. So we first present following Pitt’s
inequality for QOLCT.

######  Lemma 4.1 ( Pitt’s inequality for QOLCT).

[24]
For $f\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H}),$ and $0\leq\alpha<2$,

$\int_{{\mathbb{R}}^{2}}{{\left|\frac{w}{b}\right|}^{-\alpha}}{\left|{\mathcal{O}}^{i,j}_{A_{1},A_{2}}\left\\{f(x)\right\\}\left(w\right)\right|}^{2}dw\
\leq\frac{C_{\alpha}}{4\pi^{2}}\int_{{\mathbb{R}}^{2}}{{\left|x\right|}^{\alpha}}{\left|f(x)\right|}^{2}\
dx.$ (4.1)

With
$C_{\alpha}:=\frac{{4\pi}^{2}}{2^{\alpha}}{{[}\Gamma(\frac{2-\alpha}{4})/\Gamma(\frac{2+\alpha}{4})]}^{2}$,
and $\Gamma\left(.\right)$ is the Gamma function and
${\mathcal{S}}({\mathbb{R}}^{2},{\mathbb{H}})$ denotes the Schwartz space.

######  Theorem 4.1 (Pitt’s inequality for the $\mathbb{O}-$OLCT).

For $f\in\mathcal{S}(\mathbb{R}^{3},\mathbb{O}),$ and $0\leq\alpha<3$ and
under the assumptions of lemma 4.1, we have

$\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}{\left|{\mathcal{O}}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\left\\{f(x)\right\\}\left(w\right)\right|}^{2}dw\
\leq\frac{C_{\alpha}}{8\pi^{3}|b_{3}|}\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}|f(x)|^{2}dx.$
(4.2)

###### Proof.

We have split $\mathbb{O}-$OLCT into four QOLCT in lemma 3.3, therefore

$\displaystyle\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}{\left|{\mathcal{O}}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\left\\{f(x)\right\\}\left(w\right)\right|}^{2}dw$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\left(\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)|^{2}dw\right.$
(4.3)
$\displaystyle+\left.\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)|^{2}dw\right.$
$\displaystyle+\left.\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)|^{2}dw\right.$
$\displaystyle+\left.\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)|^{2}dw\right)$

By the Pitt’s inequality for QOLCT (4.1), we have

$\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}{\left|{\mathcal{O}}^{i,j}_{A_{1},A_{2}}\left\\{g_{e}(x)\right\\}\left(w\right)\right|}^{2}dw\
\leq\frac{C_{\alpha}}{4\pi^{2}}\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}{\left|f(x)\right|}^{2}\
dx.$ (4.4)

As similar inequalities hold for the remaining three terms. Collecting all and
inserting in (4.3), we get

$\displaystyle\int_{{\mathbb{R}}^{3}}{{\left|\frac{w}{b}\right|}^{-\alpha}}{\left|{\mathcal{O}}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\left\\{f(x)\right\\}\left(w\right)\right|}^{2}dw$
$\displaystyle\leq$
$\displaystyle\frac{C_{\alpha}}{8\pi^{3}|b_{3}|}\left(\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}{\left|g_{e}\right|}^{2}\
dx+\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}{\left|g_{o}(x)\right|}^{2}\
dx\right.$
$\displaystyle+\left.\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}{\left|h_{e}(x)\right|}^{2}\
dx+\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}{\left|h_{o}(x)\right|}^{2}\
dx\right)$ $\displaystyle=$
$\displaystyle\frac{C_{\alpha}}{8\pi^{3}|b_{3}|}\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}\left(|g_{e}(x)|^{2}+|g_{o}(x)|^{2}+|h_{e}(x)|^{2}+|h_{o}(x)|^{2}\right)dx.$
$\displaystyle=$
$\displaystyle\frac{C_{\alpha}}{8\pi^{3}|b_{3}|}\int_{{\mathbb{R}}^{3}}{{\left|x\right|}^{\alpha}}|f(x)|^{2}dx$

where last equality occurs because of (3.2).
Which completes the proof. ∎

Here $C_{\alpha}$ can’t be smaller any more. It is equal to the ordinary
complex and the quaternion cases. Thus, the inequality is sharp. If
$\alpha=0$, it changes to equality, at $\alpha=0$ differentiating the sharp
Pitt’s inequalities led to the following logarithmic uncertainty inequality
for the $\mathbb{O}-$OLCT.

######  Theorem 4.2 (Logarithmic uncertainty principle for the
$\mathbb{O}-$OLCT).

Let $f\in\mathcal{S}(\mathbb{R}^{3},\mathbb{O}),$ then the following
inequality is satisfied:

$2\pi|b_{3}|\int_{{\mathbb{R}}^{2}}\ln\left|\frac{w}{b}\right|\left|{\mathcal{O}}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\\{f\\}(w)\right|^{2}dw+\int_{{\mathbb{R}}^{2}}\ln|x||f(x)|^{2}dx\geq
D\int_{{\mathbb{R}}^{2}}|f(x)|^{2}dx$ (4.5)

with $D=\ln(2)+\Gamma^{\prime}(\frac{1}{2})/\Gamma(\frac{1}{2}).$

###### Proof.

Following the procedure of theorem 4.11 in [24] we will get desired result.

Alternatively, we can prove Logarithmic uncertainty principle for the
$\mathbb{O}-$OLCT from the Logarithmic uncertainty principle for the QOLCT
[24].

######  Lemma 4.2 (Logarithmic uncertainty principle for the QOLCT).

[24]
Let$f\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H}),$ then

$\int_{{\mathbb{R}}^{2}}\ln\left|\frac{w}{b}\right|\left|{\mathcal{O}}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{f\\}(w)\right|^{2}dw+\int_{{\mathbb{R}}^{2}}\ln|x||f(x)|^{2}dx\geq
D\int_{{\mathbb{R}}^{2}}|f(x)|^{2}dx$ (4.6)

with $D=\ln(2)+\Gamma^{\prime}(\frac{1}{2})/\Gamma(\frac{1}{2}).$

Proof of theorem 4.2
By lemma 3.3, $\mathbb{O}-$OLCT can be written in split quaternion form as

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|^{2}\right.$
$\displaystyle+\left.\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|^{2}\right).$

Therefore

$\displaystyle\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}dw$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|^{2}\right.$
$\displaystyle+\left.\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|^{2}\right)dw.$

Implies

$\displaystyle
2\pi|b_{3}|\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}dw$
$\displaystyle\quad=\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}dw+\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|^{2}dw.$
$\displaystyle\quad\quad+\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|^{2}dw+\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|^{2}dw$
(4.7)

Also by virtue of (3.2), we can write

$\displaystyle\int_{\mathbb{R}^{2}}\ln|x||f(x)|^{2}dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\ln|x||g_{e}(x)|^{2}dx+\int_{\mathbb{R}^{2}}\ln|x||g_{o}(x)|^{2}dx$
$\displaystyle+\int_{\mathbb{R}^{2}}\ln|x||h_{e}(x)|^{2}dx+\int_{\mathbb{R}^{2}}\ln|x||h_{o}(x)|^{2}dx.$

By logarithmic uncertainty principle for the QOLCT given in lemma 4.2, we get

$\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\int_{\mathbb{R}^{2}}\ln|x||g_{e}(x)|^{2}dx\geq
D\int_{\mathbb{R}^{2}}|g_{e}(x)|^{2}dx.$ (4.9)

Similarly ,

$\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\int_{\mathbb{R}^{2}}\ln|x||g_{e}(x)|^{2}dx\geq
D\int_{\mathbb{R}^{2}}|g_{o}(x)|^{2}dx,$ (4.10)
$\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\int_{\mathbb{R}^{2}}\ln|x||g_{e}(x)|^{2}dx\geq
D\int_{\mathbb{R}^{2}}|h_{e}(x)|^{2}dx.$ (4.11)
$\int_{\mathbb{R}^{2}}\ln\left|\frac{w}{b}\right|\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|^{2}+\int_{\mathbb{R}^{2}}\ln|x||g_{e}(x)|^{2}dx\geq
D\int_{\mathbb{R}^{2}}|h_{o}(x)|^{2}dx.$ (4.12)

Collecting all equations (4.9, 4.10, 4.11, 4.12) and making use of (4.1, 4.1),
we obtain the desired result

$2\pi|b_{3}|\int_{{\mathbb{R}}^{2}}\ln\left|\frac{w}{b}\right|\left|{\mathcal{O}}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{3}}\\{f\\}(w)\right|^{2}dw+\int_{{\mathbb{R}}^{2}}\ln|x||f(x)|^{2}dx\geq
D\int_{{\mathbb{R}}^{2}}|f(x)|^{2}dx.$ (4.13)

Which completes the proof ∎

###  4.2. Hausdorff-Young inequality for $\mathbb{O}-$OLCT

In this subsection we will establish Hausdorff-Young inequality which is very
important in signal processing. This inequality will be helpful for
researchers in establishing Shannon’s entropy uncertainty relation.

######  Lemma 4.3 (Hausdorff-Young inequality for QOLCT).

[26] For $1\leq p\leq 2$ and $\frac{1}{p}+\frac{1}{q}=1,$ we have

$\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2},}\\{f\\}(w)\|_{q}\leq(2\pi)^{\frac{1}{q}-\frac{1}{p}}|b_{1}b_{2}|^{\frac{1}{q}-\frac{1}{2}}\|f(x)\|_{p}.$
(4.14)

######  Theorem 4.3 (Hausdorff-Young inequality for $\mathbb{O}-$OLCT).

For $1\leq p\leq 2$ and $\frac{1}{p}+\frac{1}{q}=1,$ we have

$\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|_{q}\leq{(2\pi)^{\frac{1}{2q}-\frac{1}{p}}}|b_{1}b_{2}|^{\frac{1}{q}-\frac{1}{2}}|b_{3}|^{-\frac{1}{2q}}\|f(x)\|_{p}.$
(4.15)

###### Proof.

From (3.3), we obtain

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|_{q}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi|b_{3}|)^{\frac{1}{2q}}}\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{e}\\}(w)\|_{q}+\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{g_{o}\\}(w)\|_{q}\right.$
$\displaystyle+\left.\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{e}\\}(w)\|_{q}+\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{h_{o}\\}(w)\|_{q}\right)$

Now applying lemma 4.3 to the R.H.S of above equation, we obtain

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|_{q}$
$\displaystyle\leq$
$\displaystyle\frac{1}{(2\pi|b_{3}|)^{\frac{1}{2q}}}(2\pi)^{\frac{1}{q}-\frac{1}{p}}|b_{1}b_{2}|^{\frac{1}{q}-\frac{1}{2}}\left(\|g_{e}(x)\|_{p}+\|g_{o}(x)\|_{p}\right.$
$\displaystyle\left.+\|h_{e}(x)\|_{p}+\|h_{o}(x)\|_{p}\right).$
$\displaystyle=$
$\displaystyle{(2\pi)^{\frac{1}{2q}-\frac{1}{p}}}|b_{1}b_{2}|^{\frac{1}{q}-\frac{1}{2}}|b_{3}|^{-\frac{1}{2q}}\|f(x)\|_{p}$

where last equality follows from (3.2).
Which completes the proof ∎

###  4.3. Local uncertainty principle for $\mathbb{O}-$OLCT

Local uncertainty principle states that if $f$ is highly localized, then the
octonion Fourier transform can not be concentrated in a small neighborhood of
two or more separated points. We shall establish Local uncertainty principle
for $\mathbb{O}-$OLCT in this subsection.

######  Lemma 4.4 (Local uncertainty principle for QOLCT).

[26]
(1) For $0<\alpha<1$ and for all $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$
there is a constant $M_{\alpha}$ and all measurable set
$E\subset\mathbb{R}^{3}$ that holds

$\int_{bE}|\mathcal{O}^{\mu_{1},\mu_{2}}_{A_{1},A_{2}}\\{f\\}(w)|^{2}dw\leq
M_{\alpha}|E|^{\alpha}\||x|^{\alpha}f\|^{2}_{2}.$ (4.16)

(2) If $\alpha>1$,and for all $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$ there
is a constant $M_{\alpha}$ and all measurable set $E\subset\mathbb{R}^{3}$
that holds

$\int_{bE}|\mathcal{O}^{\mu_{1},\mu_{2}}_{A_{1},A_{2}}\\{f\\}(w)|^{2}dw\leq
M_{\alpha}|b_{1}b_{2}|^{\alpha-\frac{1}{\alpha}}|E|^{\alpha}\|f\|^{2-2\alpha}_{2}\||x|^{\alpha}f\|^{\frac{2}{\alpha}}_{2},$
(4.17)
$M_{\alpha}=\left\\{\begin{array}[]{cc}\dfrac{(1+\alpha^{2})}{\alpha^{2\alpha}}(2-2\alpha)^{\alpha-2},&0<\alpha<1,\\\
\\\
\dfrac{\pi}{\alpha\Gamma(1/2)}\Gamma\left(\frac{1}{\alpha}\right)\Gamma\left(1-\frac{1}{\alpha}\right)(\alpha-1)^{\alpha}\left(1-\frac{1}{\alpha}\right)^{-1},&\alpha>1,\\\
\\\ \end{array}\right.$ (4.18)

######  Theorem 4.4 (Local uncertainty principle for $\mathbb{O}-$OLCT).

(1) For $0<\alpha<1$ and for all $f\in L^{2}(\mathbb{R}^{3},\mathbb{O}),$
there is a constant $M_{\alpha}$ and all measurable set
$E\subset\mathbb{R}^{3}$ that holds

$\int_{bE}|\mathcal{O}^{\mu_{1},\mu_{2}}_{A_{1},A_{2}}\\{f\\}(w)|^{2}dw\leq\frac{1}{2\pi|b_{3}|}M_{\alpha}|E|^{\alpha}\||x|^{\alpha}f\|^{2}_{2}.$
(4.19)

(2) If $\alpha>1$,and for all $f\in L^{2}(\mathbb{R}^{3},\mathbb{O}),$ there
is a constant $M_{\alpha}$ and all measurable set $E\subset\mathbb{R}^{3}$
that holds

$\int_{bE}|\mathcal{O}^{\mu_{1},\mu_{2}}_{A_{1},A_{2}}\\{f\\}(w)|^{2}dw\leq\frac{1}{2\pi|b_{3}|}M_{\alpha}|b_{1}b_{2}|^{\alpha-\frac{1}{\alpha}}|E|^{\alpha}\|f\|^{2-2\alpha}_{2}\||x|^{\alpha}f\|^{\frac{2}{\alpha}}_{2},$
(4.20)
$M_{\alpha}=\left\\{\begin{array}[]{cc}\dfrac{(1+\alpha^{2})}{\alpha^{2\alpha}}(2-2\alpha)^{\alpha-2},&0<\alpha<1,\\\
\\\
\dfrac{\pi}{\alpha\Gamma(1/2)}\Gamma\left(\frac{1}{\alpha}\right)\Gamma\left(1-\frac{1}{\alpha}\right)(\alpha-1)^{\alpha}\left(1-\frac{1}{\alpha}\right)^{-1},&\alpha>1,\\\
\\\ \end{array}\right.$ (4.21)

###### Proof.

By the splitting of $\mathbb{O}-$OLCT in (3.3),we have

$\displaystyle\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}g_{e}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})c_{3}dx$
$\displaystyle+\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}h_{o}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})s_{3}dx$
$\displaystyle+\left(\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}h_{e}(x)K_{A_{1}}^{-\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{-\mu_{2}}(x_{2},w_{2})c_{3}dx\right.$
$\displaystyle-\left.\frac{1}{\sqrt{2\pi|b_{3}|}}\int_{\mathbb{R}^{3}}g_{o}(x)K_{A_{1}}^{\mu_{1}}(x_{1},w_{1})K_{A_{2}}^{\mu_{2}}(x_{2},w_{2})s_{3}dx\right)\mu_{4}.$

Setting $f_{m}(x)=g_{e}(x_{+++})+h_{o}(x_{+-+})e_{2}$ and
$f_{n}(x)x=h_{e}(x_{+++})-g_{o}(x_{+-+})e_{2}$. Then $\mathbb{O}-$OLCT can be
written as the combination of two QOLCTs as

$\displaystyle\|\mathcal{O}^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{f_{m}\\}(w)\|^{2}+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{f_{n}\\}(w)\|^{2}$

Now, by lemma 4.4 we have

$\int_{bE}|\mathcal{O}^{\mu_{1},\mu_{2}}_{A_{1},A_{2}}\\{f_{j}\\}(w)|^{2}dw\leq
M_{\alpha}|E|^{\alpha}\||x|^{\alpha}f_{j}\|^{2}_{2},\quad j=m,n$ (4.23)

Therefore from (4.3),(4.23) and using lemma 2.1,we have

$\displaystyle\int_{bE}\|O^{A_{1},A_{2},A_{3}}_{\mu_{1},\mu_{2},\mu_{4}}\\{f\\}(w)\|^{2}dw$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}\int_{bE}\left(\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{f_{m}\\}(w)\|^{2}dw+\|\mathcal{O}^{A_{1},A_{2}}_{\mu_{1},\mu_{2}}\\{f_{n}\\}(w)\|^{2}dw\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi|b_{3}|}M_{\alpha}|E|^{\alpha}\left(\||x|^{\alpha}f_{m}\|^{2}+\||x|^{\alpha}f_{n}\|^{2}_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|b_{3}|}M_{\alpha}|E|^{\alpha}\||x|^{\alpha}f\|^{2}_{2}$

which completes the proof.
Similarly we can easily prove (4.20). ∎

##  5\. Conclusions

In this paper, based on the association between the $\mathbb{O}-$OLCT the
QOLCT via split norm, we have established some basic properties of the
proposed transform including the inversion formula and energy conservation.
These results are very important for their applications in digital signal and
image processing. Finally, the uncertainty inequalities for the
$\mathbb{O}-$OLCT such as logarithmic uncertainty inequality,Hausdorff-Young
inequality and local uncertainty are obtained. In our future works, we will
discuss the physical significance and engineering background of this paper.
Moreover, we will formulate convolution and correlation theorems for the
$\mathbb{O}-$OLCT.

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# Early dark energy in $k$-essence

S. X. Tian<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal
University, 100875 Beijing, China  Zong-Hong Zhu<EMAIL_ADDRESS>Department
of Astronomy, Beijing Normal University, 100875 Beijing, China  School of
Physics and Technology, Wuhan University, 430072 Wuhan, China

###### Abstract

Early dark energy (EDE) that becomes subdominant around the epoch of matter-
radiation equality can be used to ease the Hubble tension. However, there is a
theoretical problem that why the energy scale of EDE is in coincidence with
that of matter-radiation equality when their physics are completely unrelated.
Sakstein and Trodden [Phys. Rev. Lett. 124, 161301 (2020)] proposed a
mechanism to solve this coincidence problem with $\mathcal{O}({\rm eV})$-mass
neutrino. In this paper, in order to solve the coincidence problem, we propose
a new scenario for EDE, in which the onset and ending of EDE are triggered by
the radiation-matter transition. The specific example we study is a
$k$-essence model. The cosmic evolution equations can be recast into a two-
dimensional dynamical system and its main properties are analyzed. Our results
suggest that $k$-essence seems unable to realize the new scenario for EDE.
However, an EDE model with different scenario is realized in $k$-essence. In
this model, the ending of EDE can be triggered by the radiation-matter
transition while the onset depends on the initial conditions of the scalar
field. Therefore, the obtained model can only be used to solve half of the
coincidence problem. The full resolution in the framework of our initial
proposed scenario is worthy of more research.

##  I Introduction

Today we are entering the era of precise cosmology. Many observations, such as
type Ia supernovae (SNe Ia [1]), baryon acoustic oscillations [2], cosmic
microwave background (CMB [3]), gravitational lensing systems [4],
gravitational waves [5] and so on, can give tight constraints on the
cosmological parameters. However, a discrepancy between high and low redshift
measurements of the Hubble constant $H_{0}$ appeared with the improvement of
precision. The Planck CMB data, together with the $\Lambda$CDM model, predict
$H_{0}=67.4\pm 0.6\ {\rm km}/{\rm s}/{\rm Mpc}$ [3], while the local Cepheid-
calibrated SNe Ia observations give $H_{0}=74.03\pm 1.42\ {\rm km}/{\rm
s}/{\rm Mpc}$ [6]. Statistically, there is an inconsistency of approximately
$4\sigma$ level between these two measurements. This is called the Hubble
tension and is one of the most important issue in modern cosmology [7].

If, as suggested by the existing analysis [8, 9, 10, 11], systematic errors in
observations are not responsible for the inconsistency, then the Hubble
tension may indicate new physics beyond the $\Lambda$CDM model. There are many
attempts to address the Hubble tension rely on the possible new physics, such
as Planck-scale loop quantum cosmology [12], primordial magnetic field [13],
early dark energy (EDE [14, 15, 16, 17, 18, 19, 20]), low redshift resolutions
[21, 22, 23, 24, 25] and so on [26, 27, 28]. Among these, EDE has attracted
widespread attention. A scalar field with up to approximately $10\%$ relative
energy density at a critical redshift $z_{c}\sim 3500$ seems able to bring the
high and low redshift measurements of $H_{0}$ into agreement [17]. The
underlying mechanism of the EDE resolution is that it can reduce the cosmic
sound horizon at last scattering and thus increase the measured value of
$H_{0}$ form CMB data. However, controversy about EDE’s ability to ease the
Hubble tension exists in the literature [29, 30, 31, 32, 33]. In this paper,
we ignore this controversy and assume that EDE can indeed ease the Hubble
tension. The topic of this paper is to discuss the theoretical naturalness of
EDE. More precisely, our discussion focus on the coincidence problem about
EDE, which was first recognized in Ref. [34].

Historically, there has been some works discussing EDE long before the
confirmation of the Hubble tension [35, 36, 37, 38, 39]. These works generally
described EDE with parameterized relative energy density $\Omega_{\rm EDE}$ or
equation of state $w_{\rm EDE}$, and the time when EDE appears was not
determined based on any theoretical or observational reasons. Hubble tension
determined the emergence time of EDE to be near the matter-radiation equality
[14]. However, a coincidence problem related to EDE also emerged
simultaneously [34]. Intuitively, the physics of the matter-radiation equality
is completely irrelevant to that of EDE, but why do they appear at the same
time? Technically, one can specify the energy scale111Note that the emergence
time of EDE in cosmic history is specified by its energy scale. of EDE by
introducing the cosmic expansion rate $H(z_{c})$ into the parameterized
$\Omega_{\rm EDE}$ or $w_{\rm EDE}$ [14], or even a first principle Lagrangian
[15, 17], which is proposed after the confirmation of the Hubble tension. This
can give a working EDE model, but does not help solve the related coincidence
problem. Is it possible to construct EDE around the matter-radiation equality
without introducing $H(z_{c})$? Refs. [34, 40, 41] did it. They specified the
energy scale of EDE through a nonminimal coupling between $\mathcal{O}({\rm
eV})$-mass neutrino and a scalar field. In this model, EDE appears when
neutrinos become nonrelativistic, i.e., at the energy scale of
$\mathcal{O}({\rm eV})$, which is exactly around the matter-radiation
equality. Therefore, this neutrino-assisted model can naturally solve the
coincidence problem about EDE.

In this paper, we try a different approach to solve the coincidence problem
about EDE. Look back the cosmic history, the epoch of matter-radiation
equality is special. At that time, the Universe changed from radiation-
dominated to matter-dominated. We want use the radiation-matter transition to
trigger the onset and ending of EDE. If so, the energy scale of EDE must be in
coincidence with the energy scale of matter-radiation equality. Furthermore,
we require the model only introduces Planck scale parameters and dimensionless
parameters of order unity. This requirement has been imposed to construct dark
energy models that can solve the coincidence problem related to the cosmic
late-time acceleration [42, 43]. In this scenario, no explicit energy scale
below Planck values is required, which is an essential difference between our
model and the existing models, e.g., EDE models we mentioned in the previous
two paragraphs. To achieve the above scenario, one general $k$-essence model
is discussed in Sec. II and one specific case is analyzed in Sec. III.

##  II One general model

A beautiful theoretical model should be as simple as possible. As the first
step, we only consider the minimally coupled single scalar field in this
paper. The model should belong to $k$-essence [44, 45, 46, 47]. We expect that
the EDE relative energy density $\Omega_{\rm EDE}$ has a peak near the matter-
radiation equality, and keeps small enough in the radiation or matter-
dominated Universe. For a minimally coupled EDE, its energy conservation
equation can be formally written as $\dot{\rho}_{\rm EDE}+3(1+w_{\rm
EDE})H\rho_{\rm EDE}=0$. Introducing $N=\ln(a/a_{0})$, this equation gives

$\frac{\rho_{\rm EDE}(N_{2})}{\rho_{\rm
EDE}(N_{1})}=\exp\left[-3\int_{N_{1}}^{N_{2}}[1+w_{\rm EDE}(\tilde{N})]{\rm
d}\tilde{N}\right],$ (1)

where $N_{2}$ represents the time when EDE appears, and $N_{1}$ is slightly
earlier than $N_{2}$ (see Fig. 1 for an illustration).

Figure 1: Illustration of our initial idea about the EDE (see the main text
for details).

Even if $w_{\rm EDE}$ can be less than $-1$, the right side of Eq. (1) will
not be a very large number. Therefore, if observation expects $\rho_{\rm
EDE}(N_{2})$ to be considerable compared with normal matters [14], then
$\rho_{\rm EDE}(N_{1})$ cannot be extremely small compared with normal
matters. One way to preserve a non-vanishing $\Omega_{\rm EDE}(N_{1})$ in the
radiation-dominated Universe is the scaling solution [48]. The realistic EDE
may evolve as a scaling solution with small222We require $\Omega_{\rm EDE}$ to
be small in the radiation-dominated era due to the primordial nucleosynthesis
constraints [48]. $\Omega_{\rm EDE}$ in the radiation-dominated era.
Similarly, EDE may also evolve as a scaling solution in the matter-dominated
era. Ref. [49] proved Eq. (3) is necessary for the existence of the scaling
solution in $k$-essence. Therefore, in order to preserve the scaling solution,
we adopt the action

$S=\int{\rm
d}^{4}x\sqrt{-g}\left[\frac{R}{2\kappa}-\frac{\mathcal{L}}{\kappa}\right]+S_{\rm
m},$ (2)

where $\kappa=8\pi G/c^{4}$ and

$\mathcal{L}=[1+f(y)]X+[1+g(y)]V_{\rm exp},$ (3)

where $X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, $V_{\rm
exp}=V_{0}\exp(-\lambda\phi)$, $V_{0}$ and $\lambda$ are constants,
$y=X/V_{\rm exp}$, $f(y)$ and $g(y)$ are arbitrary functions. Performing the
transformation

$g(y)=-f(y)y+h(y),$ (4)

the above Lagrangian can be rewritten as

$\mathcal{L}=X+h(y)V_{\rm exp}.$ (5)

This transformation eliminates one of the arbitrary functions in the
Lagrangian, as well as in the corresponding cosmic evolution equations. In our
conventions, $\lambda$ is dimensionless and $[V_{0}]={\rm length}^{-2}$. As
the following results do not explicitly depend on the value of $V_{0}$, we can
assume $V_{0}=\mathcal{O}(l_{\rm P}^{-2})$, where $l_{\rm P}$ is the Planck
length. Similar property exists in the exponential quintessence model [48] and
an oscillating dark energy model [42, 43]. For $S_{\rm m}$, we know its
variation $\delta S_{\rm m}=-\frac{1}{2}\int{\rm
d}^{4}x\sqrt{-g}T_{\mu\nu}\delta g^{\mu\nu}$ and $T_{\mu\nu}=\left(\rho_{\rm
m}+p_{\rm m}/c^{2}\right)u_{\mu}u_{\nu}+p_{\rm m}g_{\mu\nu}$. The equation of
state of normal matters is defined as $w_{\rm m}\equiv p_{\rm m}/(\rho_{\rm
m}c^{2})$. For the realistic Universe, normal matters include radiation
$(\rho_{\rm rad}\propto e^{-4N})$ and pressureless fluid $(\rho_{\rm
mat}\propto e^{-3N})$. Therefore, the $w_{\rm m}$ can be written as [43]

$w_{\rm m}(N)=\frac{\rho_{\rm rad}c^{2}/3+0}{(\rho_{\rm rad}+\rho_{\rm
mat})c^{2}}=\frac{1/3}{1+e^{N-N_{\rm eq}}},$ (6)

where $N_{\rm eq}$ corresponds to the matter-radiation equality. Lagrangian
similar to Eq. (5) has been used to construct dark energy for the late-time
Universe [49, 50, 51]. Setting $h(y)=0$, Eq. (5) gives the classical
exponential quintessence model [48]. Nonzero $h(y)$ can make the motion of the
scalar field more complicated when $w_{\rm m}$ goes from $1/3$(radiation) to
0(matter). We expect that the radiation-matter transition with nonzero $h(y)$
can induce the desired EDE. The above scenario is summarized in Fig. 1. Note
that this is just our initial idea about the EDE, and the final result
obtained in this paper is slightly different from it.

Variation of Eq. (2) with respect to the metric gives the gravitational field
equations

$G_{\mu\nu}=\kappa T_{\mu\nu}+\Phi_{\mu\nu},$ (7)

where
$\Phi_{\mu\nu}=\mathcal{L}_{X}\partial_{\mu}\phi\partial_{\nu}\phi-\mathcal{L}g_{\mu\nu}$
and $\mathcal{L}_{X}\equiv\partial\mathcal{L}/\partial X$. The equation of
motion of the scalar field is

$(\partial_{\mu}\mathcal{L}_{X})\partial^{\mu}\phi+\mathcal{L}_{X}\Box\phi-\mathcal{L}_{\phi}=0,$
(8)

where $\mathcal{L}_{\phi}\equiv\partial\mathcal{L}/\partial\phi$. In this
paper, we assume the Universe is described by the flat Friedmann-Lemaître-
Robertson-Walker metric

${\rm d}s^{2}=-c^{2}{\rm d}t^{2}+a^{2}({\rm d}x^{2}+{\rm d}y^{2}+{\rm
d}z^{2}),$ (9)

where $a=a(t)$. Substituting Eq. (9) and $\phi=\phi(t)$ into Eqs. (7) and (8),
we obtain the cosmic evolution equations

$\displaystyle H^{2}=\frac{\kappa c^{4}}{3}\rho_{\rm tot},$ (10)
$\displaystyle\frac{\ddot{a}}{a}=-\frac{\kappa c^{4}}{6}\left(\rho_{\rm
tot}+\frac{3p_{\rm tot}}{c^{2}}\right),$ (11)
$\displaystyle\ddot{\phi}(\mathcal{L}_{X}+2X\mathcal{L}_{XX})+3H\dot{\phi}\mathcal{L}_{X}\quad$
$\displaystyle\quad\qquad\qquad+c^{2}(\mathcal{L}_{\phi}-2X\mathcal{L}_{X\phi})=0,$
(12)

where $\rho_{\rm tot}=\rho_{\rm m}+\rho_{\phi}$, $p_{\rm tot}=p_{\rm
m}+p_{\phi}$, $\rho_{\phi}=(\mathcal{L}-2X\mathcal{L}_{X})/(\kappa c^{2})$,
$p_{\phi}=-\mathcal{L}/\kappa$,
$\mathcal{L}_{XX}\equiv\partial^{2}\mathcal{L}/\partial X^{2}$ and
$\mathcal{L}_{X\phi}\equiv\partial^{2}\mathcal{L}/(\partial\phi\partial X)$.
Equation (12) can be derived from Eqs. (10) and (11). Following Refs. [48,
42], we introduce

$x_{1}\equiv\frac{\dot{\phi}}{\sqrt{6}H},\quad x_{2}\equiv\frac{c\sqrt{V_{\rm
exp}}}{\sqrt{3}H}.$ (13)

This definition gives $x_{2}>0$. Then the cosmic evolution equations can be
written as

$\displaystyle\frac{{\rm d}x_{1}}{{\rm d}N}$
$\displaystyle=\frac{-3x_{1}(h^{\prime}+1)+\sqrt{3/2}\lambda
x_{2}^{2}(2y^{2}h^{\prime\prime}-yh^{\prime}+h+1)}{2yh^{\prime\prime}+h^{\prime}+1}$
$\displaystyle\quad-\frac{3}{2}x_{1}L,$ (14) $\displaystyle\frac{{\rm
d}x_{2}}{{\rm d}N}$ $\displaystyle=-\frac{\sqrt{6}}{2}\lambda
x_{1}x_{2}-\frac{3}{2}x_{2}L,$ (15)

where ${}^{\prime}\equiv{\rm d}/{\rm d}y$, $y=-x_{1}^{2}/x_{2}^{2}$ and
$L=2w_{\rm m}x_{1}^{2}h^{\prime}+(1+w_{\rm m})(x_{2}^{2}h+x_{2}^{2}-1)+(w_{\rm
m}-1)x_{1}^{2}$. The EDE relative energy density is defined as
$\Omega_{\phi}\equiv 8\pi G\rho_{\phi}/(3H^{2})$ and its explicit expression
is

$\Omega_{\phi}=x_{2}^{2}(-2yh^{\prime}+h-y+1).$ (16)

Note that the above results are valid for both constant and time-varying
$w_{\rm m}$.

The scenario depicted in Fig. 1 strongly depends on the scaling solution. A
scaling solution corresponds to a critical point of the above dynamic system
with $0<\Omega_{\phi}<1$. Solving ${\rm d}x_{1}/{\rm d}N=0$ and ${\rm
d}x_{2}/{\rm d}N=0$, we obtain one critical point

$\displaystyle x_{1,{\rm c}}$ $\displaystyle=\frac{\sqrt{6}(1+w_{\rm
m})}{2\lambda},$ (17) $\displaystyle x_{2,{\rm c}}$
$\displaystyle=\sqrt{\frac{3(1+w_{\rm m})(1-w_{\rm m}-2w_{\rm
m}h^{\prime})}{2\lambda^{2}(h+1)}},$ (18)

which corresponds to the scaling solution, and

$\Omega_{\phi,{\rm c}}=\frac{3(1+w_{\rm m})(1+h^{\prime})}{\lambda^{2}}.$ (19)

Note that Eq. (18) is an implicit equation as $h=h(-x_{1,{\rm
c}}^{2}/x_{2,{\rm c}}^{2})$. Without the expression of $h(y)$, one cannot
obtain the explicit coordinates, nor analyze its stability and existence
conditions.

Our original motivation was to construct a model to realize the scenario
described in Fig. 1. However, we are failed to achieve this. We tested several
$h(y)$: $h(y)=\exp(\alpha y)$, $\cos(\alpha y)$ and $-\alpha(y+\beta)^{n}$ for
$n=2,3,4$. The functions $\exp(\alpha y)$ and $-\alpha(y+\beta)^{3}$ are
monotonic with respect to $y$, while other functions are not. Contrary to Fig.
1, there is no peak between the two scaling solutions for all the tests.
$\Omega_{\phi}$ evolves almost333Generally, there is a small oscillation
around $\Omega_{\phi}=\Omega_{\phi,{\rm c},0}$. This is understandable because
the corresponding critical point is an attracting spiral (see the exponential
quintessence [48] for an example). monotonously from $\Omega_{\phi,{\rm
c},1/3}$ to $\Omega_{\phi,{\rm c},0}$, where the subscripts $1/3$ and $0$
represent the value of $w_{\rm m}$. But, in these tests, we also found some
interesting results. For some specific $h(y)$, the ratio
$r\equiv\Omega_{\phi,{\rm c},0}/\Omega_{\phi,{\rm c},1/3}$ can be extremely
small. Note that $r=3/4=\mathcal{O}(1)$ for the exponential quintessence model
(see Eq. (19) with $h=0$ or Ref. [48]). With small enough $r$, we propose a
new scenario to realize the EDE in Sec. III. The relevant coincidence problem
is also discussed.

##  III Dynamics of a specific model

In this section, we assume $h(y)=-\alpha(y+\beta)^{2}$, where $\alpha$ and
$\beta$ are constants. There is no essential difference for the cosmic
evolutions between this and other $h(y)$ functions. The function
$h(y)=-\alpha(y+\beta)^{2}$ can be regarded as a representative of the
functions we tested. Numerical results show that the expected model lies in
the parameter space of $\lambda\gg 1$, $\alpha\geq 0$ and $\beta\geq 0$.
Therefore, we only discuss this parameter space for simplicity. Substituting
the expression of $h(y)$ into Eq. (18), we can find the explicit coordinates
of the critical point. For $w_{\rm m}=1/3$, we obtain

$\displaystyle x_{1,{\rm c},1/3}=\frac{2\sqrt{6}}{3\lambda},\quad x_{2,{\rm
c},1/3}=\frac{2}{\sqrt{3}\lambda}\sqrt{\frac{1-2\alpha\beta}{1-\alpha\beta^{2}}},$
(20)

and

$\Omega_{\phi,{\rm
c},1/3}=\frac{4+16\alpha(1-\beta)}{\lambda^{2}(1-2\alpha\beta)}.$ (21)

The existence of the above critical point requires $0<\Omega_{\phi,{\rm
c},1/3}<1$ and $x_{2}$ is a positive real number. Linear stability theory [51]
is suitable for analyzing the stability of this critical point. It is stable
if the real part of the corresponding eigenvalues are all negative. The result
is that the above critical point exists and is stable when

$\left\\{\begin{array}[]{ll}0\leq\beta<1/\sqrt{\alpha}&\textrm{if
$0<\alpha<1/4$ \& $\lambda\gg 1$},\\\
0\leq\beta<\frac{\lambda^{2}-16\alpha-4}{2\alpha(\lambda^{2}-8)}&\textrm{if
$\frac{1}{4}<\alpha<\frac{\lambda^{2}-4}{16}$ \& $\lambda\gg
1$}.\end{array}\right.$ (22)

For $w_{\rm m}=0$, we obtain

$\Omega_{\phi,{\rm c},0}=\frac{3}{\lambda^{2}}\sqrt{1+4\alpha(1-\beta)},$ (23)

and the corresponding critical point exists and is stable when

$\left\\{\begin{array}[]{ll}0\leq\beta<1/\sqrt{\alpha}&\textrm{if
$0<\alpha<1/4$ \& $\lambda\gg 1$},\\\ 0\leq\beta<1+1/(4\alpha)&\textrm{if
$1/4<\alpha$ \& $\lambda\gg 1$}.\end{array}\right.$ (24)

Parameter space Eq. (24) always contains parameter space Eq. (22). Therefore,
Eq. (22) is sufficient to ensure the existence of the above two stable
critical points.

EDE means a peak of $\Omega_{\phi}$ near matter-radiation equality. Numerical
result shows that there is no peak between the two scaling solutions for the
model with $h(y)=-\alpha(y+\beta)^{2}$. Here we propose a new scenario to
realize the EDE in the framework of $h(y)=-\alpha(y+\beta)^{2}$. Inspired by
Ref. [52], we can set the initial conditions of $(x_{1},x_{2})$ to be very
close to the repulsive fixed point $(0,0)$ to suppress $\Omega_{\phi}$ in the
far past. Meanwhile, if we set $\Omega_{\phi,{\rm c},1/3}\approx 10\%$ and
$\Omega_{\phi,{\rm c},0}\ll\Omega_{\phi,{\rm c},1/3}$ with suitable
parameters, then the scalar field will evolve as the expected EDE. In Fig. 2,
we plot the evolution of $\Omega_{\phi}$ for several cases. The peak appears
near the matter-radiation equality as we expected. The relative energy density
$\Omega_{\phi}$ decreases rapidly after the matter-radiation equality. This
solves half of the coincidence problem discussed in Sec. I that why the EDE
decreases after the matter-radiation equality. Note that we solve this problem
without specifying any energy scale. However, the remaining half has not been
resolved. Our model cannot naturally explain the onset of EDE, which depends
on the initial conditions of $(x_{1},x_{2})$, i.e., the initial energy
density, in our model. In addition, one thing is worth mentioning here. Fig. 2
shows that the attractive ability of the stable critical point with $w_{\rm
m}=1/3$ is weak. Therefore, if we expect the peak value of $\Omega_{\phi}$ to
be approximately equal to $10\%$, then we may need to set $\Omega_{\phi,{\rm
c},1/3}$ slightly larger than $10\%$.

Figure 2: Evolution of the EDE relative energy density $\Omega_{\phi}$. The
model parameters are $\lambda=25$, $\alpha=4$ and $\beta=0.1$. The initial
conditions of $(x_{1},x_{2})$ are given in the legend. The equation of state
$w_{\rm m}$ is given by Eq. (6) with $N_{\rm eq}=60$. This scenario solves
half of the coincidence problem we discussed in Sec. I.

##  IV Conclusions

In this paper we have tried a new mechanism to solve the coincidence problem
about EDE, which is first discussed in Ref. [34]. Our idealized mechanism is
to use the radiation-matter transition to trigger the onset and ending of EDE,
so as to solve the related coincidence problem. The desired model is
illustrated in Fig. 1. No energy scale needs to be specified in the idealized
mechanism. As a comparison, Ref. [34] uses neutrino with $\mathcal{O}({\rm
eV})$ rest mass to specify the energy scale of EDE. We tried to use the
$k$-essence described by Eq. (5) to achieve the above mechanism. However, we
were not completely successful in constructing the model. The problem is that,
for the model we considered, if the scalar field evolves as scaling solution
in both the radiation and matter-dominated era, then no peak of
$\Omega_{\phi}$ appears around matter-radiation equality. One good news is
that the radio $r\equiv\Omega_{\phi,{\rm c},0}/\Omega_{\phi,{\rm c},1/3}$ can
be extremely small for some parameter settings. This property, together with
the method that using initial conditions to suppress $\Omega_{\phi}$ in the
far past [52], inspired us to propose a new framework to achieve EDE. Several
illustrative examples are plotted in Fig. 2. In this scenario, the onset of
EDE depends on the initial conditions, which means the coincidence problem
related the onset of EDE is not solved, while the ending is always around the
matter-radiation equality, which means the coincidence problem related the
ending of EDE can be solved naturally.

Our initial idea about EDE that using the radiation-matter transition to
trigger its onset and ending is worthy of further exploration. The $k$-essence
is a general minimally coupled model and it seems cannot realize our original
idea. Nonminimal coupling may be required. For example, the Lagrangian

$\mathcal{L}=R+\underbrace{\alpha w_{\rm m}(1/3-w_{\rm
m})R}_{\textrm{effective EDE}}$ (25)

may be able to realize our idea, where $\alpha=\mathcal{O}(1)$ is a
dimensionless constant. In this model, the effective EDE may vanish in both
radiation and matter-dominated era, but appear during the radiation-matter
transition. The details will be carried out in a separate publication.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
under Grants Nos. 11633001, 11920101003 and 12021003, the Strategic Priority
Research Program of the Chinese Academy of Sciences, Grant No. XDB23000000 and
the Interdiscipline Research Funds of Beijing Normal University. S.X.T. was
supported by the Initiative Postdocs Supporting Program under Grant No.
BX20200065.

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