# PortfolioMentor: Multimodal Generative AI Companion for Learning and
Crafting Interactive Digital Art Portfolios

Tao Long<EMAIL_ADDRESS>Columbia UniversityNew YorkNew YorkUSA and
Weirui Peng<EMAIL_ADDRESS>Columbia UniversityNew YorkNew YorkUSA

(2023)

###### Abstract.

Digital art portfolios serve as impactful mediums for artists to convey their
visions, weaving together visuals, audio, interactions, and narratives.
However, without technical backgrounds, design students often find it
challenging to translate creative ideas into tangible codes and designs, given
the lack of tailored resources for the non-technical, academic support in art
schools, and a comprehensive guiding tool throughout the mentally demanding
process. Recognizing the role of companionship in code learning and leveraging
generative AI models’ capabilities in supporting creative tasks, we present
PortfolioMentor , a coding companion chatbot for IDEs. This tool guides and
collaborates with students through proactive suggestions and responsible Q&As
for learning, inspiration, and support. In detail, the system starts with the
understanding of the task and artist’s visions, follows the co-creation of
visual illustrations, audio or music suggestions and files, click-scroll
effects for interactions, and creative vision conceptualization, and finally
synthesizes these facets into a polished interactive digital portfolio.

coding companion, creativity supports, generative AI, digital art, portfolio,
large language models, text-to-music, text-to-image

††copyright: acmcopyright††journalyear: 2023††conference: UIST ’23 SIC;
October 29 – November 1, 2023; San Francisco, CA††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Natural language
interfaces††ccs: Human-centered computing Graphical user interfaces††ccs:
Applied computing Interactive learning environments††ccs: Applied computing
Fine arts

Figure 1. PortfolioMentor System Affordances

Our system provides six affordances, including project initiation, isual
design, audio design, interaction design, creative vision narrtives, and final
delivery.

##  1\. Introduction

Digital art portfolios serve as an impactful medium for artists to convey
their creative visions, weaving dimensions like visuals, audio, and
interactions together (Leitner and Subasi, 2016; Abrami and Barrett, 2005).
However, crafting an interactive artwork from scratch is always tedious and
undersupported through its iterations, trials, and errors (Mcnutt et al.,
2023; Mitchell and Bown, 2013). Particularly, early-stage design students,
often without technical backgrounds, encounter obstacles. From our formative
study (N=5), both students and instructors shared it was challenging to
translate creative ideas into tangible designs and codes, given the lack of
academic resources, support, and a need for companionship throughout the
process.

Generative models raise huge potential in supporting the brainstorming and
exploration process for making digital art, given the domain knowledge and
contextual understanding capabilities in creative tasks. For example, various
generative AI-based tools have been used to extend design professionals’
creativity, covering visual and interface design (Petridis et al., 2023; Kim
et al., 2022; Angert et al., 2023), audio and video design (Liu et al., 2023;
Louie et al., 2020).

In response, we designed PortfolioMentor , an interactive coding companion
tool for non-technical design students. Recognizing AI’s companionship and
availability inside the coding learning and creativity support process (Gero
et al., 2023; Sarkar et al., 2022; Biermann et al., 2022), PortfolioMentor
guides and collaborates with the students with proactive coding and design
suggestions and responsible Q&A. Similar to (Long et al., 2023; MacNeil et
al., 2021; Wang et al., 2023), this tool aims to scaffold and alleviate the
tedious and mentally demanding process of building an interactive art
portfolio through comprehensive creativity and coding support. Our system
incorporates multiple affordances (See Figure 1) to foster learning and
encourage the development of unique creative ideas beyond visual
presentations. Based on various multimodal generative models, PortfolioMentor
supports the co-creation process of digital artwork including visuals,
animations, background sounds, interaction designs, creative visions and
narratives, and deployable code snippets.

##  2\. Formative Study & Design Goals

We conducted semi-structured interviews with four design students (S1-S4),
each pursuing a degree in fine art or visual design, and one instructor (I1)
with over ten years of teaching experience in coding for art students. Each
interview lasted 30 minutes, exploring personal experiences and identifying
coding-related obstacles. Here are four challenges they faced in the process
of building an interactive art portfolio, following with design goals for
PortfolioMentor :

1\. Information Barrier. All participants highlighted that there exists a
significant information barrier for non-technical students while navigating
code-learning materials and forums online.
$\bigstar$ Design Goals: PortfolioMentor should incorporate LLMs and visual
models to retarget technical content into natural, simplified, and visual
languages for non-technical art students.

2\. Lack of Community Support. Many shared concerns about the scant support
for art students learning to code in schools, including the lack of teaching
assistants (I1), experienced peer students with technical backgrounds (P2,
P4), and instructors’ availability (P3).
$\bigstar$ Design Goals: PortfolioMentor should be an easy-to-use and natural
language-friendly chatbot embedded in prevalent creative coding or
visualization editors like p5.js111https://editor.p5js.org/ or
OpenProcessing222https://openprocessing.org/sketch/create to forge constant
support for students, which 1) offers suggestions, 2) addresses questions, 3)
explains concepts and codes, 4) supports finding creative designs and
connections, and 5) motivates students’ code learning and design process.

3\. Mental Barriers with Coding. All participants expressed that the iterative
debugging process of converting artistic ideas into code is always
overwhelming and underestimated by the students given their limited technical
experiences and insufficient support.
$\bigstar$ Design Goals: PortfolioMentor should leverage LLMs to simplify the
coding process: provide natural language examples and explanations, scaffold
complicated tasks into approachable steps, and provide emotional backing,
companionship, and validation.

4\. Translation of Artistic Ideas into Code. Participants emphasize the
difficulty of translating creative visions into code due to coding
limitations. P2, P3, and P4 particularly resonate with this as they usually
search and adapt online code templates to achieve their creative objectives.
However, they find the available templates limited, particularly when
extending to interaction and music.
$\bigstar$ Design Goals: PortfolioMentor should incorporate design and various
generative models to assist students in rendering visions as code: 1) use
scaffolding to break down the art-to-code process into tangible steps, 2) use
LLMs and image-to-text models to help students formulate their initial
creative vision from scratch or existing designs, 3) use LLMs and text-to-
image models to help students find creative connections, generate unique
coding templates, and co-create visuals and interactive animations, 4) use
text-to-music models to produce background sounds for artworks, and 5) use
LLMs to finalize a creative vision, descriptions, and deployable code snippets
for portfolios.

##  3\. System Implementation & Feasibility

PortfolioMentor will be developed into a coding companion chatbot and
functioned as an add-on for leading online creative coding and visualization
editors. As a floating overlay on the browser, PortfolioMentor will use
JavaScript for interfacing with the editors while leveraging generative model
APIs and Python for backend operations, thus enabling real-time coding and
design suggestions and supports. Besides, we will build a Flask app to test
the workflow.

For a technical proof-of-concept, we break down the system into six stages and
have already tested the feasibility. Users can interact with the overlaying
chatbot about any of these stages for dedicated help, or they can navigate
through a complete cycle. See Appendix for a detailed walkthrough and demo of
technical feasibility:

Stage #1: Project Ideation
Begin artwork conceptualization using textual or visual inputs.
\- Input: Rough ideas, sketches, or online images.
\- Models: Claude (Anthropic, [n. d.]), image-to-text (MiniGPT-4 (Zhu et al.,
2023), BLIP-2 (Li et al., 2023)).
\- Output: Descriptive text and design suggestions for artwork.

Stage #2: Visual Design
Brainstorm artwork and generate coding templates.
\- Input: Text description of artwork (from Stage #1 or user).
\- Models: Claude (Anthropic, [n. d.]), text-to-image (CLIP (Radford et al.,
2021), VQGAN-CLIP (Crowson et al., 2022)).
\- Output: Visuals, creative connections, code templates, technical
feasibility.

Stage #3: Audio Design
Suggest and craft background music for artwork.
\- Input: Artwork design (from earlier stages or user).
\- Models: Claude (Anthropic, [n. d.]), text-to-music (MusicLM (Agostinelli et
al., 2023), MusicGen (Copet et al., 2023), CLAP (Wu et al., 2023)).
\- Output: Music recommendations and sample audio files (.mp3).

Stage #4: Interaction Design
Develop user interactivity in sync with audio elements.
\- Input: Artwork design (from earlier stages or user).
\- Models: Claude (Anthropic, [n. d.]).
\- Output: Coding editor updates with interactive feature templates.

Stage 5 & 6: Creative Vision & Portfolio Delivery
Finalize creative visions, narratives, and coding for portfolio.
\- Input: Artwork design (from earlier stages or user).
\- Models: Claude (Anthropic, [n. d.]).
\- Output: Polished vision, design narratives, code snippets (.html, .js,
.css).

Q&A, Proactive Suggestions, Code Learning, Companionship
Analyze on-screen content, give suggestions, handle Q&A, offer code
explanations and learning guidance, and provide feedback for motivation.
\- Input: Artwork design and on-screen materials.
\- Models: Claude (Anthropic, [n. d.]), DOM Manipulation, Screen Capture API.
\- Output: Chat interactions, screen highlights, and code explanations.

##  4\. Next Steps

We will refine our stage and system designs by co-designing with 10 more non-
technical art students. This involves prompt engineering, feature testing, and
natural chat testing to optimize the interaction. Besides, we will implement
sentimental understanding and expand the system’s art contextual knowledge to
make it more useful. After user testing, PortfolioMentor will be introduced to
real portfolio studios, positioning it as a hands-on tool for teaching and
learning.

## Appendix: Prototype Links

  * •

link Design File

  * •

link Design Prototype

  * •

link Design Video

## References

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[1,2]Bing-Zhao Li

1]School of Mathematics and Statistics, Beijing Institute of Technology,
Beijing, 100081, China

2]Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing,
100081, China

# Linear canonical space-time transform and convolution theorems

Yi-Qiao Xu<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

Following the idea of the fractional space-time Fourier transform, a linear
canonical space-time transform for 16-dimensional space-time
$C\ell_{3,1}$-valued signals is investigated in this paper. First, the
definition of the proposed linear canonical space-time transform is given, and
some related properties of this transform are obtained. Second, the
convolution operator and the corresponding convolution theorem are proposed.
Third, the convolution theorem associated with the two-sided linear canonical
space-time transform is derived.

###### keywords:

Space-time algebra, Fractional space-time Fourier transform, Linear canonical
transform, Linear canonical space-time transform, Convolution theorem

##  1 Introduction

Space-time algebra, exemplified by the Clifford $C\ell_{3,1}$ algebra, is an
associative algebra developed within the $\mathbb{R}^{3,1}$ space-time vector
space equipped with the Minkowski metric [1]. Introducing the transforms on
space-time algebra is becoming one of the hottest research topics. Hitzer is
one of the pioneers who introduce the Fourier transform on space-time algebra.
He first explained and formulated the space-time Fourier transform [2]. The
space-time Fourier transform (SFT) is a hypercomplex non-commutative Fourier
transform operating on functions defined on $\mathbb{R}^{3,1}$ and taking
values in Clifford’s geometric algebra $C\ell_{3,1}$ [3, 4]. Following this
direction, other researchers investigated the definitions and properties
associated with space-time Fourier transform [5, 6, 7]. Recently, inspired by
the orthogonal splitting of space-time algebra, the Heisenberg’s uncertainty
principle and the Donoho-Stark uncertainty principle for the space-time
Fourier transform were established successfully [8, 4].

In order to find the flexibility in representing information in the space-time
domain, researchers try to apply the other transforms on space-time algebra.
Hitzer generalized the SFT to a special affine space-time Fourier transform
(SASFT) for 16-dimensional space-time multivector $C\ell_{3,1}$-valued signals
over the domain of space-time $\mathbb{R}^{3,1}$ [9]. Zayed proposed a space-
time fractional Fourier transform (FrSFT) by generalizing the fractional
Fourier transform (FrFT) into the space-time algebra [1]. These newly defined
transforms are the first in literature to offer analogues of the FrFT and the
special affine Fourier transform (SAFT) for Clifford algebras. This opens a
new research direction on the theories and applications for new transforms
associated with space-time algebra.

The linear canonical transform (LCT) has three free parameters, which can
enable the rotation and expansion of the time-frequency plane, this
flexibility makes it particularly well-suited for the analysis and processing
of non-stationary functions [10]. The FT, FrFT, and Fresnel transform are
special cases of the LCT. Based on this fact and inspired by the FrSFT and the
SASFT, we investigate the linear canonical transform associated with space-
time algebra.

Convolution theory is of great significance in various fields such as
mathematics, signal processing, image processing, and optics, especially in
the design and implementation of multiplier filters [11]. In recent years,
great efforts have been made to popularize classical convolution theory and
extend its applicability to a wider range of fields. Therefore, many
convolution theorems related to linear canonical transform (LCT) have been
proposed [12, 13, 14]. Researchers have also shown significant interest in
convolution theory in the field of Clifford algebra. The classical convolution
and Mustard convolution of the general two-sided quaternion Fourier transform
and the general steerable two-sided Clifford Fourier transform were explored
in [15, 16]. Aajaz A. Teali and Firdous A. Shah extended the convolution
theorem to Clifford-valued linear canonical transform and two-sided Clifford-
valued linear canonical transform [17, 18]. Their contributions provide new
tools and methods for signal processing, image processing, and related fields.

The article is structured as follows: Section 2 gives an overview of space-
time algebra. In section 3, we introduce the concept of linear canonical
space-time transform (LCST) and learn its basic properties. Section 4 presents
the convolution theorem. Section 5 presents the convolution theorem for the
two-sided LCST. Finally, this article ends with the 6 section.

##  2 Preliminaries

###  2.1 Space-time algebra

The space-time algebra, is the Clifford algebra $C\ell_{3,1}$ derived from the
geometric product of all $\mathbb{R}^{3,1}$ and a 16-dimensional basis
algebra:

$C\ell_{3,1}=\rm{span}\\{1,e_{t},e_{1},e_{2},e_{3},e_{12},e_{13},e_{23},e_{t1},e_{t2},e_{t3},i_{3},e_{t12},e_{t13},e_{t23},i_{st}\\}.$
(1)

It is composed of scalar, vector, bivector, trivector, and pseudoscalar
elements.

The associative geometric multiplication of the basis vectors obeys the rule:

$e_{i}e_{j}+e_{j}e_{i}=2\epsilon_{j}\delta(i-j),\qquad i,j\in\\{t,1,2,3\\}.$
(2)

In the above the Kronecker symbol $\delta(i-j)$ is $1$ if $i=j$, zero
otherwise. And $\epsilon_{j}=-1$, for $j=\\{1,2,3\\}$, and $\epsilon_{j}=-1$,
for $j=t$.

Here, we will use conventional index notations such as $e_{12}=e_{1}e_{2}$,
$e_{t12}=e_{t}e_{12}$, etc. Here, $i_{3}=e_{1}e_{2}e_{3}$ is the unit volume
of the 3-vector, $i_{st}=e_{t}i_{3}$ is a pseudo-scalar, both squared by $-1$.
That is, ${i_{3}}^{2}={i_{st}}^{2}=-1$.

Each spatial element $h$ can be expressed as a real linear combination of
elementary edges given by (1). This notation allows $h$ to be expressed in the
general form:

$h=\left\langle h\right\rangle_{0}+\left\langle
h\right\rangle_{1}+\left\langle h\right\rangle_{2}+\left\langle
h\right\rangle_{3}+\left\langle h\right\rangle_{4},$ (3)

$\left\langle h\right\rangle_{0}=h_{0}$, $\left\langle
h\right\rangle_{1}=h_{t}e_{t}+h_{1}e_{1}+h_{2}e_{2}+h_{3}e_{3}$, $\left\langle
h\right\rangle_{2}=h_{12}e_{12}+h_{13}e_{13}+h_{23}e_{23}+h_{t1}e_{t1}+h_{t2}e_{t2}+h_{t3}e_{t3}$,
$\left\langle
h\right\rangle_{3}=h_{123}i_{3}+h_{t12}e_{t12}+h_{t13}e_{t13}+h_{t23}e_{t23}$,
$\left\langle h\right\rangle_{4}=h_{t123}i_{st}$, with
$h_{0},h_{t},h_{1},h_{2},h_{3},h_{12},h_{13},h_{23},h_{t1},h_{t2},h_{t3},h_{123},h_{t12},h_{t13},h_{t23},h_{t123}\in\mathbb{R}$.
$\left\langle h\right\rangle_{0}$, $\left\langle h\right\rangle_{1}$,
$\left\langle h\right\rangle_{2}$, $\left\langle h\right\rangle_{3}$ and
$\left\langle h\right\rangle_{4}$ are respectively called the 0th degree
scalar part, the 1st degree vector part, the 2nd degree bivector part, the 3rd
degree three vector part and the pseudo-scalar part.

A useful tool for us will be the following split of space-time multi-vector
$h\in C\ell_{3,1}$:

$h=h_{+}+h_{-},\ where\ h_{\pm}=\frac{1}{2}(h\pm e_{t}\ h\ i_{3}).$ (4)

It is worth noting that the decomposition (4) yields the norm identity:

$|h|^{2}=|h_{+}|^{2}+|h_{-}|^{2}.$ (5)

We have the following key identities for shifting:

$exp\\{\alpha
e_{t}\\}h_{\pm}exp\\{\alpha^{\prime}i_{3}\\}=h_{\pm}exp\\{(\alpha^{\prime}\mp\alpha)i_{3}\\}=exp\\{(\alpha\mp\alpha^{\prime})e_{t}\\}h_{\pm},\
\forall\alpha,\alpha^{\prime}\in\mathbb{R}$ (6)

The principal reverse of $h\in C\ell_{3,1}$ is defined by

$\displaystyle\widetilde{h}$
$\displaystyle=h_{0}e_{0}-h_{t}e_{t}+\sum_{i=1}^{3}h_{i}e_{i}-h_{12}e_{12}-h_{13}e_{13}-h_{23}e_{23}+\sum_{i=1}^{3}h_{ti}e_{ti}$
(7) $\displaystyle-
h_{3}i_{3}+h_{t12}e_{t12}+h_{t13}e_{t13}+h_{t23}e_{t23}-h_{t123}i_{st}.$

In particular, each space-time element has $h_{1}$ and $h_{2}$

$\widetilde{h_{1}+h_{2}}=\widetilde{h_{1}}+\widetilde{h_{2}};\widetilde{h_{1}h_{2}}=\widetilde{h_{2}}\widetilde{h_{1}}\mathrm{~{};~{}}\widetilde{\widetilde{h_{1}}}=h_{1}.$
(8)

The trace of each element $h$ of $C\ell_{3,1}$ is defined as the scalar part
of its decomposition, as shown in (3). That is, it is expressed by the
following equation:

$Tr(h)\overset{\underset{\mathrm{def}}{}}{=}h_{0}.$ (9)

The symmetry of cyclic multiplication can be easily verified in the following
way:

$Tr(h_{1}h_{2}h_{3})~{}=~{}Tr(h_{3}h_{1}h_{2}),~{}\forall
h_{1},~{}h_{2},~{}h_{3}\in C\ell_{3,1}.$ (10)

Additionally, we can easily prove that

$\int_{\mathbb{R}}\mathbf{e}^{ht(x-x_{0})}dt=2\pi\delta(x-x_{0}),$ (11)

Let $h\in C\ell_{3,1}$, $\alpha,\alpha^{\prime}\in\mathbb{R}$, one has

$|e_{t}hi_{3}|=|h|\mathrm{~{},~{}}\left|\mathbf{e}^{\alpha
e_{t}}h\mathbf{e}^{\alpha^{\prime}i_{3}}\right|=|h|.$ (12)

Based on the linearity and properties (10) of the traces (9), the
orthogonality of the spatial partitioning can be directly verified in the
following sense

Let $f,g\in C\ell_{3,1},\alpha\in\mathbb{R}$, then

$Tr\left(\mathbf{e}^{\alpha i_{3}}\widetilde{f_{\pm}}g_{\mp}\right)=0.$ (13)

###  2.2 Related works

Next, we review the definition of the space-time Fourier transform (SFT).

######  Definition 1.

The space-time Fourier transform of $f\in L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$
is defined as [2]

$\mathcal{F}_{\mathrm{S}FT}[f](\bm{w})=\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}f(\bm{x})\exp\left\\{-i_{3}\underline{x}\cdot\underline{w}\right\\}d^{4}\bm{x},$
(14)

where
$\bm{w}=w_{t}e_{t}+\underline{w}\in\mathbb{R}^{3,1},\underline{w}=w_{1}e_{1}+w_{2}e_{2}+w_{3}e_{3}\in\mathbb{R}^{3}$
is the space-time frequency vector, $\begin{aligned}
\bm{x}&=te_{t}+\underline{x}\in\mathbb{R}^{3,1},\underline{x}=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\in\mathbb{R}^{3}\end{aligned}$
is the space-time vector, and
$\underline{x}\cdot\underline{w}={\textstyle\sum_{k=1}^{k=3}x_{k}w_{k}}$ is
the natural $\mathbb{R}^{3}-scalar$ product.

The inversion formula for the SFT is given by

$f(\bm{x})=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{3,1}}\exp{\\{e_{t}w_{t}t\\}}~{}\mathcal{F}_{SFT}[f](\bm{w})~{}\exp{\\{i_{3}\underline{x}\cdot\underline{w}\\}}~{}d^{4}\bm{w}.$
(15)

Based on the above research, Zayed first introduced the fractional Fourier
transform (FrFT) to space-time algebra [1]. Next, we review the definition of
the fractional space-time Fourier transform (FrSFT).

######  Definition 2.

The fractional space-time Fourier transform of $f\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ with respect to a parameter $\alpha$, is
defined as [1]

$\mathcal{F}_{rSFT}^{\alpha}[f](\bm{w})=\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}\left.f(\bm{x})\mathcal{K}_{i_{3}}^{\alpha}(\underline{x},\underline{w})\right.d^{4}\bm{x},$
(16)

the right kernel factor $\mathcal{K}_{i_{3}}^{\alpha}$ is

$\mathcal{K}_{i_{3}}^{\alpha}(\underline{x},\underline{w})=\begin{cases}(\csc\alpha)^{\frac{3}{2}}\bm{e}^{i_{3}\left((\underline{x}^{2}+\underline{w}^{2})\frac{\cot\alpha}{2}-\underline{x}\cdot\underline{w}\csc\alpha+\frac{2\alpha-\pi}{4}\right)},&\alpha\neq
n\pi\\\ \delta(\underline{w}-\underline{x}),&\alpha=2n\pi\\\
\delta(\underline{w}+\underline{x}),&\alpha=(2n+1)\pi\end{cases},$ (17)

where $sin\alpha>0$. It is assumed that $\alpha\neq n\pi$.

The inversion formula for the FrSFT is given by

$f(\bm{x})=\frac{(\csc\alpha)^{\frac{3}{2}}}{(2\pi)^{4}}\int_{\mathbb{R}^{3,1}}\bm{e}^{e_{t}w_{t}t}\mathcal{F}_{rSFT}^{\alpha}[f](\bm{w})\mathcal{K}_{-i_{3}}^{\alpha}(\underline{x},\underline{w})d^{4}\bm{w},$
(18)

where the space-time frequency volume
$d^{4}\bm{w}=dw_{t}dw_{1}dw_{2}dw_{3}dw_{4}$.

Based on the existing research results, we introduce the concept of linear
canonical space-time transform (LCST) by replacing the fractional Fourier
kernel in the fractional space-time Fourier transform with a linear canonical
kernel, and propose the fundamental properties of the proposed linear
canonical space-time transform.

##  3 Linear canonical space-time transform

###  3.1 Definition

######  Definition 3.

The linear canonical space-time transform of $f\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ with respect to a parameter matrix
$A=(a,b;c,d)\in\mathbb{R}^{2\times 2}$, is defined as

$\mathcal{L}_{A}[f](\bm{w})=\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}\left.f(\bm{x})\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})\right.d^{4}\bm{x},$
(19)

here we modify the right kernel factor $\mathcal{K}_{i_{3}}^{A}$ to

$\displaystyle\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})=\begin{cases}\frac{1}{(2\pi
b)^{\frac{3}{2}}}\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{\underline{x}\cdot\underline{w}}{b}\right)},&b\neq
0\\\
\sqrt{d}\bm{e}^{i_{3}\left(\frac{cd}{2}\underline{w}^{2}\right)}\delta(\underline{x}-d\underline{w}),&b=0\\\
\end{cases}.$

where $a,b,c,d\in\mathbb{R}$, and $ad-bc=1$.

###  3.2 Properties

######  Theorem 1.

(Inversion of LCST) Let $f,~{}\mathcal{L}_{A}[f]\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$, then $f$ can be recovered from
$~{}\mathcal{L}_{A}[f]$ by using the formula

$f(\bm{x})=\frac{1}{2\pi}\int_{\mathbb{R}^{3,1}}\bm{e}^{e_{t}w_{t}t}\mathcal{L}_{A}[f](\bm{w})\mathcal{K}_{-i_{3}}^{A}(\underline{x},\underline{w})d^{4}\bm{w},$
(20)

where the space-time frequency volume
$d^{4}\bm{w}=dw_{t}dw_{1}dw_{2}dw_{3}dw_{4}$.

###### Proof.

Based on identity (11) and Definition 3, we obtain

$\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}^{3,1}}\bm{e}^{e_{t}w_{t}t}\mathcal{L}_{A}[f](\bm{w})\mathcal{K}_{-i_{3}}^{A}(\underline{x},\underline{w})d^{4}\bm{w}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\bm{e}^{e_{t}w_{t}t}\bm{e}^{-e_{t}w_{t}s}f(\bm{y})\mathcal{K}_{i_{3}}^{A}(\underline{y},\underline{w})\mathcal{K}_{-i_{3}}^{A}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{w}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}^{3,1}}\left[\int_{\mathbb{R}}\bm{e}^{w_{t}(t-s)e_{t}}dw_{t}\right]f(y)\bm{e}^{\frac{-i_{3}a}{2b}\left(\underline{x}^{2}-\underline{y}^{2}\right)}$
$\displaystyle\times\left[\int_{\mathbb{R}^{3}}\bm{e}^{\frac{i_{3}}{b}\left((\underline{x}-\underline{y})\cdot\underline{w}\right)}dw_{1}dw_{2}dw_{3}\right]d^{4}\bm{y}$
$\displaystyle=\frac{1}{(2\pi)^{4}b^{3}}\int_{\mathbb{R}^{3,1}}2\pi\delta(t-s)\left.f(\bm{y})\right.e^{\frac{-i_{3}a}{2b}\left(\underline{x}^{2}-\underline{y}^{2}\right)}\left(2\pi
b\right)^{3}\delta(\underline{x}-\underline{y})d^{4}\bm{y}$
$\displaystyle=f(\bm{x}).$

∎

######  Theorem 2.

(Properties) Let $f,g\in L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$, with
$\bm{x},\bm{w}\in\mathbb{R}^{3,1}$, constants
$\bm{u}=(u_{t},\underline{u}),\bm{y}=(y_{t},\underline{y})\in\mathbb{R}^{3,1}$.
We have

(1). Right-hand linearity that commute with $i_{3}$ and left-hand linearity
that commute with $e_{t}$:

$\mathcal{L}_{A}[Mf+gN](\bm{w})=M\mathcal{L}_{A}[f](\bm{w})+\mathcal{L}_{A}[g](\bm{w})N,$

with$M=M_{0}+M_{t}e_{t}+M_{12}e_{12}+M_{23}e_{23}+M_{13}e_{13}+M_{t12}e_{t12}+M_{t23}e_{t23}+M_{t13}e_{t13}$,
$N=N_{0}+N_{1}e_{1}+N_{2}e_{2}+N_{3}e_{3}+N_{12}e_{12}+N_{23}e_{23}+N_{13}e_{13}+N_{123}i_{3}$.

(2). Modulation covariance:

$\displaystyle\mathcal{L}_{A}{\left[\bm{e}^{-e_{t}u_{t}t}\left.f(\bm{x})\right.\bm{e}^{-i_{3}\underline{x}\cdot\underline{u}}\right]}(\bm{w})$
$\displaystyle=\mathcal{L}_{A}{\left[f(\bm{x})\right]}\left(\bm{w}+(u_{t}e_{t}+b\underline{u})\right)$
$\displaystyle\times\bm{e}^{\frac{-i_{3}}{2}d\left(b\underline{u}^{2}+2\underline{w}\cdot\underline{u}\right)}.$

(3). Translation covariance:

$\begin{gathered}\mathcal{L}_{A}\bigl{[}f(\bm{x}-\bm{y})\bigr{]}(\bm{w})\left.=\bm{e}^{-e_{t}y_{t}w_{t}}\mathcal{L}_{A}[f]\left(w_{t}e_{t}+\underline{w}-a\right.\underline{y}\right)\\\
\times\bm{e}^{i_{3}\frac{1-a}{b}\left(\frac{a\underline{y}^{2}}{2}-\underline{y}\cdot\underline{w}\right)}.\end{gathered}$

(4). Reflection:
$\mathcal{L}_{A}[f(-\bm{x})](\bm{w})=\mathcal{L}_{A}[f(\bm{x})](-\bm{w})$.

###### Proof.

We only give the proof of (2) as an example.

$\displaystyle\mathcal{L}_{A}{\left[f(\bm{x})\right]}\left(\bm{w}+(u_{t}e_{t}+b\underline{u})\right)\times\bm{e}^{\frac{-i_{3}}{2}d\left(b\underline{u}^{2}+2\underline{w}\cdot\underline{u}\right)}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}(w_{t}+u_{t})t}\left.f(\bm{x})\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w}+b\underline{u})\right.d^{4}\bm{x}\times\bm{e}^{\frac{-i_{3}}{2}d\left(b\underline{u}^{2}+2\underline{w}\cdot\underline{u}\right)}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}(w_{t}+u_{t})t}\left.f(\bm{x})\frac{1}{(2\pi
b)^{\frac{3}{2}}}\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}(\underline{w}+b\underline{u})^{2}-\frac{\underline{x}\cdot(\underline{w}+b\underline{u})}{b}\right)}\right.d^{4}\bm{x}\times\bm{e}^{\frac{-i_{3}}{2}d\left(b\underline{u}^{2}+2\underline{w}\cdot\underline{u}\right)}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}[\bm{e}^{-e_{t}u_{t}t}\left.f(\bm{x})\bm{e}^{-i_{3}\underline{x}\cdot\underline{u}}]\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})\right.d^{4}\bm{x}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{A}{\left[\bm{e}^{-e_{t}u_{t}t}\left.f(\bm{x})\right.\bm{e}^{-i_{3}\underline{x}\cdot\underline{u}}\right]}(\bm{w}).$

∎

######  Theorem 3.

(Plancherel theorem) Let $\mathcal{L}_{A}[f_{1}]$ and $\mathcal{L}_{A}[f_{2}]$
be the LCSTs of $f_{1}$ and $f_{2}\in L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})$,
respectively. Then, we have

$\left\langle\mathcal{L}_{A}[f_{1}],\mathcal{L}_{A}[f_{2}]\right\rangle_{L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})}=2\pi\langle
f_{1},f_{2}\rangle_{L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})}.$ (21)

###### Proof.

Using the definition of 3 and Fubini replacement theorem, we get the following

$\displaystyle\big{\langle}\mathcal{L}_{A}\big{[}f_{1}\big{]},\mathcal{L}_{A}\big{[}f_{2}\big{]}\big{\rangle}_{L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})}$
$\displaystyle=\frac{1}{(2\pi
b)^{3}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}}Tr\bigg{[}\bm{e}^{-e_{t}w_{t}t}f_{1}(\mathbf{x})\exp\left\\{\frac{i_{3}}{2}\left(\underline{x}^{2}-\underline{y}^{2}\right)\frac{a}{b}\cot\alpha\right\\}$
$\displaystyle\left.\times\int_{\mathbb{R}^{3}}\exp\left\\{-i_{3}\left((\underline{x}-\underline{y})\cdot\underline{w}\right)\frac{1}{b}\right\\}dw_{1}dw_{2}dw_{3}\widetilde{f_{2}(\mathbf{y})}\exp\left\\{e_{t}w_{t}s\right\\}\right]dw_{t}d^{4}\mathbf{x}d^{4}\mathbf{y}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq16})}}{{=}}\frac{1}{(2\pi
b)^{3}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}}Tr\bigg{[}\bm{e}^{-e_{t}w_{t}t}f_{1}(\mathbf{x})\exp\left\\{\frac{i_{3}}{2}\left(\underline{x}^{2}-\underline{y}^{2}\right)\frac{a}{b}\right\\}$
$\displaystyle\left.\times\left(2\pi
b\right)^{3}\delta(\underline{x}-\underline{y})\widetilde{f_{2}(\mathbf{y})}\exp\left\\{e_{t}w_{t}s\right\\}\right]dw_{t}d^{4}\mathbf{x}d^{4}\mathbf{y}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq12})}}{{=}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}}Tr\bigg{[}\exp{\\{-e_{t}w_{t}(t-s)\\}}f_{1}(\mathbf{x})$
$\displaystyle\left.\times\exp\left\\{\frac{i_{3}}{2}\left(\underline{x}^{2}-\underline{y}^{2}\right)\frac{a}{b}\right\\}\delta(\underline{x}-\underline{y})\widetilde{f_{2}(\mathbf{y})}\right]dw_{t}d^{4}\mathbf{x}d^{4}\mathbf{y}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}Tr\bigg{[}\int_{\mathbb{R}}\exp\left\\{-e_{t}w_{t}(t-s)\right\\}dw_{t}f_{1}(\mathbf{x})$
$\displaystyle\left.\times\exp\left\\{\frac{i_{3}}{2}\left(\underline{x}^{2}-\underline{y}^{2}\right)\frac{a}{b}\right\\}\delta(\underline{x}-\underline{y})\widetilde{f_{2}(\mathbf{y})}\right]d^{4}\mathbf{x}d^{4}\mathbf{y}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq16})}}{{=}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}Tr\Big{[}2\pi\delta(t-s)f_{1}(\mathbf{x})\exp\left\\{\frac{i_{3}}{2}\left(\underline{x}^{2}-\underline{y}^{2}\right)\frac{a}{b}\cot\alpha\right\\}$
$\displaystyle\times\left.\delta(\underline{x}-\underline{y})\widetilde{f_{2}(\mathbf{y})}\right]d^{4}\mathbf{x}d^{4}\mathbf{y}$
$\displaystyle=2\pi\int_{\mathbb{R}^{3,1}}Tr\left[f_{1}(\mathbf{x})\widetilde{f_{2}(\mathbf{x})}\right]d^{4}\mathbf{x}$
$\displaystyle=2\pi\left<f_{1},f_{2}\right>_{L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})}.$

∎

######  Theorem 4.

(Parseval formula) If $f_{1}=f_{2}=f$, then Parseval’s identity

$\left\|\mathcal{L}_{A}[f]\right\|_{2}=\sqrt{2\pi}\left\|f\right\|_{2}.$ (22)

holds.

######  Theorem 5.

(Partial derivatives) For $k=1,2,3$, if $\begin{aligned} f,\frac{\partial
f}{\partial t},\frac{\partial f}{\partial x_{k}},\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})\end{aligned}$, provided that the
derivatives exist, then

(1). $\mathcal{L}_{A}\left[\frac{\partial f(\bm{x})}{\partial
t}\right](\bm{w})=e_{t}w_{t}\mathcal{L}_{A}[f](\bm{w})$.

(2). $\mathcal{L}_{A}\left[\frac{\partial f(\bm{x})}{\partial
x_{k}}\right](\bm{w})=\left[w_{k}\mathcal{L}_{A}[f](\bm{w})~{}-~{}a~{}\mathcal{L}_{A}[x_{k}~{}f(\bm{x})](\bm{w})\right]\times\frac{i_{3}}{b}$.

(3). $\frac{\partial}{\partial
w_{t}}\Bigl{\\{}\mathcal{L}_{A}\left[f\right](\bm{w})\Bigr{\\}}=-e_{t}\mathcal{L}_{A}[t\left.f(\bm{x})\right](\bm{w})$.

(4). $\frac{\partial}{\partial
w_{k}}\Big{\\{}\mathcal{L}_{A}[f](\bm{w})\Big{\\}}=\Big{[}a\mathcal{L}_{A}\Big{[}w_{k}f(\bm{x})\Big{]}(\bm{w})-\mathcal{L}_{A}\Big{[}x_{k}f(\bm{x})\Big{]}(\bm{w})\Big{]}\times\frac{i_{3}}{b}$.

###### Proof.

We will just prove (2). For $k=1,2,3$, we have

$\displaystyle\mathcal{L}_{A}\left[\frac{\partial f(x)}{\partial
x_{k}}\right](\bm{w})$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}\frac{\partial
f(\bm{x})}{\partial
x_{k}}\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})d^{4}\bm{x}$
$\displaystyle=\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}\left(\int_{\mathbb{R}}\frac{\partial}{\partial
x_{k}}f(\bm{x})\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})dx_{k}\right)d^{3}\bm{x}$
$\displaystyle=\frac{1}{(2\pi
b)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}$
$\displaystyle\times\left(\int_{\mathbb{R}}\frac{\partial}{\partial
x_{k}}\left.f(\bm{x})\right.\bm{e}^{i_{3}(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{1}{b}\underline{x}\cdot\underline{w})}dx_{k}\right)d^{3}\bm{x}$
$\displaystyle=\frac{1}{(2\pi
b)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}$
$\displaystyle\times\left(\int_{\mathbb{R}}f\left(\bm{x}\right)\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{1}{b}\underline{x}\cdot\underline{w}\right)}\right|_{x_{k}=-\infty}^{x_{k}=\infty}$
$\displaystyle\left.-\left.f(\bm{x})\frac{(ax_{k}-w_{k})i_{3}}{b}\right.\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{1}{b}\underline{x}\cdot\underline{w}\right)}dx_{k}\right)d^{3}\bm{x}$
$\displaystyle=-\frac{1}{(2\pi
b)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}$
$\displaystyle\times\left(\int_{\mathbb{R}}f\left(\bm{x}\right)\frac{\left(ax_{k}-w_{k}\right)i_{3}}{b}\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{1}{b}\underline{x}\cdot\underline{w}\right)}dx_{k}\right)d^{3}\bm{x}$
$\displaystyle=\frac{1}{(2\pi
b)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}$
$\displaystyle\times\left(\int_{\mathbb{R}}f(\bm{x})(w_{k}-ax_{k})\bm{e}^{i_{3}\left(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{1}{b}\underline{x}\cdot\underline{w}\right)}dx_{k}\right)d^{3}\bm{x}\frac{i_{3}}{b}$
$\displaystyle=\int_{\mathbb{R}^{3}}\bm{e}^{-e_{t}w_{t}t}\left(\int_{\mathbb{R}}f(\bm{x})(w_{k}-x_{k}\cos\alpha)\mathcal{K}_{i_{3}}^{\alpha}(\underline{x},\underline{w})dx_{k}\right)d^{3}\bm{x}i_{3}\frac{i_{3}}{b}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\bm{e}^{-\bm{e}_{t}w_{t}t}f(\bm{x})(w_{k}-ax_{k})\mathcal{K}_{i_{3}}^{A}(\underline{x},\underline{w})d^{4}\bm{x}\frac{i_{3}}{b}$
$\displaystyle=\left[w_{k}\mathcal{L}_{A}\bigl{[}f\bigr{]}(\bm{w})-a\mathcal{L}_{A}\bigl{[}x_{k}f(\bm{x})\bigr{]}(\bm{w})\right]\frac{i_{3}}{b}.$

∎

##  4 Convolution theorems for the LCST

Convolution stands as a fundamental concept in linear time-invariant systems.
It characterizes the output of a continuous-time linear time-invariant system
as the convolution of the input signal and the system’s impulse response.
Essentially, convolution entails an integral operation and finds extensive
application across fields like numerical analysis, finite impulse response,
pattern recognition, and signal processing. Consequently, delving deeper into
the convolution operation’s form and the associated convolution theorem within
the domain of LCT under space-time algebra bears significant theoretical and
practical relevance.

We define the convolution of two space-time signals $a,b\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ as

$(a\star
b)(\bm{x})=\int_{\mathbb{R}^{3,1}}a(\bm{y})b(\bm{x}-\bm{y})d^{4}\bm{y},$ (23)

provided that the integral exists.

The Mustard convolution of two space-time signals $a,b\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ is defined as

$(a\star_{M}b)(\bm{x})=\mathcal{F}^{-1}(\mathcal{F}\\{a\\}\mathcal{F}\\{b\\}).$
(24)

provided that the integral exists.

######  Theorem 6.

(Mustard convolution in terms of standard convolution)[19]. The Mustard
convolution (24) of two space-time functions $a,b\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ can be expressed in terms of eight
standard convolutions (23) as

$\displaystyle a\star_{M}b(\bm{x})$ $\displaystyle=[a_{+}\star
b_{+}(\bm{x})]_{+}+[a_{+}\star b_{+}^{(1,1)}(t_{1},-\underline{x})]_{-}$ (25)
$\displaystyle+[a_{+}\star b_{-}^{(1,0)}(\bm{x})]_{+}+[a_{+}\star
b_{-}^{(0,1)}(t_{1},-\underline{x})]_{-}$ $\displaystyle+[a_{-}\star
b_{+}^{(0,1)}(t_{1},-\underline{x})]_{+}+[a_{-}\star
b_{+}^{(1,0)}(\bm{x})]_{-}$ $\displaystyle+[a_{-}\star
b_{-}^{(1,1)}(t_{1},-\underline{x})]_{+}+[a_{-}\star b_{-}(\bm{x})]_{-}.$

The computation of a typical term in Theorem 6 can be illustrated by writing
out, eg, the third term in full detail

+ $\displaystyle=\frac{1}{2}\Bigg{[}\int_{\mathbb{R}^{3,1}}\frac{1}{2}(a(\bm{y})+e_{t}a(\bm{y})i_{3})$ (26) $\displaystyle\frac{1}{2}(b(-(t_{x}-t_{y}),\underline{x}-\underline{y})-e_{t}b(-(t_{x}-t_{y}),\underline{x}-\underline{y})i_{3})d^{4}\bm{y}$ $\displaystyle+e_{t}\int_{\mathbb{R}^{3,1}}\frac{1}{2}(a(\bm{y})+e_{t}a(\bm{y})i_{3})$ $\displaystyle\frac{1}{2}(b(-(t_{x}-t_{y}),\underline{x}-\underline{y})-e_{t}b(-(t_{x}-t_{y}),\underline{x}-\underline{y})i_{3})d^{4}\bm{y}i_{3}\Bigg{]}.$

We define a new convolution structure for LCST based on the relationship
between the LCST and the SFT, which states that a modified ordinary
convolution in the time domain is equivalent to a simple multiplication
operation for LCST and SFT. Based on the definition of the ordinary SFT (14),
the relationship between the LCST and the SFT is given below:

######  Lemma 1.

Let $f\in L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ and $\alpha\neq n\pi$, then

$\mathcal{L}_{A}\big{[}f\big{]}(\bm{w})=B\mathcal{F}_{SFT}[f(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}](w_{t},\frac{\underline{w}}{b})\bm{e}^{\frac{i_{3}d}{2b}\underline{w}^{2}},$
(27)

and

$\left|B\mathcal{L}_{A}\bigl{[}f\bigr{]}(w_{t},\underline{w}b)\right|=\left|\mathcal{F}_{SFT}\biggl{[}f(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}\biggr{]}(\bm{w})\biggr{|}.\right.$
(28)

where $B=\frac{1}{(2\pi b)^{3/2}}$.

######  Definition 4.

For any space-time function $f(\bm{x})\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$, let us define the function
$\tilde{f}(\bm{x})=f(\bm{x})e^{i_{3}\frac{a}{2b}\underline{x}^{2}}$. For any
two functions $f$ and $g$, we define the convolution operation $\odot$ by

$h(\bm{x})=(f\odot
g)(\bm{x})=e^{i_{3}\frac{a}{2b}\underline{x}^{2}}\left(\tilde{f}\star_{M}g\right)(\bm{x}).$
(29)

where $\star_{M}$ is the Mustard convolution operation for the SFT as defined
by (24).

Now we state and prove our new convolution theorem.

######  Theorem 7.

Let $h(\bm{x})=(f\odot g)(\bm{x})$ and $\mathcal{L}_{A}[h](\bm{w})$,
$\mathcal{L}_{A}[f](\bm{w})$ denote the LCST of $h(\bm{x})$ and $f(\bm{x})$.
$\mathcal{F}_{SFT}[g](\bm{w})$ denotes the SFT of $g(\bm{x})$. Then

$\mathcal{L}_{A}[h(\bm{x})](\bm{w})=\mathcal{L}_{A}[f(\bm{x})](\bm{w})\mathcal{F}_{SFT}[g(\bm{x})](w_{t},\frac{\underline{w}}{b}).$
(30)

###### Proof.

From the definition of the LCST and Definition 4, we have

$\displaystyle\mathcal{L}_{A}[h]$
$\displaystyle=B\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}h(\bm{x})\bm{e}^{i_{3}(\frac{a}{2b}\underline{x}^{2}+\frac{d}{2b}\underline{w}^{2}-\frac{\underline{x}\cdot\underline{w}}{b})}d^{4}\bm{x}$
(31)
$\displaystyle=B\mathcal{F}_{SFT}[h(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}](w_{t},\frac{\underline{w}}{b})\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}$
$\displaystyle=B\mathcal{F}_{SFT}[(\tilde{f}\star_{M}g)(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}](w_{t},\frac{\underline{w}}{b})\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}.$

According to convolution theorem for SFT and (27), we have

$\displaystyle\mathcal{L}_{A}[h]$
$\displaystyle=B\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}\mathcal{F}_{SFT}[f(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}\star_{M}g(\bm{x})](w_{t},\frac{\underline{w}}{b})$
(32)
$\displaystyle=B\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}\mathcal{F}_{SFT}[f(\bm{x})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}](w_{t},\frac{\underline{w}}{b})\mathcal{F}_{SFT}[g(\bm{x})](w_{t},\frac{\underline{w}}{b})$
$\displaystyle=\mathcal{L}_{A}[f(\bm{x})]\mathcal{F}_{SFT}[g(\bm{x})](w_{t},\frac{\underline{w}}{b}),$

and this completes the proof of (30), that is

$\mathcal{L}_{A}[f\star_{M}g(\bm{x})](\bm{w})=\mathcal{L}_{A}[f(\bm{x})](\bm{w})\mathcal{F}_{SFT}[g(\bm{x})](w_{t},\frac{\underline{w}}{b}).$
(33)

∎

######  Definition 5.

For any space-time function $f(\bm{x})\in
L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$, let us define the function
$\hat{f}(\bm{x})=f(\bm{x})e^{-i_{3}\frac{d}{2b}\underline{w}^{2}}$. For any
two functions $f$ and $g$, we define the convolution operation $\otimes$ by

$h(\bm{x})=(f\otimes
g)(\bm{x})=e^{i_{3}\frac{d}{2b}\underline{x}^{2}}\left(\hat{f}\star
g\right)(\bm{x}).$ (34)

where $\star$ is the conventional convolution operation for the SFT as defined
by (23).

Now we state and prove our new product theorem.

######  Theorem 8.

Let $h(\bm{x})=(f\otimes g)(\bm{x})$ and $\mathcal{L}_{A}[h](\bm{w})$,
$\mathcal{L}_{A}[f](\bm{w})$ denote the LCST of $h(\bm{x})$ and $f(\bm{x})$.
$\mathcal{F}_{SFT}[g](\bm{w})$ denotes the SFT of $g(\bm{x})$. Then

$\mathcal{L}_{A}[f(\bm{x})g(\frac{\bm{x}}{b})]=\frac{1}{(2\pi)^{4}}(\mathcal{L}_{A}[f]\otimes\mathcal{F}_{SFT}[g]).$
(35)

###### Proof.

According to Definition 5, we have

$\displaystyle\mathcal{L}_{A}[f]\otimes\mathcal{F}_{SFT}[g]$
$\displaystyle=\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}\hat{\mathcal{L}_{A}}[f]\star\mathcal{F}_{SFT}[g]$
(36)
$\displaystyle=\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}\mathcal{L}_{A}[f(\bm{y})]\bm{e}^{-i_{3}\frac{d}{2b}\underline{w}^{2}}\mathcal{F}_{SFT}[g(\bm{x}-\bm{y})]d^{4}\bm{y}.$

Making use of the definition of the LCST, we obtain

$\displaystyle\mathcal{L}_{A}[f]\otimes\mathcal{F}_{SFT}[g]$
$\displaystyle=\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}B\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}f(\bm{x})\bm{e}^{i_{3}(\frac{a}{2b}\underline{x}^{2}-\frac{\underline{x}\cdot\underline{w}}{b})}d^{4}\bm{x}$
(37)
$\displaystyle\int_{\mathbb{R}^{3,1}}\mathcal{F}_{SFT}[g(\bm{x}-\bm{y})]d^{4}\bm{y}$
$\displaystyle=\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}B\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}f(\bm{x})\bm{e}^{i_{3}(\frac{a}{2b}\underline{x}^{2})}\bm{e}^{-i_{3}\frac{\underline{x}\cdot(\underline{w}-\bm{u})}{b}}d^{4}\bm{x}$
$\displaystyle\int_{\mathbb{R}^{3,1}}\mathcal{F}_{SFT}[g(\bm{u})]d^{4}\bm{u}$
$\displaystyle=(2\pi)^{4}\int_{\mathbb{R}^{3,1}}\bm{e}^{-e_{t}w_{t}t}f(\bm{x})g(\frac{\bm{x}}{b})\bm{e}^{i_{3}\frac{a}{2b}\underline{x}^{2}}\bm{e}^{-i_{3}\frac{\underline{x}\cdot\underline{w}}{b}}\bm{e}^{i_{3}\frac{d}{2b}\underline{w}^{2}}d^{4}\bm{x}$
$\displaystyle=(2\pi)^{4}\mathcal{L}_{A}[f(\bm{x})g(\frac{\bm{x}}{b})].$

∎

##  5 Convolution theorem for the two-sided LCST

######  Definition 6.

For any real matrix $M_{k}=\begin{pmatrix}A_{k}&B_{k}\\\
C_{k}&D_{k}\end{pmatrix}$ with $A_{k}D_{k}-B_{k}C_{k}=1$, $k=1,2$. The two-
sided linear canonical space-time transform maps 16-dimensional space-time
algebra functions $h\in L^{1}(\mathbb{R}^{3,1},C\ell_{3,1})$ to 16-dimensional
spectrum functions $\mathcal{L}_{A}[h]:\mathbb{R}^{3,1}\to C\ell_{3,1}$. It is
defined as

$\mathcal{L}_{A}[h](\bm{w})=\int_{\mathbb{R}^{3,1}}\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t)h(\bm{x})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{x},$
(38)

We give the left and the right kernel factors of the two-sided LCST as follows

$\displaystyle\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t)=\frac{1}{\sqrt{2\pi
B_{1}}}\bm{e}^{\frac{e_{t}}{2B_{1}}\left(A_{1}t^{2}+D_{1}w_{t}^{2}-2tw_{t}\right)},$
(39)
$\displaystyle\mathcal{K}_{i_{3}}^{M_{2}}(\underline{x},\underline{w})=\frac{1}{\sqrt{2\pi
B_{2}}}\bm{e}^{\frac{i_{3}}{2B_{2}}\left(A_{2}\underline{x}^{2}+D_{2}\underline{w}^{2}-2\underline{x}\cdot\underline{w}\right)}.$

The inverse transformation of the proposed two-sided linear canonical space-
time transform is given by the following theorem.

######  Theorem 9.

Let $\mathcal{L}_{A}[h](\bm{w})$ be the two-sided linear canonical space-time
transform of any space-time valued function $h\in
L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})$. Then, $h(\bm{x})$ can be completely
reconstructed via the formula

$h(\bm{x})=\int_{\mathbb{R}^{3,1}}\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t)\mathcal{L}_{A}[h](\bm{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{w}.$
(40)

We define a new convolution of two space-time valued functions.

######  Definition 7.

The new convolution of $f$ and $g$ is denoted by $f\star_{N}g$ and is defined
as

$\displaystyle(f\star_{N}g)(\bm{x})$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})f(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})$
(41)
$\displaystyle\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w},$

where

$\lambda_{e_{t}}(\bm{w})=(2\pi)^{2}B_{1}^{\frac{1}{2}}B_{2}^{\frac{7}{2}}\bm{e}^{e_{t}(\frac{D_{1}w_{t}^{2}}{2B_{1}}+\frac{D_{2}\underline{w}^{2}}{2B_{2}})}.$
(42)

We are now ready to present the main result of the section, which gives the
expression of the proposed convolution (41) in terms of the classical
convolutions. In the proof of this result, we shall use the splitting
technique. Moreover, for the shifting of kernels, we shall make the use of the
following general identity:

$\mathcal{K}_{e_{t}}^{M_{i}}h_{\pm}\mathcal{K}_{i_{3}}^{M_{i}}=\mathcal{K}_{e_{t}}^{M_{i}}\mathcal{K}_{\mp{e_{t}}}^{M_{i}}h_{\pm}=h_{\pm}\mathcal{K}_{i_{3}}^{M_{i}}\mathcal{K}_{\mp{e_{t}}}^{M_{i}}.$
(43)

Furthermore, for any function $h:\mathbb{R}^{3,1}\to C\ell_{3,1}$ and a multi-
index $(\phi_{1},\phi_{2})$ with $\phi_{1},\phi_{2}\in\left\\{0,1\right\\}$,
we set

$h^{(\phi_{1},\phi_{2})}(\bm{x})=h\Big{(}(-1)^{\phi_{1}}t_{1},(-1)^{\phi_{2}}\underline{x}\Big{)}.$
(44)

######  Theorem 10.

The newly defined convolution (41) of two space-time valued functions $f,g\in
L^{2}(\mathbb{R}^{3,1},C\ell_{3,1})$ can be expressed in terms of the eight
standard convolutions (23) as

$\displaystyle f\star_{N}g(x)$ $\displaystyle=[f_{+}\star
g_{+}(\bm{x})]_{+}+[f_{+}\star g_{+}^{(1,1)}(t_{1},-\underline{x})]_{-}$ (45)
$\displaystyle+[f_{+}\star g_{-}^{(1,0)}(\bm{x})]_{+}+[f_{+}\star
g_{-}^{(0,1)}(t_{1},-\underline{x})]_{-}$ $\displaystyle+[f_{-}\star
g_{+}^{(0,1)}(t_{1},-\underline{x})]_{+}+[f_{-}\star
g_{+}^{(1,0)}(\bm{x})]_{-}$ $\displaystyle+[f_{-}\star
g_{-}^{(1,1)}(t_{1},-\underline{x})]_{+}+[f_{-}\star g_{-}(\bm{x})]_{-}.$

###### Proof.

Using the definition of two-sided linear canonical space-time transform, we
have

$\displaystyle f\star_{N}g(x)$
$\displaystyle=\mathcal{L}_{A}^{-1}(\lambda_{e_{t}}(\bm{w})\mathcal{L}_{A}[f(\bm{x})](\bm{w})\mathcal{L}_{A}[g(\bm{x})](\bm{w}))$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\lambda_{e_{t}}(\bm{w})\mathcal{L}_{A}[f(\bm{y})](\bm{w})\mathcal{L}_{A}[g(\bm{z})](\bm{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{w}$
$\displaystyle=\lambda_{e_{t}}(\bm{w})\int_{\mathbb{R}^{3,1}}\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\int_{\mathbb{R}^{3,1}}\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})f(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})d^{4}\bm{y}$
$\displaystyle\times\int_{\mathbb{R}^{3,1}}\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g(\bm{z})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})d^{4}\bm{z}\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})f(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})$
$\displaystyle\times\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g(\bm{z})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}.$

Implementing the general $\pm$ splitting technique
$f(\bm{y})=f_{+}(\bm{y})+f_{-}(\bm{y})$ and
$g(\bm{z})=g_{+}(\bm{z})+g_{-}(\bm{z})$, we obtain

$\displaystyle f\star_{N}g(x)$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})$
(46)
$\displaystyle\times[f_{+}(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g_{+}(\bm{z})$
$\displaystyle+f_{+}(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g_{-}(\bm{z})$
$\displaystyle+f_{-}(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g_{+}(\bm{z})$
$\displaystyle+f_{-}(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})g_{-}(\bm{z})]$
$\displaystyle\times\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}.$

Applying the identities (43) to shift inner kernel factors towards the left
and right, we get

$\displaystyle f\star_{N}g(x)$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})$
(47)
$\displaystyle\times[\mathcal{K}_{-e_{t}}^{M_{2}}(\underline{y},\underline{w})(f_{+}(\bm{y})g_{+}(\bm{z}))\mathcal{K}_{-i_{3}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})$
$\displaystyle+\mathcal{K}_{-e_{t}}^{M_{2}}(\underline{y},\underline{w})(f_{+}(\bm{y})g_{-}(\bm{z}))\mathcal{K}_{i_{3}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})$
$\displaystyle+\mathcal{K}_{e_{t}}^{M_{2}}(\underline{y},\underline{w})(f_{-}(\bm{y})g_{+}(\bm{z}))\mathcal{K}_{-i_{3}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})$
$\displaystyle+\mathcal{K}_{e_{t}}^{M_{2}}(\underline{y},\underline{w})(f_{-}(\bm{y})g_{-}(\bm{z}))\mathcal{K}_{i_{3}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})]$
$\displaystyle\times\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}.$

Again utilizing the general $\pm$ split and the notation similar to
$f_{+}(\bm{y})g_{-}(\bm{z})=[f_{+}(\bm{y})g_{-}(\bm{z})]_{+}+[f_{+}(\bm{y})g_{-}(\bm{z})]_{-}$
for the second time split, and then applying the identities (43) for shifting
the right-sided kernels towards the left. Abbreviating
$\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}$ to
$\iiint$, we obtain

$\displaystyle f\star_{N}g(x)$
$\displaystyle=\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},e_{t}}^{M_{1}}\mathcal{K}_{-i_{3},-i_{3},i_{3}}^{M_{2}}[f_{+}\bm{y})g_{+}(\bm{z})]_{+}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},-e_{t}}^{M_{1}}\mathcal{K}_{-i_{3},i_{3},-i_{3}}^{M_{2}}[f_{+}\bm{y})g_{+}(\bm{z})]_{-}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},-e_{t}}^{M_{1}}\mathcal{K}_{-i_{3},-i_{3},i_{3}}^{M_{2}}[f_{+}\bm{y})g_{-}(\bm{z})]_{+}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},e_{t}}^{M_{1}}\mathcal{K}_{-i_{3},i_{3},-i_{3}}^{M_{2}}[f_{+}\bm{y})g_{-}(\bm{z})]_{-}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},e_{t}}^{M_{1}}\mathcal{K}_{i_{3},-i_{3},i_{3}}^{M_{2}}[f_{-}\bm{y})g_{+}(\bm{z})]_{+}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},-e_{t}}^{M_{1}}\mathcal{K}_{i_{3},i_{3},-i_{3}}^{M_{2}}[f_{-}\bm{y})g_{+}(\bm{z})]_{-}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},-e_{t}}^{M_{1}}\mathcal{K}_{i_{3},-i_{3},i_{3}}^{M_{2}}[f_{-}\bm{y})g_{-}(\bm{z})]_{+}$
$\displaystyle+\iiint\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t},e_{t},e_{t}}^{M_{1}}\mathcal{K}_{i_{3},i_{3},-i_{3}}^{M_{2}}[f_{-}\bm{y})g_{-}(\bm{z})]_{-}.$

We now only show explicitly how to simplify the second triple integral, the
others follow the same pattern.

$\displaystyle\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})$
$\displaystyle\times\mathcal{K}_{-e_{t}}^{M_{2}}(\underline{y},\underline{w})\mathcal{K}_{e_{t}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-e_{t}}^{M_{2}}(\underline{x},\underline{w})[f_{+}(\bm{y})g_{+}(\bm{z})]_{-}d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}(2\pi)^{2}B_{1}^{\frac{1}{2}}B_{2}^{\frac{7}{2}}\bm{e}^{e_{t}(\frac{D_{1}w_{t}^{2}}{2B_{1}}+\frac{D_{2}\underline{w}^{2}}{2B_{2}})}$
$\displaystyle\times(2\pi)^{-\frac{3}{2}}(B_{1})^{-\frac{3}{2}}\times\bm{e}^{\frac{e_{t}}{2B_{1}}(A_{1}(t_{1}^{\prime
2}-t_{1}^{2}-t_{1}^{\prime\prime
2})-2(t_{1}^{\prime}-t_{1}-t_{1}^{\prime\prime})w_{t}-D_{1}w_{t}^{2})}$
$\displaystyle\times(2\pi)^{-\frac{9}{2}}(B_{2})^{-\frac{9}{2}}\times\bm{e}^{\frac{e_{t}}{2B_{2}}(A_{2}(\underline{z}^{2}-\underline{x}^{2}-\underline{y}^{2})-2(\underline{z}-\underline{x}-\underline{y})\cdot\underline{w}-D_{2}\underline{w}^{2})}$
$\displaystyle\times[f_{+}(\bm{y})g_{+}(\bm{z})]_{-}d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}(2\pi)^{-1}(B_{1})^{-1}\bm{e}^{\frac{e_{t}}{2B_{1}}[A_{1}(t_{1}^{\prime
2}-t_{1}^{2}-t_{1}^{\prime\prime
2})]}\int_{\mathbb{R}}\bm{e}^{\frac{-e_{t}}{B_{1}}(t_{1}^{\prime}-t_{1}-t_{1}^{\prime\prime})w_{t}}dw_{t}$
$\displaystyle\times(2\pi)^{-3}(B_{2})^{-1}\bm{e}^{\frac{e_{t}}{2B_{2}}[A_{2}(\underline{z}^{2}-\underline{x}^{2}-\underline{y}^{2})]}\int_{\mathbb{R}^{3}}\bm{e}^{\frac{-e_{t}}{B_{2}}(\underline{z}-\underline{x}-\underline{y})\cdot\underline{w}}d\underline{w}$
$\displaystyle\times[f_{+}(\bm{y})g_{+}(\bm{z})]_{-}d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\bm{e}^{\frac{e_{t}}{2B_{1}}[A_{1}(t_{1}^{\prime
2}-t_{1}^{2}-t_{1}^{\prime\prime
2})]}\delta(t_{1}^{\prime}-t_{1}-t_{1}^{\prime\prime})$
$\displaystyle\times\bm{e}^{\frac{e_{t}}{2B_{2}}[A_{2}(\underline{z}^{2}-\underline{x}^{2}-\underline{y}^{2})]}\delta(\underline{z}-\underline{x}-\underline{y})d^{4}\bm{y}d^{4}\bm{z}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}[f_{+}(\bm{y})g_{+}(-(t_{1}-t_{1}^{\prime}),\underline{x}+\underline{y})]_{-}d^{4}\bm{y}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}[f_{+}(\bm{y})g_{+}^{(1,1)}(t_{1}-t_{1}^{\prime},\underline{x}+\underline{y})]_{-}d^{4}\bm{y}$
$\displaystyle=[f_{+}\star g_{+}^{(1,1)}(t_{1},-\underline{x})]_{-}.$

Simplifying the other seven triple integrals in similar manner, we obtain the
the desired decomposition for the proposed convolution. This completes the
proof of Theorem 10. ∎

Now we state and prove our new convolution theorem.

######  Theorem 11.

Let $p(\bm{x})=(f\star_{N}g)(\bm{x})$ and $\mathcal{L}_{A}[p](\bm{w})$,
$\mathcal{L}_{A}[f](\bm{w})$ and $\mathcal{L}_{A}[g](\bm{w})$ denote the two-
sided LCST of $p(\bm{x})$, $f(\bm{x})$ and $g(\bm{x})$. Then

$\mathcal{L}_{A}[p(\bm{x})](\bm{w})=\lambda_{e_{t}}(\bm{w})\mathcal{L}_{A}[f(\bm{x})](\bm{w})\mathcal{L}_{A}[g(\bm{x})](\bm{w}).$
(48)

###### Proof.

$\displaystyle(f\star_{N}g)(\bm{x})$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})f(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})$
(49)
$\displaystyle\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{y}d^{4}\bm{z}d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\lambda_{e_{t}}(\bm{w})\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\int_{\mathbb{R}^{3,1}}\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime})f(\bm{y})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{y},\underline{w})d^{4}\bm{y}$
$\displaystyle\times\int_{\mathbb{R}^{3,1}}\mathcal{K}_{e_{t}}^{M_{1}}(w_{t},t_{1}^{\prime\prime})\mathcal{K}_{i_{3}}^{M_{2}}(\underline{z},\underline{w})d^{4}\bm{z}\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{w}$
$\displaystyle=\int_{\mathbb{R}^{3,1}}\mathcal{K}_{-e_{t}}^{M_{1}}(w_{t},t_{1})\lambda_{e_{t}}(\bm{w})\mathcal{L}_{A}[f(\bm{y})](\bm{w})\mathcal{L}_{A}[g(\bm{z})](\bm{w})\mathcal{K}_{-i_{3}}^{M_{2}}(\underline{x},\underline{w})d^{4}\bm{w}$
$\displaystyle=\mathcal{L}_{A}^{-1}[\lambda_{e_{t}}(\bm{w})\mathcal{L}_{A}[f(\bm{x})](\bm{w})\mathcal{L}_{A}[g(\bm{x})](\bm{w})].$

∎

##  6 Conclusion

After an overview of space-time algebra and considering the fractional space-
time Fourier transform (FrSFT) and some of its properties, we propose the
concept of linear canonical space-time transform (LCST) as a broad
generalization of FrSFT. Finally, we propose a convolution operator and the
corresponding convolution theorem. We hope that applied mathematics will
continue to investigate this new class of hyper-complex transformations.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
[No. 62171041].

## Declarations

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data Availability Statement

Not applicable.

## Code Availability Statement

Not applicable.

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