# Uncertainty principles for the windowed Opdam–Cherednik transform

Shyam Swarup Mondal  and  Anirudha Poria  Department of Mathematics, Indian
Institute of Technology Guwahati, Guwahati 781039, India
<EMAIL_ADDRESS>Department of Mathematics, Bar-Ilan University,
Ramat-Gan 5290002, Israel<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study a few versions of the uncertainty principle for the
windowed Opdam–Cherednik transform. In particular, we establish the
uncertainty principle for orthonormal sequences, Donoho–Stark’s uncertainty
principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty
principle and local uncertainty inequality for this transform. We also obtain
the Heisenberg-type uncertainty inequality using the $k$-entropy of the
windowed Opdam–Cherednik transform.

###### Key words and phrases:

Opdam–Cherednik transform; windowed Opdam–Cherednik transform; Heisenberg-type
uncertainty inequality; Donoho–Stark’s uncertainty principle; Benedicks-type
uncertainty principle

###### 2010 Mathematics Subject Classification:

Primary 44A15; Secondary 42C20, 33C45.

Research supported by ERC Starting Grant No. 713927.

##  1\. Introduction

Uncertainty principles play a fundamental role in the field of mathematics,
physics, in addition to some engineering areas such as signal processing,
image processing, quantum theory, optics, and many other well known areas [37,
17, 12, 6]. In quantum physics, it says that a particle’s speed and position
cannot both be measured with infinite precision. The classical Heisenberg
uncertainty principle was established in the Schrödinger space (the square-
integrable function space). It states that a non-zero function and its Fourier
transform cannot be both sharply localized, i.e., it is impossible for a non-
zero function and its Fourier transform to be simultaneously small. This
phenomenon has been under intensive study for almost a century now and
extensively investigated in different settings. One can formulate different
forms of the uncertainty principle depending on various ways of measuring the
localization of a function. Uncertainty principles can be divided into two
categories: quantitative and qualitative uncertainty principles. Quantitative
uncertainty principles are some special types of inequalities that tells us
how a function and its Fourier transform relate. For example, Benedicks [5],
Donoho and Stark [12], and Slepian and Pollak [39] gave quantitative
uncertainty principles for the Fourier transforms. On the other hand,
qualitative uncertainty principles imply the vanishing of a function under
some strong conditions on the function. In particular, Hardy [20], Morgan
[32], Cowling and Price [9], and Beurling [24] theorems are the examples of
qualitative uncertainty principles. For a more detailed study on the history
of the uncertainty principle, and for many other generalizations and
variations of the uncertainty principle, we refer to the book of Havin and
Jöricke [21], and the excellent survey of Folland and Sitaram [14].

Considerable attention has been devoted to discovering generalizations to new
contexts for quantitative uncertainty principles. For example, quantitative
uncertainty principles were studied in [29] for the generalized Fourier
transform. Over the years, discovering new mathematical formulations of the
uncertainty principle for the windowed Fourier transform have drawn
significant attention among many researchers, see for example [7, 40, 21] and
the references therein. Uncertainty principles are further extended and
studied in different settings by many authors. For instance, one can look [30]
for the Dunkl Gabor transform, [8] for two sided quaternion windowed Fourier
transform, [4, 18] for the continuous Hankel transform, [15, 16] for the
windowed Hankel transform, [19] for the Hankel–Gabor transform, and [26] for
the Heckman–Opdam transform.

As a generalization of Euclidean uncertainty principles for the Fourier
transform, Daher et al. [10] have obtained some qualitative uncertainty
principles for the Cherednik transform. Also, qualitative uncertainty
principles for the Opdam–Cherednik transform have been studied by Mejjaoli in
[27] using classical uncertainty principles for the Fourier transform and
composition properties of the Opdam–Cherednik transform. These results have
been further extented to modulation spaces by the second author in [35].
Moreover, the Benedicks-type uncertainty principle for the Opdam–Cherednik
transform has been investigated by Achak and Daher in [1]. Recently, the
second author introduced the windowed Opdam–Cherednik transform and discussed
the time-frequency analysis of localization operators associated with the
windowed Opdam–Cherednik transform in [36]. However, upto our knowledge,
quantitative uncertainty principles have not been studied for the windowed
Opdam–Cherednik transform. In this paper, we extend some quantitative
uncertainty principles for this transform.

The motivation to prove these types of quantitative uncertainty principles for
the windowed Opdam–Cherednik transform in the framework of the Opdam–Cherednik
transform arises from the classical uncertainty principles for the windowed
Fourier transform and the remarkable contribution of the Opdam–Cherednik
transform in harmonic analysis (see [2, 33, 34, 38]). Another important
motivation to study the Jacobi–Cherednik operators arises from their relevance
in the algebraic description of exactly solvable quantum many-body systems of
Calogero–Moser–Sutherland type (see [11, 23]) and they provide a useful tool
in the study of special functions with root systems (see[13, 22]). These
describe algebraically integrable systems in one dimension and have gained
considerable interest in mathematical physics. Other motivation for the
investigation of the Jacobi–Cherednik operator and the windowed
Opdam–Cherednik transform is to generalize the previous subjects which are
bound with the physics. For a more detailed discussion, one can see [31].

The main aim of this paper is to study a few uncertainty principles related to
the windowed Opdam–Cherednik transform. More preciously, we prove the
uncertainty principle for orthonormal sequences, Donoho–Stark’s uncertainty
principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty
principle and local uncertainty inequality for the windowed Opdam–Cherednik
transform. We study the version of Donoho–Stark’s uncertainty principle and
show that the windowed Opdam–Cherednik transform cannot be concentrated in any
small set. Also, we obtain an estimate for the size of the essential support
of this transform under the condition that the windowed Opdam–Cherednik
transform of a non-zero function is time-frequency concentrated on a
measurable set. Then, we investigate the Benedicks-type uncertainty principle
and show that the windowed Opdam–Cherednik transform cannot be concentrated
inside a set of measures arbitrarily small. Further, we study the Heisenberg-
type uncertainty inequality for a general magnitude and provide the result
related to the $L^{2}(\mathbb{R},A_{\alpha,\beta})$-mass of the windowed
Opdam–Cherednik transform outside sets of finite measure. Finally, we obtain
the Heisenberg-type uncertainty inequality using the $k$-entropy and study the
localization of the $k$-entropy of the windowed Opdam–Cherednik transform.

The presentation of this manuscript is divided into four sections apart from
the introduction. In Section 2, we present some preliminaries related to the
Opdam–Cherednik transform. In Section 3, we recall some essential properties
related to the windowed Opdam–Cherednik transform. Some different types of
uncertainty principles associated with the windowed Opdam–Cherednik transform
are provided in Section 4. In particular, we prove the uncertainty principle
for orthonormal sequences, Donoho–Stark’s uncertainty principle, Benedicks-
type uncertainty principle, Heisenberg-type uncertainty principle and local
uncertainty inequality for the windowed Opdam–Cherednik transform. We conclude
the paper with the Heisenberg-type uncertainty inequality using the
$k$-entropy of this transform.

##  2\. Harmonic analysis and the Opdam–Cherednik transform

In this section, we recall some necessary definitions and results from the
harmonic analysis related to the Opdam–Cherednik transform, which will be used
frequently. A complete account of harmonic analysis related to this transform
can be found in [2, 28, 33, 34, 38, 25, 35, 36]. However, we will use the
notations given in [35].

Let $T_{\alpha,\beta}$ denote the Jacobi–Cherednik differential–difference
operator (also called the Dunkl–Cherednik operator)

$T_{\alpha,\beta}f(x)=\frac{d}{dx}f(x)+\Big{[}(2\alpha+1)\coth
x+(2\beta+1)\tanh x\Big{]}\frac{f(x)-f(-x)}{2}-\rho f(-x),$

where $\alpha,\beta$ are two parameters satisfying
$\alpha\geq\beta\geq-\frac{1}{2}$ and $\alpha>-\frac{1}{2}$, and
$\rho=\alpha+\beta+1$. Let $\lambda\in{\mathbb{C}}$. The Opdam hypergeometric
functions $G^{\alpha,\beta}_{\lambda}$ on $\mathbb{R}$ are eigenfunctions
$T_{\alpha,\beta}G^{\alpha,\beta}_{\lambda}(x)=i\lambda
G^{\alpha,\beta}_{\lambda}(x)$ of $T_{\alpha,\beta}$ that are normalized such
that $G^{\alpha,\beta}_{\lambda}(0)=1$. The eigenfunction
$G^{\alpha,\beta}_{\lambda}$ is given by

$G^{\alpha,\beta}_{\lambda}(x)=\varphi^{\alpha,\beta}_{\lambda}(x)-\frac{1}{\rho-i\lambda}\frac{d}{dx}\varphi^{\alpha,\beta}_{\lambda}(x)=\varphi^{\alpha,\beta}_{\lambda}(x)+\frac{\rho+i\lambda}{4(\alpha+1)}\sinh
2x\;\varphi^{\alpha+1,\beta+1}_{\lambda}(x),$

where
$\varphi^{\alpha,\beta}_{\lambda}(x)={}_{2}F_{1}\left(\frac{\rho+i\lambda}{2},\frac{\rho-i\lambda}{2};\alpha+1;-\sinh^{2}x\right)$
is the classical Jacobi function.

For every $\lambda\in{\mathbb{C}}$ and $x\in\mathbb{R}$, the eigenfunction
$G^{\alpha,\beta}_{\lambda}$ satisfy

$|G^{\alpha,\beta}_{\lambda}(x)|\leq
C\;e^{-\rho|x|}e^{|\text{Im}(\lambda)||x|},$

where $C$ is a positive constant. Since $\rho>0$, we have

$|G^{\alpha,\beta}_{\lambda}(x)|\leq C\;e^{|\text{Im}(\lambda)||x|}.$ (2.1)

Let us denote by $C_{c}(\mathbb{R})$ the space of continuous functions on
$\mathbb{R}$ with compact support. The Opdam–Cherednik transform is the
Fourier transform in the trigonometric Dunkl setting, and it is defined as
follows.

######  Definition 2.1.

Let $\alpha\geq\beta\geq-\frac{1}{2}$ with $\alpha>-\frac{1}{2}$. The
Opdam–Cherednik transform $\mathcal{H}_{\alpha,\beta}(f)$ of a function $f\in
C_{c}(\mathbb{R})$ is defined by

${\mathcal{H}}_{\alpha,\beta}(f)(\lambda)=\int_{\mathbb{R}}f(x)\;G^{\alpha,\beta}_{\lambda}(-x)\;A_{\alpha,\beta}(x)dx\quad\text{for
all }\lambda\in{\mathbb{C}},$

where $A_{\alpha,\beta}(x)=(\sinh|x|)^{2\alpha+1}(\cosh|x|)^{2\beta+1}$. The
inverse Opdam–Cherednik transform for a suitable function $g$ on $\mathbb{R}$
is given by

${\mathcal{H}}_{\alpha,\beta}^{-1}(g)(x)=\int_{\mathbb{R}}g(\lambda)\;G^{\alpha,\beta}_{\lambda}(x)\;d\sigma_{\alpha,\beta}(\lambda)\quad\text{for
all }x\in\mathbb{R},$

where

$d\sigma_{\alpha,\beta}(\lambda)=\left(1-\dfrac{\rho}{i\lambda}\right)\dfrac{d\lambda}{8\pi|C_{\alpha,\beta}(\lambda)|^{2}}$

and

$C_{\alpha,\beta}(\lambda)=\dfrac{2^{\rho-i\lambda}\Gamma(\alpha+1)\Gamma(i\lambda)}{\Gamma\left(\frac{\rho+i\lambda}{2}\right)\;\Gamma\left(\frac{\alpha-\beta+1+i\lambda}{2}\right)},\quad\lambda\in{\mathbb{C}}\setminus
i\mathbb{N}.$

The Plancherel formula is given by

$\int_{\mathbb{R}}|f(x)|^{2}A_{\alpha,\beta}(x)dx=\int_{\mathbb{R}}{\mathcal{H}}_{\alpha,\beta}(f)(\lambda)\overline{{\mathcal{H}}_{\alpha,\beta}(\check{f})(-\lambda)}\;d\sigma_{\alpha,\beta}(\lambda),$
(2.2)

where $\check{f}(x):=f(-x)$.

Let $L^{p}(\mathbb{R},A_{\alpha,\beta})$ (resp.
$L^{p}(\mathbb{R},\sigma_{\alpha,\beta})$), $p\in[1,\infty]$, denote the
$L^{p}$-spaces corresponding to the measure $A_{\alpha,\beta}(x)dx$ (resp.
$d|\sigma_{\alpha,\beta}|(x)$).

The generalized translation operator associated with the Opdam–Cherednik
transform is defined by [3]

$\tau_{x}^{(\alpha,\beta)}f(y)=\int_{\mathbb{R}}f(z)\;{d\mu}_{\;x,y}^{(\alpha,\beta)}(z),$
(2.3)

where ${d\mu}_{\;x,y}^{(\alpha,\beta)}$ is given by

${d\mu}_{\;x,y}^{(\alpha,\beta)}(z)=\begin{cases}\mathcal{K}_{\alpha,\beta}(x,y,z)\;A_{\alpha,\beta}(z)\;dz&\text{if}\;\;xy\neq
0\\\ d\delta_{x}(z)&\text{if}\;\;y=0\\\
d\delta_{y}(z)&\text{if}\;\;x=0\end{cases}$ (2.4)

and

$\displaystyle\mathcal{K}_{\alpha,\beta}{}$
$\displaystyle(x,y,z)=M_{\alpha,\beta}|\sinh x\cdot\sinh y\cdot\sinh
z|^{-2\alpha}\int_{0}^{\pi}g(x,y,z,\chi)_{+}^{\alpha-\beta-1}$
$\displaystyle\times\left[1-\sigma^{\chi}_{x,y,z}+\sigma^{\chi}_{x,z,y}+\sigma^{\chi}_{z,y,x}+\frac{\rho}{\beta+\frac{1}{2}}\coth
x\cdot\coth y\cdot\coth
z(\sin\chi)^{2}\right]\times(\sin\chi)^{2\beta}\;d\chi$

if $x,y,z\in\mathbb{R}\setminus\\{0\\}$ satisfy the triangular inequality
$||x|-|y||<|z|<|x|+|y|$, and $\mathcal{K}_{\alpha,\beta}(x,y,z)=0$ otherwise.
Here

$\sigma^{\chi}_{x,y,z}=\begin{cases}\frac{\cosh x\cdot\cosh y-\cosh
z\cdot\cos\chi}{\sinh x\cdot\sinh y}&\text{if}\;\;xy\neq 0\\\
0&\text{if}\;\;xy=0\end{cases}\quad\text{for}\;x,y,z\in\mathbb{R},\;\chi\in[0,\pi],$

$g(x,y,z,\chi)=1-\cosh^{2}x-\cosh^{2}y-\cosh^{2}z+2\cosh x\cdot\cosh
y\cdot\cosh z\cdot\cos\chi$, and

$g_{+}=\begin{cases}g&\text{if}\;\;g>0\\\ 0&\text{if}\;\;g\leq 0.\end{cases}$

The kernel $\mathcal{K}_{\alpha,\beta}(x,y,z)$ satisfies the following
symmetry properties:

$\mathcal{K}_{\alpha,\beta}(x,y,z)=\mathcal{K}_{\alpha,\beta}(y,x,z)$,
$\mathcal{K}_{\alpha,\beta}(x,y,z)=\mathcal{K}_{\alpha,\beta}(-z,y,-x)$,
$\mathcal{K}_{\alpha,\beta}(x,y,z)=\mathcal{K}_{\alpha,\beta}(x,-z,-y)$.

For every $x,y\in\mathbb{R}$, we have

$\tau_{x}^{(\alpha,\beta)}f(y)=\tau_{y}^{(\alpha,\beta)}f(x),$ (2.5)

and

${\mathcal{H}}_{\alpha,\beta}(\tau_{x}^{(\alpha,\beta)}f)(\lambda)=G_{\lambda}^{\alpha,\beta}(x)\;{\mathcal{H}}_{\alpha,\beta}(f)(\lambda),$
(2.6)

for $f\in C_{c}(\mathbb{R})$.

If $f\in L^{1}(\mathbb{R},A_{\alpha,\beta})$, then

$\int_{\mathbb{R}}\tau_{x}^{(\alpha,\beta)}f(y)\;A_{\alpha,\beta}(y)\;dy=\int_{\mathbb{R}}f(y)\;A_{\alpha,\beta}(y)\;dy.$
(2.7)

For every $f\in L^{p}(\mathbb{R},A_{\alpha,\beta})$ and every
$x\in\mathbb{R}$, the function $\tau_{x}^{(\alpha,\beta)}f$ belongs to the
space $L^{p}(\mathbb{R},A_{\alpha,\beta})$ and

$\left\|\tau_{x}^{(\alpha,\beta)}f\right\|_{L^{p}(\mathbb{R},A_{\alpha,\beta})}\leq
C_{\alpha,\beta}\left\|f\right\|_{L^{p}(\mathbb{R},A_{\alpha,\beta})},$ (2.8)

where $C_{\alpha,\beta}$ is a positive constant.

The convolution product associated with the Opdam–Cherednik transform is
defined for two suitable functions $f$ and $g$ by [3]

$(f*_{\alpha,\beta}g)(x)=\int_{\mathbb{R}}\tau_{x}^{(\alpha,\beta)}f(-y)\;g(y)\;A_{\alpha,\beta}(y)\;dy$

and

${\mathcal{H}}_{\alpha,\beta}(f*_{\alpha,\beta}g)={\mathcal{H}}_{\alpha,\beta}(f)\;{\mathcal{H}}_{\alpha,\beta}(g).$
(2.9)

##  3\. The windowed Opdam–Cherednik transform

In this section, we collect the necessary definitions and results from the
harmonic analysis related to the windowed Opdam–Cherednik transform. For a
detailed discussion on this transform, we refer to [36].

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ and $\xi\in\mathbb{R}$, the
modulation operator of $g$ associated with the Opdam–Cherednik transform is
defined by

${\mathcal{M}}^{(\alpha,\beta)}_{\xi}g={\mathcal{H}}^{-1}_{\alpha,\beta}\left(\sqrt{\tau_{\xi}^{(\alpha,\beta)}|{\mathcal{H}}_{\alpha,\beta}(g)|^{2}}\right).$
(3.1)

Then, for every $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ and
$\xi\in\mathbb{R}$, by using the Plancherel formula (2.2) and the translation
invariance of the Plancherel measure $d\sigma_{\alpha,\beta}$, we obtain

$\left\|{\mathcal{M}}^{(\alpha,\beta)}_{\xi}g\right\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=\left\|g\right\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$
(3.2)

Now, for a non-zero window function $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$
and $(x,\xi)\in\mathbb{R}^{2}$, we consider the function
$g_{x,\xi}^{(\alpha,\beta)}$ defined by

$g_{x,\xi}^{(\alpha,\beta)}=\tau_{x}^{(\alpha,\beta)}{\mathcal{M}}^{(\alpha,\beta)}_{\xi}g.$
(3.3)

For any function $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, we define the
windowed Opdam–Cherednik transform by

${\mathcal{W}}^{(\alpha,\beta)}_{g}(f)(x,\xi)=\int_{\mathbb{R}}f(s)\;\overline{g_{x,\xi}^{(\alpha,\beta)}(-s)}\;A_{\alpha,\beta}(s)\;ds,\quad(x,\xi)\in\mathbb{R}^{2},$
(3.4)

which can be also written in the form

${\mathcal{W}}^{(\alpha,\beta)}_{g}(f)(x,\xi)=\left(f*_{\alpha,\beta}\overline{{\mathcal{M}}^{(\alpha,\beta)}_{\xi}g}\right)(x).$
(3.5)

We define the measure $A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}$ on
$\mathbb{R}^{2}$ by

$d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)=A_{\alpha,\beta}(x)dx\;d\sigma_{\alpha,\beta}(\xi).$
(3.6)

The windowed Opdam–Cherednik transform satisfies the following properties.

######  Proposition 3.1.

[36] Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window
function. Then we have

$(1)$ $($Plancherel’s formula$)$ For every $f\in
L^{2}(\mathbb{R},A_{\alpha,\beta})$,

$\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}=\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$
(3.7)

$(2)$ $($Orthogonality relation$)$ For every $f,h\in
L^{2}(\mathbb{R},A_{\alpha,\beta})$, we have

$\iint_{\mathbb{R}^{2}}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)(x,\xi)\;\overline{{\mathcal{W}}^{(\alpha,\beta)}_{g}(h)(x,\xi)}\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)=\|g\|^{2}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\int_{\mathbb{R}}f(s)\overline{h(s)}\;A_{\alpha,\beta}(s)\;ds.$
(3.8)

$(3)$ $($Reproducing kernel Hilbert space$)$ The space
${\mathcal{W}}^{(\alpha,\beta)}_{g}(L^{2}(\mathbb{R},A_{\alpha,\beta}))$ is a
reproducing kernel Hilbert space in
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$ with
kernel function $K_{g}$ defined by

$\displaystyle K_{g}((x^{\prime},\xi^{\prime});(x,\xi))$ $\displaystyle=$
$\displaystyle\frac{1}{\|g\|^{2}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}\;\left(g_{x,\xi}^{(\alpha,\beta)}(-\;\cdot)*_{\alpha,\beta}\overline{{\mathcal{M}}^{(\alpha,\beta)}_{\xi^{\prime}}g}\right)(x^{\prime})$
(3.9) $\displaystyle=$
$\displaystyle\frac{1}{\|g\|^{2}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}\;{\mathcal{W}}^{(\alpha,\beta)}_{g}\left(g_{x,\xi}^{(\alpha,\beta)}(-\;\cdot)\right)(x^{\prime},\xi^{\prime}).$

Furthermore, the kernel is pointwise bounded

$\left|K_{g}((x^{\prime},\xi^{\prime});(x,\xi))\right|\leq 1,\quad\text{for
all}\;(x,\xi);(x^{\prime},\xi^{\prime})\in\mathbb{R}^{2}.$ (3.10)

The windowed Opdam–Cherednik transform also satisfies the following
boundedness properties.

######  Proposition 3.2.

[36] Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window
function. Then for every $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, we have

$(1)$

$\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{\infty}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\leq\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$
(3.11)

$(2)$ The function ${\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\in
L^{p}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$,
$p\in[2,\infty)$ and

$\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{p}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\leq\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$
(3.12)

##  4\. Uncertainty principles for the windowed Opdam–Cherednik transform

In this section, we obtain various uncertainty principles for the windowed
Opdam–Cherednik transform.

###  4.1. Uncertainty principle for orthonormal sequences

In this subsection, we establish the uncertainty principle for orthonormal
sequences associated with the windowed Opdam–Cherednik transform. First, we
consider the following orthogonal projections:

  1. (1)

Let $P_{g}$ be the orthogonal projection from
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$ onto
${\mathcal{W}}_{g}^{(\alpha,\beta)}\left(L^{2}\left(\mathbb{R},A_{\alpha,\beta}\right)\right)$
and $\operatorname{Im}P_{g}$ denotes the range of $P_{g}$.

  2. (2)

Let $P_{\Sigma}$ be the orthogonal projection on
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$ defined
by

$\displaystyle P_{\Sigma}F=\chi_{\Sigma}F,\quad F\in
L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}),$ (4.1)

where $\Sigma\subset\mathbb{R}^{2}$ and $\operatorname{Im}P_{\Sigma}$ is the
range of $P_{\Sigma}$.

Also, we define

$\left\|P_{\Sigma}P_{g}\right\|=\sup\left\\{\left\|P_{\Sigma}P_{g}(F)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}:F\in
L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}),\|F\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}=1\right\\}.$

We first need the following result.

######  Theorem 4.1.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function.
Then for any $\Sigma\subset\mathbb{R}^{2}$ of finite measure
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$, the operator
$P_{\Sigma}P_{g}$ is a Hilbert–Schmidt operator. Moreover, we have the
following estimation

$\left\|P_{\Sigma}P_{g}\right\|^{2}\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma).$

###### Proof.

Since $P_{g}$ is a projection onto a reproducing karnel Hilbert space, for any
function $F\in
L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$, the
orthogonal projection $P_{g}$ can be expressed as

$P_{g}(F)(x,\xi)=\iint_{\mathbb{R}^{2}}F(x^{\prime},\xi^{\prime})K_{g}((x^{\prime},\xi^{\prime});(x,\xi))\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x^{\prime},\xi^{\prime}),$

where $K_{g}((x^{\prime},\xi^{\prime});(x,\xi))$ is given by (3.9). Using the
relation (4.1), we obtain

$P_{\Sigma}P_{g}(F)(x,\xi)=\iint_{\mathbb{R}^{2}}\chi_{\Sigma}(x,\xi)F(x^{\prime},\xi^{\prime})K_{g}((x^{\prime},\xi^{\prime});(x,\xi))\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x^{\prime},\xi^{\prime}).$

This shows that the operator $P_{\Sigma}P_{g}$ is an integral operator with
kernel
$K((x^{\prime},\xi^{\prime});(x,\xi))=\chi_{\Sigma}(x,\xi)K_{g}((x^{\prime},\xi^{\prime});(x,\xi))$.
Using the relation (3.9), Plancherel’s formula (3.7), and Fubini’s theorem, we
have

$\displaystyle\left\|P_{\Sigma}P_{g}\right\|_{HS}^{2}$
$\displaystyle=\iint_{\mathbb{R}^{2}}\iint_{\mathbb{R}^{2}}\left|K((x^{\prime},\xi^{\prime});(x,\xi))\right|^{2}\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x^{\prime},\xi^{\prime})\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle=\iint_{\mathbb{R}^{2}}\iint_{\mathbb{R}^{2}}\left|\chi_{\Sigma}(x,\xi)\right|^{2}\left|K_{g}((x^{\prime},\xi^{\prime});(x,\xi))\right|^{2}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x^{\prime},\xi^{\prime})d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle=\frac{1}{\|g\|^{4}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}\iint_{\Sigma}\left(\iint_{\mathbb{R}^{2}}\left|{\mathcal{W}}^{(\alpha,\beta)}_{g}\left(g_{x,\xi}^{(\alpha,\beta)}(-\;\cdot)\right)(x^{\prime},\xi^{\prime})\right|^{2}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x^{\prime},\xi^{\prime})\right)$
$\displaystyle\qquad\qquad\quad\qquad\times
d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle\leq\frac{\|g\|^{4}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}{\|g\|^{4}_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)=A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma).$

Thus, the operator $P_{\Sigma}P_{g}$ is a Hilbert–Schmidt operator. Now, the
proof follows from the fact that
$\left\|P_{\Sigma}P_{g}\right\|\leq\left\|P_{\Sigma}P_{g}\right\|_{HS}$. ∎

In the following, we obtain the uncertainty principle for orthonormal
sequences associated with the windowed Opdam–Cherednik transform.

######  Theorem 4.2.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
and $\left\\{\phi_{n}\right\\}_{n\in\mathbb{N}}$ be an orthonormal sequence in
$L^{2}(\mathbb{R},A_{\alpha,\beta})$. Then for any
$\Sigma\subset\mathbb{R}^{2}$ of finite measure
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty,$ we have

$\sum_{n=1}^{N}\left(1-\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\right)\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma),$

for every $N\in\mathbb{N}.$

###### Proof.

Let $\left\\{e_{n}\right\\}_{n\in\mathbb{N}}$ be an orthonormal basis for
$L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)$.
Since $P_{\Sigma}P_{g}$ is a Hilbert–Schmidt operator, from Theorem 4.1, we
get

$tr\left(P_{g}P_{\Sigma}P_{g}\right)=\sum_{n\in\mathbb{N}}\left\langle
P_{g}P_{\Sigma}P_{g}e_{n},e_{n}\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}=\left\|P_{\Sigma}P_{g}\right\|_{HS}^{2}\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma),$

where $tr\left(P_{g}P_{\Sigma}P_{g}\right)$ denotes the trace of the operator
$P_{g}P_{\Sigma}P_{g}$ and
$\langle\cdot,\cdot\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
denotes the inner product of
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$. Since
$\left\\{\phi_{n}\right\\}_{n\in\mathbb{N}}$ be an orthonormal sequence in
$L^{2}\left(\mathbb{R},A_{\alpha,\beta}\right)$, from the orthogonality
relation (3.8), we obtain that
$\\{{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\\}_{n\in\mathbb{N}}$
is also an orthonormal sequence in
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$.
Therefore

$\displaystyle\sum_{n=1}^{N}\left\langle
P_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
$\displaystyle=\sum_{n=1}^{N}\left\langle
P_{g}P_{\Sigma}P_{g}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
$\displaystyle\leq tr\left(P_{g}P_{\Sigma}P_{g}\right).$

Thus, we have

$\sum_{n=1}^{N}\left\langle
P_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma).$

Moreover, for any $n$ with $1\leq n\leq N$, using the Cauchy–Schwarz
inequality, we get

$\displaystyle\left\langle
P_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
$\displaystyle=1-\left\langle
P_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
$\displaystyle\geq
1-\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}.$

Therefore

$\displaystyle\sum_{n=1}^{N}\left(1-\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\right)$
$\displaystyle\leq\sum_{n=1}^{N}\left\langle
P_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right),{\mathcal{W}}_{g}^{(\alpha,\beta)}\left(\phi_{n}\right)\right\rangle_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}$
$\displaystyle\leq A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma).$

This completes the proof of the theorem. ∎

###  4.2. Donoho–Stark’s uncertainty principle for the windowed
Opdam–Cherednik transform

Here, we study the version of Donoho–Stark’s uncertainty principle for the
windowed Opdam–Cherednik transform. In particular, we investigate the case
where $f$ and ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ are close to zero
outside measurable sets. We start with the following result.

######  Theorem 4.3.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
and $f\in{L^{2}\left(\mathbb{R},A_{\alpha,\beta}\right)}$ such that $f\neq 0$.
Then for any $\Sigma\subset\mathbb{R}^{2}$ and $\varepsilon\geq 0$ such that

$\iint_{\Sigma}\left|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)\right|^{2}\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)\geq(1-\varepsilon)\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2},$

we have

$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq 1-\varepsilon.$

###### Proof.

Using the relation (3.11), we get

$\displaystyle(1-\varepsilon)\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\iint_{\Sigma}\left|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)\right|^{2}\;d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle\leq\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|^{2}_{L^{\infty}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)$
$\displaystyle\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

Therefore, $A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq
1-\varepsilon.$ ∎

The following proposition shows that the windowed Opdam–Cherednik transform
cannot be concentrated in any small set.

######  Proposition 4.4.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function.
Then for any function $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ and for any
$\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<1$, we have

$\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}\geq\sqrt{1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)}\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$

###### Proof.

For any function $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, using the relation
(3.11), we get

$\displaystyle\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}=\left\|\chi_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)+\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}$
$\displaystyle=\left\|\chi_{\Sigma}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}+\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}$
$\displaystyle\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}+\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}$
$\displaystyle\leq
A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}+\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}.$

Thus, using Plancherel’s formula (3.7), we obtain

$\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}\geq\sqrt{1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)}\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$

∎

######  Definition 4.5.

Let $E$ be a measurable subset of $\mathbb{R}$ and $0\leq\varepsilon_{E}<1$.
Then we say that a function $f\in L^{p}(\mathbb{R},A_{\alpha,\beta}),1\leq
p\leq 2,$ is $\varepsilon_{E}$-concentrated on $E$ in
$L^{p}(\mathbb{R},A_{\alpha,\beta})$-norm, if

$\left\|\chi_{E^{c}}f\right\|_{L^{p}(\mathbb{R},A_{\alpha,\beta})}\leq\varepsilon_{E}\|f\|_{L^{p}(\mathbb{R},A_{\alpha,\beta})}.$

If $\varepsilon_{E}=0$, then $E$ contains the support of $f$ .

######  Definition 4.6.

Let $\Sigma$ be a measurable subset of $\mathbb{R}^{2}$ and
$0\leq\varepsilon_{\Sigma}<1$. Let $f,g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$
be two non-zero functions. We say that ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$
is $\varepsilon_{\Sigma}$-time-frequency concentrated on $\Sigma$, if

$\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}\leq\varepsilon_{\Sigma}\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}.$

If ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is $\varepsilon_{\Sigma}$-time-
frequency concentrated on $\Sigma$, then in the following, we obtain an
estimate for the size of the essential support of the windowed Opdam–Cherednik
transform.

######  Theorem 4.7.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
and $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ such that $f\neq 0$. Let
$\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$ and
$\varepsilon_{\Sigma}\geq 0$. If ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is
$\varepsilon_{\Sigma}$-time-frequency concentrated on $\Sigma,$ then

$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq(1-\varepsilon_{\Sigma}^{2}).$

###### Proof.

Since ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is $\varepsilon_{\Sigma}$-time-
frequency concentrated on $\Sigma$, using Plancherel’s formula (3.7), we
deduce that

$\displaystyle\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle=\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle=\left\|\chi_{\Sigma^{c}}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq\varepsilon_{\Sigma}^{2}\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}+\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}.$

Hence, using the relation (3.11), we obtain

$\displaystyle(1-\varepsilon_{\Sigma}^{2})\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
(4.2)
$\displaystyle\leq\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}\;A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)$
$\displaystyle\leq\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma),$

which completes the proof. ∎

######  Theorem 4.8.

Let $\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$,
$\varepsilon_{\Sigma}\geq 0$, $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a
non-zero window function, and $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ such
that $f\neq 0$. If ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is
$\varepsilon_{\Sigma}$-time-frequency concentrated on $\Sigma$, then for every
$p>2$, we have

$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq(1-\varepsilon_{\Sigma}^{2})^{\frac{p}{p-2}}.$

###### Proof.

Since ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is $\varepsilon_{\Sigma}$-time-
frequency concentrated on $\Sigma$, from (4.2), we have

$\displaystyle(1-\varepsilon_{\Sigma}^{2})\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}.$

Again, aplying Hölder’s inequality for the conjugate exponents $\frac{p}{2}$
and $\frac{p}{p-2}$, we get

$\displaystyle\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{p}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma))^{\frac{p-2}{p}}.$

Now, using the realtion (3.12), we obtain

$\displaystyle\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma))^{\frac{p-2}{p}}.$

Hence,

$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq(1-\varepsilon_{\Sigma}^{2})^{\frac{p}{p-2}}.$

∎

######  Theorem 4.9.

Let $\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$, $g\in
L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function, and $f\in
L^{1}(\mathbb{R},A_{\alpha,\beta})\cap L^{2}(\mathbb{R},A_{\alpha,\beta})$
such that
$\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}=1$.
Let $E\subset\mathbb{R}$ such that $A_{\alpha,\beta}(E)<\infty$. If $f$ is
$\varepsilon_{E}$-concentrated on $E$ in
$L^{1}(\mathbb{R},A_{\alpha,\beta})$-norm and
${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is $\varepsilon_{\Sigma}$-time-
frequency concentrated on $\Sigma$, then

$A_{\alpha,\beta}(E)\geq(1-\varepsilon_{E})^{2}\;\|f\|_{L^{1}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2},$

and

$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\geq(1-\varepsilon_{\Sigma}^{2}).$

In particular,

$A_{\alpha,\beta}(E)~{}A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\geq(1-\varepsilon_{E})^{2}\;(1-\varepsilon_{\Sigma}^{2})\;\|f\|_{L^{1}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

###### Proof.

Since ${\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$ is $\varepsilon_{\Sigma}$-time-
frequency concentrated on $\Sigma$, from (4.2), we have

$\displaystyle(1-\varepsilon_{\Sigma}^{2})\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}.$

Since
$\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}=1,$
using the relation (3.11) and Plancherel’s formula (3.7), we obtain

$\displaystyle(1-\varepsilon_{\Sigma}^{2})$
$\displaystyle\leq\left\|\chi_{\Sigma}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{\infty}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}~{}A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)$
$\displaystyle\leq\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}~{}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma).$
(4.3)

Similarly, since $f$ is $\varepsilon_{E}$-concentrated on $E$ in
$L^{1}(\mathbb{R},A_{\alpha,\beta})$-norm, using the Cauchy–Schwarz inequality
and the fact that
$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=1$,
we get

$\displaystyle(1-\varepsilon_{E})\|f\|_{L^{1}(\mathbb{R},A_{\alpha,\beta})}\leq\left\|\chi_{E}f\right\|_{L^{1}(\mathbb{R},A_{\alpha,\beta})}\leq\left\|f\right\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}~{}A_{\alpha,\beta}(E)^{\frac{1}{2}}=\frac{A_{\alpha,\beta}(E)^{\frac{1}{2}}}{\left\|g\right\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}.$
(4.4)

Finally, from (4.2) and (4.4), we have

$A_{\alpha,\beta}(E)~{}A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\geq(1-\varepsilon_{E})^{2}\;(1-\varepsilon_{\Sigma}^{2})\;\|f\|_{L^{1}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

This completes the proof of the theorem. ∎

###  4.3. Benedicks-type uncertainty principle for the windowed
Opdam–Cherednik transform

In this subsection, we study Benedicks-type uncertainty principle for the
windowed Opdam–Cherednik transform. The following proposition shows that this
transform cannot be concentrated inside a set of measure arbitrarily small.

######  Proposition 4.10.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
and $\Sigma\subset\mathbb{R}^{2}$. If $\left\|P_{\Sigma}P_{g}\right\|<1$, then
there exists a constant $c(\Sigma,g)>0$ such that for every $f\in
L^{2}(\mathbb{R},A_{\alpha,\beta}),$ we have

$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\leq
c(\Sigma,g)\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}.$

###### Proof.

Since $P_{\Sigma}$ is an orthogonal projection on
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$, for any
$F\in L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$, we
get

$\|P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}=\|P_{\Sigma}P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}+\|P_{\Sigma^{c}}P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}.$

Again, using the identity $P_{\Sigma}P_{g}(F)=P_{\Sigma}P_{g}\cdot P_{g}(F)$,
we get

$\|P_{\Sigma}P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}\leq\|P_{\Sigma}P_{g}\|^{2}\|P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}.$

Hence,

$\displaystyle\|P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}\leq\frac{1}{1-\|P_{\Sigma}P_{g}\|^{2}}\|P_{\Sigma^{c}}P_{g}(F)\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}^{2}.$
(4.5)

Since $P_{g}$ is an orthogonal projection from
$L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})$ onto
${\mathcal{W}}_{g}^{(\alpha,\beta)}\left(L^{2}\left(\mathbb{R},A_{\alpha,\beta}\right)\right)$,
for any $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, using the relation (4.5)
and Plancherel’s formula (3.7), we get

$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\leq\frac{1}{\sqrt{1-\|P_{\Sigma}P_{g}\|^{2}}}\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}.$

The desired result follows by choosing the canstant
$c(\Sigma,g)=\frac{1}{\sqrt{1-\|P_{\Sigma}P_{g}\|^{2}}}$. ∎

Next, we obtain Benedicks-type uncertainty principle for the windowed
Opdam–Cherednik transform.

######  Theorem 4.11.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
such that

$\sigma_{\alpha,\beta}\left(\left\\{{\mathcal{H}}_{\alpha,\beta}(g)\neq
0\right\\}\right)<\infty.$

Then for any $\Sigma\subset\mathbb{R}^{2}$ such that for almost every
$\xi\in\mathbb{R},\int_{\mathbb{R}}\chi_{\Sigma}(x,\xi)~{}A_{\alpha,\beta}(x)\;dx<\infty$,
we have

${\mathcal{W}}^{(\alpha,\beta)}_{g}\left(L^{2}(\mathbb{R},A_{\alpha,\beta})\right)\cap\operatorname{Im}P_{\Sigma}=\\{0\\}.$

###### Proof.

Let
$F\in{\mathcal{W}}^{(\alpha,\beta)}_{g}\left(L^{2}(\mathbb{R},A_{\alpha,\beta})\right)\cap\operatorname{Im}P_{\Sigma}$
be a non-trivial function. Then there exists a function $f\in
L^{2}(\mathbb{R},A_{\alpha,\beta})$ such that
$F={\mathcal{W}}^{(\alpha,\beta)}_{g}(f)$ and $SuppF\subset\Sigma$. For any
$\xi\in\mathbb{R}$, we consider the function

$F_{\xi}(x)={\mathcal{W}}^{(\alpha,\beta)}_{g}(f)(x,\xi),\qquad
x\in\mathbb{R}.$

Then, we get $SuppF_{\xi}\subset\\{x\in\mathbb{R}:(x,\xi)\in\Sigma\\}.$ Since
for almost every
$\xi\in\mathbb{R},\int_{\mathbb{R}}\chi_{\Sigma}(x,\xi)A_{\alpha,\beta}(x)\;dx<\infty$,
we have $A_{\alpha,\beta}\left(SuppF_{\xi}\right)<\infty.$ Using the relation
(3.5), we get

${\mathcal{H}}_{\alpha,\beta}(F_{\xi})={\mathcal{H}}_{\alpha,\beta}(f)\;{\mathcal{H}}_{\alpha,\beta}({\mathcal{M}}^{(\alpha,\beta)}_{\xi}g)\quad\;a.e.$

Hence,

$Supp{\mathcal{H}}_{\alpha,\beta}(F_{\xi})\subset
Supp\tau_{\xi}^{(\alpha,\beta)}|{\mathcal{H}}_{\alpha,\beta}(g)|^{2},$

and from the hypothesis, we get
$\sigma_{\alpha,\beta}\left(\left\\{{\mathcal{H}}_{\alpha,\beta}(F_{\xi})\neq
0\right\\}\right)<\infty$. From Benedicks-type result for the Opdam–Cherednik
transform[1], we conclude that, for every $\xi\in\mathbb{R},F(\cdot,\xi)=0$,
which eventually implies that $F=0.$ ∎

######  Remark 4.12.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
such that

$\sigma_{\alpha,\beta}\left(\left\\{{\mathcal{H}}_{\alpha,\beta}(g)\neq
0\right\\}\right)<\infty.$

Then, for any non-zero function $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, we
have
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(Supp{\mathcal{W}}^{(\alpha,\beta)}_{g}(f))=\infty$,
i.e., the support of ${\mathcal{W}}^{(\alpha,\beta)}_{g}(f)$ cannot be of
finite measure.

######  Proposition 4.13.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
such that

$\sigma_{\alpha,\beta}\left(\left\\{{\mathcal{H}}_{\alpha,\beta}(g)\neq
0\right\\}\right)<\infty.$

Let $\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$, then there
exists a constant $c(\Sigma,g)>0$ such that

$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\leq
c(\Sigma,g)\left\|\chi_{\Sigma^{c}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}.$

###### Proof.

Assume that

$\left\|P_{\Sigma}P_{g}(F)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}=\|F\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)},\;F\in
L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}).$

Since $P_{\Sigma}$ and $P_{g}$ are orthogonal projections, we obtain
$P_{\Sigma}(F)=P_{g}(F)=F.$ Again, since
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$, for almost
every
$\xi\in\mathbb{R},\int_{\mathbb{R}}\chi_{\Sigma}(x,\xi)\;A_{\alpha,\beta}(x)\;dx<\infty$
and from Theorem 4.11, we get $F=0.$ Hence, for $F\neq 0$, we obtain

$\left\|P_{\Sigma}P_{g}(F)\right\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}<\|F\|_{L^{2}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}.$

Since $P_{\Sigma}P_{g}$ is a Hilbert–Schmidt operator, we obtain that the
largest eigenvalue $|\lambda|$ of the operator $P_{\Sigma}P_{g}$ satisfy
$|\lambda|<1$ and $\left\|P_{\Sigma}P_{g}\right\|=|\lambda|<1$. Finally, using
Proposition 4.10, we get the desire result. ∎

###  4.4. Heisenberg-type uncertainty principle for the windowed
Opdam–Cherednik transform

This subsection is devoted to study Heisenberg-type uncertainty inequality for
the windowed Opdam–Cherednik transform for general magnitude $s>0.$ Indeed, we
have the following result.

######  Theorem 4.14.

Let $s>0$. Then there exists a constant $c(s,\alpha,\beta)>0$ such that for
all $f,g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$, we have

$\displaystyle\left\|x^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+\left\|\xi^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\geq
c(s,\alpha,\beta)\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

###### Proof.

Let $r>0$ and $B_{r}=\\{(x,\xi)\in\mathbb{R}^{2}:~{}|(x,\xi)|<r\\}$ be the
ball with centre at origin and radius $r$ in $\mathbb{R}^{2}$. Fix
$\varepsilon_{0}\leq 1$ small enough such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})<1$. From
Proposition 4.4, we have

$\displaystyle\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\frac{1}{\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]}\iint_{|(x,\xi)|\geq\varepsilon_{0}}\left|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)\right|^{2}~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle\leq\frac{1}{\varepsilon_{0}^{2s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]}\iint_{|(x,\xi)|\geq\varepsilon_{0}}|(x,\xi)|^{2s}\left|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)\right|^{2}~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle\leq\frac{1}{\varepsilon_{0}^{2s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]}\iint_{\mathbb{R}^{2}}|(x,\xi)|^{2s}\left|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)\right|^{2}~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi).$

Consequentely,

$\displaystyle\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}\geq\varepsilon_{0}^{2s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$
(4.6)

Finally, using the fact that $|a+b|^{s}\leq 2^{s}\left(|a|^{s}+|b|^{s}\right)$
and from (4.6), we get

$\displaystyle\varepsilon_{0}^{2s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle\leq\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq
2^{s}\left\|x^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+2^{s}\left\|\xi^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}.$

Hence, we get the desire result with
$c(s,\alpha,\beta)=\varepsilon_{0}^{2s}2^{-s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right].$
∎

###  4.5. Local uncertainty inequality for the windowed Opdam–Cherednik
transform

Here, we discuss results related to the
$L^{2}(\mathbb{R},A_{\alpha,\beta})$-mass of the windowed Opdam–Cherednik
transform outside sets of finite measure. Indeed, we establish the following
result.

######  Theorem 4.15.

Let $s>0$. Then there exists a constant $c(s,\alpha,\beta)>0$ such that for
any $f,g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ and any
$\Sigma\subset\mathbb{R}^{2}$ such that
$A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)<\infty$, we have

$\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\Sigma,A_{\alpha,\beta}\right)}\leq
c(s,\alpha,\beta)\left[A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\right]^{\frac{1}{2}}\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}.$

###### Proof.

Since
$\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\|_{L^{2}\left(\Sigma,A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}\leq\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\|_{L^{\infty}\left(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}\right)}~{}[A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)]^{\frac{1}{2}},$
using the relation (3.11), we get

$\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\Sigma,A_{\alpha,\beta}\right)}\leq\left[A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\right]^{\frac{1}{2}}\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}~{}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$

Thus, using Heisenberg’s inequality (4.6), we get

$\displaystyle\left\|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}\left(\Sigma,A_{\alpha,\beta}\right)}\leq\frac{\left[A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(\Sigma)\right]^{\frac{1}{2}}}{\varepsilon_{0}^{s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]^{\frac{1}{2}}}\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}.$

This completes the proof of the theorem with
$c(s,\alpha,\beta)=\varepsilon_{0}^{-s}\left[1-A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{\varepsilon_{0}})\right]^{-\frac{1}{2}}.$
∎

The following result shows that, Theorem 4.15 gives a general form of
Heisenberg-type inequality with a different constant.

######  Corollary 4.16.

Let $s>0.$ Then there exists a constant $c_{s,\alpha,\beta}>0$ such that, for
all $f,g\in$ $L^{2}(\mathbb{R},A_{\alpha,\beta})$, we have

$\displaystyle\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}\geq
c_{s,\alpha,\beta}~{}\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}.$
(4.7)

In particular,

$\displaystyle\left\|x^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+\left\|\xi^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\geq\frac{c_{s,\alpha,\beta}^{2}}{2^{s}}\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

###### Proof.

Let $r>0$ and $B_{r}=\\{(x,\xi)\in\mathbb{R}^{2}:~{}|(x,\xi)|<r\\}$ be the
ball with centre at origin and radius $r$ in $\mathbb{R}^{2}$. Using
Plancherel’s formula (3.7) and Theorem 4.15, we obtain

$\displaystyle\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}$
$\displaystyle=\left\|{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle=\left\|\chi_{B_{r}}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+\left\|\chi_{B_{r}^{c}}{\mathcal{W}}^{(\alpha,\beta)}_{g}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\leq
c(s,\alpha,\beta)^{2}\left[A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta}(B_{r})\right]\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\qquad+r^{-2s}\left\||(x,\xi)|^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}.$

We get the inequality (4.7) by minimizing the right-hand side of the above
inequality over $r>0$. Now, proceeding similarly as in the proof of Theorem
4.14, we obtain

$\displaystyle
2^{s}\left\|x^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}+2^{s}\left\|\xi^{s}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)\right\|_{L^{2}(\mathbb{R}^{2},A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})}^{2}$
$\displaystyle\geq
c_{s,\alpha,\beta}^{2}\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}~{}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2},$

and this completes the proof. ∎

###  4.6. Heisenberg-type uncertainty inequality via the $k$-entropy

In this subsection, we study the localization of the $k$-entropy of the
windowed Opdam–Cherednik transform over the space $\mathbb{R}^{2}$. Before
going to prove the main result, we first need the following definition.

######  Definition 4.17.

  1. (1)

A probability density function $\rho$ on $\mathbb{R}^{2}$ is a non-negative
measurable function on $\mathbb{R}^{2}$ satisfying

$\iint_{\mathbb{R}^{2}}\rho(x,\xi)~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)=1.$

  2. (2)

Let $\rho$ be a probability density function on $\mathbb{R}^{2}$. Then the
$k$-entropy of $\rho$ is defined by

$E_{k}(\rho):=-\iint_{\mathbb{R}^{2}}\ln(\rho(x,\xi))~{}\rho(x,\xi)~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi),$

whenever the integral on the right-hand side is well defined.

In the following, we prove the main result of this subsection.

######  Theorem 4.18.

Let $g\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ be a non-zero window function
and $f\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ such that $f\neq 0$. Then, we
have

$\displaystyle
E_{k}(|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|^{2})\geq-2\ln\left(\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\right)\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$
(4.8)

###### Proof.

First, we assume that
$\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=1$.
For any $(x,\xi)\in\mathbb{R}^{2}$, using the relation (3.11), we get

$|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)(x,\xi)|\leq\;\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=1.$

Consequently, $\ln(|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|)\leq 0$ and
therefore $E_{k}\big{(}|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|\big{)}\geq 0$.
The desire inequality (4.8) holds trivially if the entropy
$E_{k}\big{(}|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|\big{)}$ is infinite. Now,
suppose that the entropy
$E_{k}\big{(}|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|\big{)}$ is finite. Let
$f$ and $g$ be two non-zero functions in $L^{2}(\mathbb{R},A_{\alpha,\beta})$.
We define

$\phi=\frac{f}{\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}~{}\text{ and
}~{}\psi=\frac{g}{\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}.$

Then $\phi,~{}\psi\in L^{2}(\mathbb{R},A_{\alpha,\beta})$ with
$\|\phi\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=\|\psi\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}=1$
and consequently

$\displaystyle
E_{k}\big{(}|{\mathcal{W}}_{\psi}^{(\alpha,\beta)}(\phi)|\big{)}\geq 0.$ (4.9)

Since
${\mathcal{W}}_{\psi}^{(\alpha,\beta)}(\phi)=\frac{1}{\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}}{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)$,
we have

$\displaystyle
E_{k}\big{(}|{\mathcal{W}}_{\psi}^{(\alpha,\beta)}(\phi)|^{2}\big{)}=$
$\displaystyle-\iint_{\mathbb{R}^{2}}\ln(|{\mathcal{W}}_{\psi}^{(\alpha,\beta)}(\phi)(x,\xi)|^{2})~{}|{\mathcal{W}}_{\psi}^{(\alpha,\beta)}(\phi)(x,\xi)|^{2}~{}d(A_{\alpha,\beta}\otimes\sigma_{\alpha,\beta})(x,\xi)$
$\displaystyle=\frac{1}{\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}}E_{k}(|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|^{2})$
$\displaystyle\qquad+2\ln\big{(}\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\big{)}.$

Finally, from (4.9), we obtain

$E_{k}(|{\mathcal{W}}_{g}^{(\alpha,\beta)}(f)|^{2})\geq-2\ln\left(\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}\right)\|f\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}\;\|g\|_{L^{2}(\mathbb{R},A_{\alpha,\beta})}^{2}.$

This completes the proof of the theorem. ∎

## Acknowledgments

The first author gratefully acknowledges the support provided by IIT Guwahati,
Government of India. The second author is deeply indebted to Prof. Nir Lev for
several fruitful discussions and generous comments.

##  5\. Data availability statement

The authors confirm that the data supporting the findings of this study are
available within the article and its supplementary materials.

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# What If We Only Use Real Datasets for Scene Text Recognition?
Toward Scene Text Recognition With Fewer Labels

Jeonghun Baek Yusuke Matsui Kiyoharu Aizawa
The University of Tokyo
{baek, matsui<EMAIL_ADDRESS>
###### Abstract

Scene text recognition (STR) task has a common practice: All state-of-the-art
STR models are trained on large synthetic data. In contrast to this practice,
training STR models only on fewer real labels (STR with fewer labels) is
important when we have to train STR models without synthetic data: for
handwritten or artistic texts that are difficult to generate synthetically and
for languages other than English for which we do not always have synthetic
data. However, there has been implicit common knowledge that training STR
models on real data is nearly impossible because real data is insufficient. We
consider that this common knowledge has obstructed the study of STR with fewer
labels. In this work, we would like to reactivate STR with fewer labels by
disproving the common knowledge. We consolidate recently accumulated public
real data and show that we can train STR models satisfactorily only with real
labeled data. Subsequently, we find simple data augmentation to fully exploit
real data. Furthermore, we improve the models by collecting unlabeled data and
introducing semi- and self-supervised methods. As a result, we obtain a
competitive model to state-of-the-art methods. To the best of our knowledge,
this is the first study that 1) shows sufficient performance by only using
real labels and 2) introduces semi- and self-supervised methods into STR with
fewer labels. Our code and data are available: https://github.com/ku21fan/STR-
Fewer-Labels.

##  1 Introduction

Reading text in natural scenes is generally divided into two tasks: detecting
text regions in scene images and recognizing the text in the regions. The
former is referred to as scene text detection (STD), and the latter as scene
text recognition (STR). Since STR can serve as a substitute for manual typing
performed by humans, we frequently employ STR for various purposes:
translation by recognizing foreign languages, street sign recognition for
autonomous driving, various card recognition to input personal information,
etc. Unlike optical character recognition (OCR), which focuses on reading
texts in cleaned documents, STR also addresses irregular cases in our lives,
such as curved or perspective texts, occluded texts, texts in low-resolution
images, and texts written in difficult font.

To address these irregular cases, prior works have developed STR models
comprising deep neural networks. For example, to address curved or perspective
texts, image transformation modules have been proposed to normalize them into
horizontal images [ASTER, ESIR, ScRN]. Qiao et al. [SE-ASTER] has integrated a
pretrained language model into STR models to recognize occluded text. Wang et
al. [wang2020scene-SR-ECCV] and Mou et al. [plugnet] have introduced a super-
resolution module into STR models to handle low-resolution images.

While prior works have improved STR models, the study of training STR models
only on fewer real labels (STR with fewer labels) is insufficient. After
emerging large synthetic data [MJSynth] in 2014, the study of STR with fewer
labels has decreased. All state-of-the-art methods use large synthetic data to
train STR models instead of sole real data [CRNN, ASTER, SAR, ESIR, Mask-
textspotter, TRBA, ScRN, DAN, textscanner, SRN, scatter, SE-ASTER,
robustscanner, plugnet]. Implicit common knowledge has been made; training STR
models only on real data results in low accuracy because the amount of real
data is very small. This common knowledge may have hindered studies on STR
with fewer labels.

STR with fewer labels is important when we have to train STR models without
synthetic data. In practical applications, generating synthetic data close to
real data can be difficult depending on the target domain, such as handwritten
text or artistic text. In the other case, when we have to recognize languages
other than English, there are not always synthetic data for them. Generating
appropriate synthetic data for them is difficult for those who do not know
target languages.

In this paper, we would like to reactivate STR with fewer labels for such
cases. As a first step, we disprove the common knowledge by showing that we
can train STR models satisfactorily only with real labels. This is not
previously feasible. Because the real data was small, STR models trained on
real data had low accuracy, as shown in Figure 1. However, the public real
data are accumulated every two years. We consolidate accumulated real data
(276K), and find that the accuracy of STR models [CRNN, TRBA] trained on them
is close to that of synthetic data (16M). Namely, we can train STR models only
on real data instead of synthetic data. It is high time to change the
prevalent perspective from “We don’t have enough real data to train STR
models” to “We have enough real data to train STR models”. It is also a good
time to study STR with fewer labels.

To improve STR with fewer labels, we find simple yet effective data
augmentations to fully exploit real data. In addition, we collect unlabeled
data and introduce a semi- and self-supervised framework into the STR. With
extensive experiments, we analyze the contribution of them and demonstrate
that we can obtain a competitive model to state-of-the-art methods by only
using real data. Furthermore, we investigate if our method is also useful when
we have both synthetic and real data.

Figure 1: Accuracy vs. number of accumulated real labeled data. Every two
years, public real data has been accumulated. In our experiments, we find that
accuracy obtained using real data approaches that obtained using synthetic
data, along with increment of real data. CRNN [CRNN] and TRBA [TRBA] are VGG-
based and ResNet-based STR models, respectively.

##  2 Common Practice in STR Dataset

According to a benchmark study [TRBA], obtaining enough real data is difficult
because of the high labeling cost. Thus, STR models are generally trained on
large synthetic data instead of real data. Real data has been used for
evaluation.

###  2.1 Synthetic Datasets for Training

There are two major synthetic datasets.

(a) MJ word boxes

(b) ST scene image

Figure 2: Examples of two major synthetic datasets.

MJSynth  (MJ) [MJSynth] is generated for STR, and it contains 9M word boxes.
Each word is generated from a 90K English lexicon and over 1,400 Google Fonts,
as shown in Figure 2(a).

SynthText  (ST) [SynthText] is originally generated for scene text detection.
The texts are rendered onto scene images, as shown in Figure 2(b). For STR, we
crop the texts in scene images and use them for training. ST has 7M word
boxes.

###  2.2 Real Benchmark Datasets for Evaluation

Six real datasets have been used to evaluate STR models. Street View Text
(SVT) [SVT] is collected from Google Street View, and contains texts in street
images. It contains 257 images for training and 647 images for evaluation.

IIIT5K-Words  (IIIT) [IIIT5K] is crawled from Google image searches with query
words such as “billboards” and “movie posters.” It contains 2,000 images for
training and 3,000 images for evaluation.

ICDAR2013  (IC13) [IC13] is created for the ICDAR 2013 Robust Reading
competition. It contains 848 images for training and 1,015 images for
evaluation.

ICDAR2015  (IC15) [IC15] is collected by people who wear Google Glass, and
thus, many of them contain perspective texts and some of them are blurry. It
contains 4,468 images for training and 2,077 images for evaluation.

SVT Perspective  (SP) [SVTP] is collected from Google Street View, similar to
SVT. Unlike SVT, SP contains many perspective texts. It contains 645 images
for evaluation.

CUTE80  (CT) [CUTE80] is collected for curved text. The images are captured by
a digital camera or collected from the Internet. It contains 288 cropped
images for evaluation.

They are generally divided into regular (SVT, IIIT, IC13) and irregular (IC15,
SP, CT) datasets. The former mainly contains horizontal texts, while the
latter mainly contains perspective or curved texts.

(a) Year 2011

(b) Year 2013

(c) Year 2015

(d) Year 2017

(e) Year 2019

Figure 3: Examples of accumulated real labeled data. More examples are
provided in the supplementary materials.

##  3 Consolidating Public Real Datasets

Recently, public real data has been sufficiently accumulated to train STR
models, as shown in Figure 1. We consolidate the training set of public real
datasets from 2011 to 2019. Table LABEL:tab:dataset lists datasets. Figure 3
shows the examples of word boxes. Before using the original data directly for
training, we conduct some preprocessing on datasets for our task. We summarize
the processes in §LABEL:sec:preprocessing, and details are in the
supplementary materials.

###  3.1 Real Labeled Datasets Have Increased

Recently, many irregular texts are accumulated as shown in Year 2015, 2017,
and 2019 in Figure 3. They can make STR models more robust. Many real labeled
datasets are released from ICDAR competitions: IC13, IC15, RCTW, ArT, LSVT,
MLT19, and ReCTS (7 of 11 datasets in Table LABEL:tab:dataset). ICDAR
competitions are held every two years, and real labeled datasets have also
increased in number. We summarize real labeled datasets for every two years.

  1. (a)

Year 2011 (SVT) and (b) Year 2013 (IIIT, IC13): Most of images are horizontal
texts in the street.

  1. (c)

Year 2015 (IC15): Images captured by Google Glass under movement of the
wearer, and thus many are perspective texts, blurry, or low-resolution images.

  2. (d)

Year 2017 (COCO, RCTW, Uber):

COCO-Text (COCO) [COCO] is created from the MS COCO dataset [MSCOCO]. As the
MS COCO dataset is not intended to capture text, COCO contains many occluded
or low-resolution texts.

RCTW [RCTW] is created for Reading Chinese Text in the Wild competition. Thus
many are Chinese text.

Uber-Text (Uber) [Uber] is collected from Bing Maps Streetside. Many are house
number, and some are text on signboards.

  3. (e)

Year 2019 (ArT, LSVT, MLT19, ReCTS):

ArT [ArT] is created to recognize Arbitrary-shaped Text. Many are perspective
or curved texts. It also includes Totaltext [totaltext] and CTW1500 [CTW1500],
which contain many rotated or curved texts.

LSVT [LSVT, sun2019LSVT] is a Large-scale Street View Text dataset, collected
from streets in China, and thus many are Chinese text.

MLT19 [MLT19] is created to recognize Multi-Lingual Text. It consists of seven
languages: Arabic, Latin, Chinese, Japanese, Korean, Bangla, and Hindi.

ReCTS [ReCTS] is created for the Reading Chinese Text on Signboard
competition. It contains many irregular texts arranged in various layouts or
written with unique fonts.


# Predicting spatial distribution of Palmer Drought Severity Index*
††thanks: Identify applicable funding agency here. If none, delete this.

Vsevolod Grabar  Skolkovo Institute of Science
and Technology
<EMAIL_ADDRESS>Aleksandr Lukashevich  Skolkovo Institute of Science
and Technology
<EMAIL_ADDRESS>Alexey Zaytcev  Skolkovo Institute of Science
and Technology
<EMAIL_ADDRESS>
###### Abstract

The probability of a drought for a particular region is crucial when making
decisions related to agriculture. Forecasting this probability is critical for
management and challenging at the same time. The prediction model should
consider multiple factors with complex relationships across the region of
interest and neighbouring regions.

We approach this problem by presenting an end-to-end solution based on a
spatio-temporal neural network. The model predicts the Palmer Drought Severity
Index (PDSI) for subregions of interest. Predictions by climate models provide
an additional source of knowledge of the model leading to more accurate
drought predictions.

Our model has better accuracy than baseline Gradient boosting solutions, as
the $R^{2}$ score for it is $0.90$ compared to $0.85$ for Gradient boosting.
Specific attention is on the range of applicability of the model. We examine
various regions across the globe to validate them under different conditions.

We complement the results with an analysis of how future climate changes for
different scenarios affect the PDSI and how our model can help to make better
decisions and more sustainable economics.

###### Index Terms:

weather, climate, forecasting, neural networks

##  I Introduction

Why is it so important to monitor and forecast droughts? They are natural
climate events that could happen in any type of terrain - starting from
deserts and ending with tropical forests. At the same time, droughts are very
costly, frequently happening, and affect the population and multiple economic
sectors. Their impact is not limited by restricted geographical areas, such as
riverbeds, coastlines or hurricane valleys.

The development of drought is slow, while the consequences can be severe.
Thus, it is a good candidate for prediction on the base of past values on
temperatures, precipitation, and overall conditions of soil and ground waters.
To streamline the quantification of this event, practitioners use a multitude
of indices [10]. Among them, we consider, Palmer Drought Severity Index (PDSI)
[2] and Selyaninov’s Hydrothermal Coefficient (HTC) [6], as they are relevant
and easy to evaluate using available data and, thus, our methodology can be
reused for other regions and conditions.

#### Palmer Drought Severity Index

PDSI was developed in the 60s to monitor droughts that could affect the
agricultural industry. It requires values of temperature, precipitation and
soil water holding capacity. The soil water holding capacity could be replaced
by default values in case of its non-availability. PDSI is widely used, and
numerous examples of its applications (with code of PDSI estimation) can be
found publicly. Finally, PDSI is not prone to seasonal problems. On the other
hand, the estimation of this index requires the absence of missing data. PDSI
is also lagging; sometimes, lags could reach up to 2 or 3 months, so we could
not detect fast-evolving droughts.

#### Selyaninov’s Hydrothermal Coefficient

HTC is an indicator widely used in Russia and tailored well to its climate
conditions. It is rather straightforward to calculate, as it uses temperature
and precipitation values:

$HTC=10\dfrac{\sum P_{i}}{\sum T_{i}}$ (1)

where $\sum P_{i}$ and $\sum T_{i}$ are sums of precipitation and temperature
over periods when the average temperature is over 10 degrees Celsius. It is
flexible enough to be used in both monthly and decadal applications. The main
disadvantage of it is that HTC should be tuned to a specific region and it
also doesn’t use information about soil water. HTC is also sensitive to dry
conditions specific to the climate regime being monitored.

#### Prediction

While drought predictions and prediction of PDSI also happen in recent
literature, the researchers focus on methods that ignore the spatio-temporal
nature of the problem or aim at a selection of relevant features limiting the
range of applicability of a prediction model to a single region [3].

#### Contribution

We managed to predict these indices using convolutional-temporal neural
networks. They take into account both spatial and temporal dependencies in the
data. Moreover, they can proceed large data volumes and don’t require feature
engineering for a specific problem.

##  II Data

To evaluate our methods, we have used geospatial data that is available
publicly - for example, Google Earth Engine [7]. To get the PDSI data, we
employed TerraClimate Monthly dataset [1]. It has data for all of the Earth
from 1958 up to 2021. For the calculation of HTC, we need daily data on
precipitation and temperature, so we downloaded samples from the ERA5
collection [4].

Extracted datasets are preprocessed. Our input looks like a 3d tensor, where
one dimension is temporal (monthly), and two others are spatial (x and y
coordinates of the grid). Datasets could be processed at various resolutions,
so our grid dimensions vary from 9 by 16 and up to 40 by 200.

##  III Methods

We compare our approach with classic methods. Among them, one of the most
powerful is gradient boosting.

###  III-A Gradient boosting

As a baseline, we consider gradient boosting of decision trees ensemble, which
is implemented using the well-known library XGBoost [5]. Treating every grid
cell as a separate value, we narrow our task to a typical task of time series
forecasting. XGBoost is both fast and efficient on a wide range of predictive
modelling tasks and is a favourite among data science competition winners. It
is an ensemble of decision trees, where new trees fix errors of those trees
that are already part of the model. Trees are added until no further
improvements can be made to the model. Such an approach works well for
geospatial data [11] and can deal with imbalanced problems [9] related to the
drought prediction problem.

To improve our baseline and exploit the spatial information, we propose to use
a modification called spatial gradient boosting. For this model, we add
information about neighbouring cells. For the sake of efficiency and
simplicity of the model, we consider a 3x3 neighbourhood. So, we include eight
additional time series. For ”edge” cells, some of them are complete zeros.

###  III-B ConvLSTM

Our model is inspired by paper [8] and is a version of Covolutional LSTM
architecture from [12]. We adopt recurrent neural networks to reflect temporal
dependencies. In particular, we use LSTM, a type of RNN which uses an
additional state cell to enable long-term memory. There we employ two-
dimensional feature maps as hidden states compared to one-dimensional hidden
states in common LSTM, as we require grid-to-grid transformations. Next, we
process a grid of PDSI values for various grid dimensions, from 9x16 to
40x200. To benefit from spatial dependencies between drought severity values,
we use convolution neural networks, which efficiently work with images and
other two-dimensional signals like geospatial data. Our approach combines RNN
and CNN architectures as we pass information through RNN in the form of a
feature map obtained using CNN.

###  III-C Our architecture

Our main architecture depicted in Figure 1 follows the pipeline:

  1. 1.

We represent data as a sequence of grids: for each cell, we specify a value of
float drought index on a particular month; the input grid at each time moment
has various dimensions (e.g. 9x16 or 40x200)

  2. 2.

We pass the input grid through a convolutional network to create an embedding
of grid dimensionality size with 16 channels. As an output of LSTM at each
time moment, we have a hidden representation (short term memory) of size
$32\times grid_{h}\times grid_{w}$, cell (long term memory) representation of
a similar size, and the output of size $32\times grid_{h}\times grid_{w}$.

  3. 3.

We transform the output to the size $1\times grid_{h}\times grid_{w}$ using
convolution $1\times 1$ to receive values of our indicator for each cell as a
final prediction. As an additional hyperparameter, we vary the forecasting
horizon - i.e. we can forecast PDSI for the next month or $k$-th month.

Figure 1: The proposed version of ConvLSTM architecture

##  IV Results

###  IV-A Evaluation procedure

For evaluation of the model, we use $R^{2}$ score. Higher values of $R^{2}$
scores correspond to better models, with the perfect value being $1$ and the
value for a random prediction being around $1$.

###  IV-B Main results

We have observed that our model performed better than the baseline gradient
boosting tree algorithm, reaching $R^{2}$ score of $0.9$ as compared with a
common Gradient boosting score of $0.85$ and a spatial Gradient boosting score
of $0.85$.

###  IV-C Prediction and errors for a particular region

We have taken the approximate region of Iowa state to track the performance of
our algorithm. The distribution of $R^{2}$ scores is at Figure 2.

We observe that the distribution of $R^{2}$ is non-uniform across the cells:
the standard deviation is quite large, and individual values vary from
negative ones to perfect, equaling almost $1.0$.

Figure 2: Spatial distribution of $R^{2}$, Iowa state, ConvLSTM

As a sanity check, we also study how average $R^{2}$ changes with the increase
of the forecasting horizon. The dynamic of $R^{2}$ scores is given in Figure
3.

As expected, longer prediction horizons lead to lower model quality. While the
1-month look ahead is close to $1$ being approximately $0.95$, it gradually
falls to $0.75$ at the 6-month horizon.

Figure 3: Dynamics of $R^{2}$ with respect to forecast horizon

##  V Conclusion

Droughts are severe natural disasters affecting the economy through various
channels, including agriculture and population well-being. Due to climate
change, droughts happen more frequently around the world, and their amplitude
has also increased, which could be witnessed from the summer of 2022 in the
Northern Hemisphere.

Thus, predicting future droughts accurately and preparing for their effects
would help to mitigate some of the negative effects of climate change. We have
observed from our research study that combining convolutional neural nets with
recurrent ones significantly increased the forecasting abilities of various
drought indicators (as compared with previous algorithms), including PDSI.

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