# PF-cpGAN: Profile to Frontal Coupled GAN for Face Recognition in the Wild

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###### Abstract

In recent years, due to the emergence of deep learning, face recognition has
achieved exceptional success. However, many of these deep face recognition
models perform relatively poorly in handling profile faces compared to frontal
faces. The major reason for this poor performance is that it is inherently
difficult to learn large pose invariant deep representations that are useful
for profile face recognition. In this paper, we hypothesize that the profile
face domain possesses a gradual connection with the frontal face domain in the
deep feature space. We look to exploit this connection by projecting the
profile faces and frontal faces into a common latent space and perform
verification or retrieval in the latent domain. We leverage a coupled
generative adversarial network (cpGAN) structure to find the hidden
relationship between the profile and frontal images in a latent common
embedding subspace. Specifically, the cpGAN framework consists of two GAN-
based sub-networks, one dedicated to the frontal domain and the other
dedicated to the profile domain. Each sub-network tends to find a projection
that maximizes the pair-wise correlation between two feature domains in a
common embedding feature subspace. The efficacy of our approach compared with
the state-of-the-art is demonstrated using the CFP, CMU Multi-PIE, IJB-A, and
IJB-C datasets.

##  1 Introduction

Due to the emergence of deep learning, face recognition has achieved
exceptional success in recent years [Cao2018PoseRobustFR]. However, many of
these deep face recognition models perform relatively poorly in handling
profile faces compared to frontal faces [sengupta_wacv_cfp]. In other words,
face recognition in the wild, or unconstrained face recognition, is a
challenging problem. Pose, expression, and lighting variations are considered
to be major obstacles in attaining high unconstrained face recognition
performance. Some methods [Masi_pose_aware_2016, Cao2018PoseRobustFR] attempt
to address pose-variation issue by learning pose-invariant features, while
some other methods [Yim2015RotatingYF, Yin2017TowardsLF,
Cole2017SynthesizingNF, FNM, Tran_disentangled_cvpr_17] try to normalize
images (along with identity-preservation) to a single frontal pose before
recognition. However, there are three major difficulties related to face
frontalization or normalization in unconstrained environment:

  * •

Complicated face-variations besides pose: In comparison to a controlled
environment, there are more complex face variations, e.g., lighting, head
pose, expression, in real-world scenarios. It is difficult to directly warp
the input face to a normalized view [FNM].

  * •

Unpaired data: Undoubtedly, obtaining a strictly normalized face is expensive
and time-consuming, but getting an effective pair of target normalized face
(i.e., frontal-facing, neutral expression) and an input face is difficult due
to highly imbalanced datasets[FNM].

  * •

Presence of artifacts: Synthesized ‘frontal’ faces contain artifacts caused by
occlusions and non-rigid expressions.

Figure 1: Frontal and Profile Images from Celebrity in Frontal Profile (CFP)
Dataset.

In this paper, we hypothesize that the profile face domain possesses a gradual
connection with the frontal face domain in a latent deep feature subspace. We
aim to exploit this connection by projecting the profile faces and frontal
faces into a common latent subspace and perform verification or retrieval in
this latent domain. We propose an embedding model for profile to frontal face
verification based on a deep coupled learning framework which uses a
generative adversarial network (GAN) to find the hidden relationship between
the profile face features and frontal face features in a latent common
embedding subspace.

Our work is conceptually related to the embedding category of super-resolution
[shekhar2017synthesis, zhang2015coupled, jiang2016cdmma, Li_2019_LRFRW] in
that our approach also performs verification of profile and frontal face in
the latent space but not in the image space. From our experiments, we observe
that transforming profile and frontal face features to a latent embedding
subspace could yield higher performance than image-level face frontalization,
which is susceptible to the negative influence of artifacts as a result of
image synthesis. To our best knowledge, this study is the first attempt to
perform profile-to-frontal face verification in the latent embedding subspace
using generative modeling.

This paper makes the following contributions:

  * •

A novel profile to frontal face recognition model using coupled GAN framework
with multiple loss functions is developed.

  * •

Comprehensive experiments using different datasets and a comparison of the
proposed method with the state-of-the-art methods have been performed,
indicating the efficacy of the proposed GAN framework.

  * •

The proposed framework can potentially be used to improve the performance of
traditional face recognition methods by integrating it as a preprocessing
procedure for face-frontalization schema.

##  2 Related Work

Face recognition using Deep Learning: Before the advent of deep learning,
traditional methods for face recognition (FR) used one or more layer
representations, such as the histogram of the feature codes, filtering
responses, or distribution of the dictionary atoms [Wang2018DeepFR]. FR
research was concentrated more toward separately improving preprocessing,
local descriptors, and feature transformation; however, overall improvement in
FR accuracy was very slow. This all changed with the advent of deep learning,
and now deep learning is the prominent technique used for FR.

Various network architectures such as AlexNet[Krizhevsky2012AlexNet], VGGNet
[simonyan2014very], GoogleNet[2015_GoogleNet], and Resnet[He2015_Resnet] are
used as baseline model for FR. Simultaneously, various loss functions have
been explored and used in FR. These loss functions can be categorized as the
Euclidean-distance-based loss, angular/cosine-margin-based loss, and softmax
loss and its variations. The contrastive loss and the triplet loss are the
commonly used Euclidean-distance-based loss functions [sun2015deepid3,
Sun_deep_learning_face_representation_nips, Sun2015SparsifyingNN,
Sankaranarayanan_triplet_16, schroff2015facenet]. For avoiding
misclassification of difficult samples, the learned face features need to be
well separated. Angular/cosine-margin based loss [Deng_2019_CVPR,
feng_wang_additive_margin, Liu_large_margin_ICML, Liu2017SphereFaceDH] are
commonly used to make the learned features more separable with a larger
angular/cosine distance. Finally, in the category of softmax loss and its
variants for FR [Hasnat2017vonMM, Wang_NormFace_ICM, Liu2017RethinkingFD], the
softmax loss is modified to improve the FR performance as in
[Liu2017RethinkingFD], where the cosine distance among data features is
optimized along with normalization of features and weights.

Profile-Frontal Face Recognition: Face recognition with pose variation in an
unconstrained environment is a very challenging problem. Existing methods
focus on the pose variation problem by training separate models for learning
pose-invariant features [Masi_pose_aware_2016, Cao2018PoseRobustFR], elaborate
dense 3D facial landmark detection and warping [taigman_deepface], and
synthesizing a frontal, neutral expression face from a single image
[Tran_disentangled_cvpr_17, Yim2015RotatingYF, Yin2017TowardsLF,
Cole2017SynthesizingNF, FNM]. For instance, Cao _et al_. [Cao2018PoseRobustFR]
exploit the inherent mapping between profile and frontal faces, and transform
a deep profile face representation to a canonical pose by adaptively adding
residuals. FF-GAN [Yin2017TowardsLF] solves the problem of large-pose face
frontalization in the wild by incorporating a 3D face model into a GAN.
Considering photorealistic and identity preserving frontal view synthesis, a
domain adaptation strategy for pose invariant face recognition is discussed in
[Zhao_PIM_cvpr]. Tran _et al_. [Tran_disentangled_cvpr_17] propose a GAN
framework to rotate the face and disentangle the identity representation by
using the pose code. In [FNM], a face normalization model (FNM) uses a
generative adversarial network (GAN) network with 3 distinct losses for
generating canonical-view and expression-free frontal images.

##  3 Generative Adversarial Network

GAN was first introduced by Goodfellow _et al_. [NIPS_GAN] and has drawn
considerable attention from the deep learning and computer vision research
community due to its remarkable performance on generative tasks. The GAN
framework is based on two competing networks — a generator network G and a
discriminator network D. The generator $G(z;\theta_{g})$ is a differentiable
function which maps the noise variable $z$ from training noise distribution
$p_{z}(z)$ to a data space with distribution $p_{data}$ using the network
parameters $\theta_{g}$. On the other hand, the discriminator
$D(.;\theta_{d})$ is also a differentiable function, which discriminates
between the real data $y$ and the generated fake data $G(z)$ using a binary
classification model. Specifically, the min-max two-player game between the
generator and the discriminator provides a simple and powerful way to estimate
target distribution and generate novel image samples [FNM]. The loss function
$L(D,G)$ for GAN is given as:

$\begin{split}L(D,G)&=E_{y\sim P_{data}(y)}[\log D(y)]\\\ &+E_{z\sim
P_{z}(z)}[\log(1-D(G(z)))].\end{split}$ (1)

The objective (two player minimax game) for GAN is as:

$\begin{split}\min_{G}\max_{D}L(D,G)&=\min_{G}\max_{D}[E_{y\sim
P_{data}(y)}[\log D(y)]\\\ &+E_{z\sim P_{z}(z)}[\log(1-D(G(z)))]].\end{split}$
(2)

Another variant of GAN is the Conditional GAN, which was introduced by Mirza
and Osindero [mirza_2014_conditional]. In conditional GAN, both the generator
and discriminator are conditioned on an additional variable $x$. This
additional variable could be any kind of auxiliary information such as
discrete labels [mirza_2014_conditional], text [RAYLLS_16], or images
[Isola_2017_ImagetoImageTW]. The loss function for the conditional GAN is
given as:

$\begin{split}L_{c}(D,G)&=E_{y\sim P_{data}(y)}[\log D(y|x)]\\\ &+E_{z\sim
P_{z}(z)}[\log(1-D(G(z|x)))].\end{split}$ (3)

Hereafter, we will denote the objective for the conditional GAN as
$F_{cGAN}(D,G,y,x)$, which is given by:

$\begin{split}F_{cGAN}(D,G,y,x)&=\min_{G}\max_{D}[E_{y\sim P_{data}(y)}[\log
D(y|x)]\\\ &+E_{z\sim P_{z}(z)}[\log(1-D(G(z|x)))]].\end{split}$ (4)

##  4 Proposed Method

Here, we describe our method for profile to frontal face recognition. In
contrast to the face normalization methods, we do not perform pose
normalization (i.e., frontalization) on each profile image before matching.
Instead, we seek to project the profile and frontal face images to a common
latent low-dimensional embedding subspace using generative modeling. Inspired
by the success of GANs [NIPS_GAN], we explore adversarial networks to project
profile and frontal images to a common subspace for recognition.

The framework of proposed profile to frontal coupled generative adversarial
network (PF-cpGAN; shown in Fig. 2) consists of two modules, where each module
contains a GAN architecture made of a generator and a discriminator. The
generators that we have used in both modules are U-net auto-encoders that are
coupled together using a contrastive loss function. In addition to
adversarial, and contrastive loss, we propose to guide the generators using a
perceptual loss [Johnson2016PerceptualLF] based on the VGG 16 architecture, as
well as an $L_{2}$ reconstruction error. The perceptual loss helps in sharp
and realistic reconstruction of the images.

Figure 2: Block diagram of PF-cpGAN.

###  4.1 Profile to Frontal Coupled GAN

The main objective of PF-cpGAN is the recognition of profile face images with
respect to a gallery of frontal face images, which have not been seen during
the training. The matching of the profile and the frontal face images is
performed in a common embedding subspace. PF-cpGAN consists of two modules: a
profile GAN module and a frontal GAN module, both consisting of a GAN
(generator + discriminator), and a perceptual network based on VGG-16.

For the generators, we use a U-Net [ronneberger_2015_unet] auto-encoder
architecture. The primary reason for using U-Net is that the encoder-decoder
structure tends to extract global features and generates images by leveraging
this overall information, which is very useful for global shape transformation
tasks such as profile to frontal image conversion. Moreover, for many image
translation problems, there is a significant amount of low-level information
that needs to be shared between the input and output, and it is desirable to
pass this information directly across all the layers including the bottleneck.
Therefore, using skip-connections, as in U-net, provides a means for the
encoder-decoder structure to circumvent the bottleneck and pass the
information over to other layers.

For discriminators, we have used patch-based discriminators
[Isola_2017_ImagetoImageTW], which are trained iteratively along with the
respective generators. $L_{1}$ loss performs very well when trying to preserve
the low frequency details but fails to preserve the high-frequency details.
However, using a patch-based discriminator that penalizes structure at the
scale of the patches, ensures the preservation of high-frequency details,
which are usually lost when only $L_{1}$ loss is used.

The final objective of PF-cpGAN is to find the hidden relationship between the
profile face features and frontal face features in a latent common embedding
subspace. To find this common subspace between the two domains, we couple the
two generators via a contrastive loss function, $L_{cont}$
[Chopra2005LearningAS].

This loss function $(L_{cont})$ is a distance-based loss function, which tries
to ensure that semantically similar examples (genuine pairs i.e., a profile
image of a subject with its corresponding frontal image) are embedded closely
in the common embedding subspace, and, simultaneously, semantic dissimilar
examples (impostor pairs i.e., a profile image of a subject and a frontal
image of a different subject) are pushed away from each other in the common
embedding subspace. The contrastive loss function is defined as:

$\begin{split}L_{cont}(z_{1}&(x^{i}_{PR}),z_{2}(x^{j}_{FR}),Y)=\\\
&(1-Y)\frac{1}{2}(D_{z})^{2}+(Y)\frac{1}{2}(\mbox{max}(0,m-D_{z}))^{2},\end{split}$
(5)

where $x^{i}_{PR}$ and $x^{j}_{FR}$ denote the input profile and frontal face
image, respectively. The variable $Y$ is a binary label, which is equal to 0
if $x^{i}_{PR}$ and $x^{j}_{FR}$ belong to the same class (i.e., genuine
pair), and equal to 1 if $x^{i}_{PR}$ and $x^{j}_{FR}$ belong to the different
class (i.e., impostor pair). $z_{1}(.)$ and $z_{2}(.)$ denote only the
encoding functions of the U-Net auto-encoder to transform $x^{i}_{PR}$ and
$x^{j}_{FR}$, respectively into a common latent embedding subspace. The value
$m$ is the contrastive margin and is used to “tighten” the constraint. $D_{z}$
denotes the Euclidean distance between the outputs of the functions
$z_{1}(x^{i}_{PR})$ and $z_{2}(x^{j}_{FR})$.

$D_{z}=\left\lVert z_{1}(x^{i}_{PR})-z_{2}(x^{j}_{FR})\right\rVert_{2}.$ (6)

Therefore, if $Y=0$ (i.e., genuine pair), then the contrastive loss function
$(L_{cont})$ is given as:

$L_{cont}(z_{1}(x^{i}_{PR}),z_{2}(x^{j}_{FR}),Y)=\frac{1}{2}\left\lVert
z_{1}(x^{i}_{PR})-z_{2}(x^{j}_{FR})\right\rVert^{2}_{2},$ (7)

and if $Y=1$ (i.e., impostor pair), then contrastive loss function
$(L_{cont})$ is :

$\begin{split}L_{cont}(z_{1}(x^{i}_{PR}),&z_{2}(x^{j}_{FR}),c(i,j))=\\\
&\frac{1}{2}\mbox{max}\biggl{(}0,m-\left\lVert
z_{1}(x^{i}_{PR})-z_{2}(x^{j}_{FR})\right\rVert^{2}_{2}\biggr{)}.\end{split}$
(8)

Thus, the total loss for coupling the profile generator and the frontal
generator is denoted by $L_{cpl}$ and is given as:

$\begin{split}L_{cpl}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}L_{cont}(z_{1}(x^{i}_{PR}),z_{2}(x^{j}_{FR}),Y),\end{split}$
(9)

where N is the number of training samples. The contrastive loss in the above
equation can also be replaced by some other distance-based metric, such as the
Euclidean distance. However, the main aim of using the contrastive loss is to
be able to use the class labels implicitly and find the discriminative
embedding subspace, which may not be the case with some other metric such as
the Euclidean distance. This discriminative embedding subspace would be useful
for matching of the profile images with the frontal images.

###  4.2 Generative Adversarial Loss

Let the generators (profile generator and frontal generator) that reconstruct
the corresponding profile and frontal image from the input profile and frontal
image, be denoted as $G_{PR}$ and $G_{FR}$, respectively. The patch-based
discriminators used for the profile and frontal GANs are denoted as $D_{PR}$
and $D_{FR}$. For the proposed method, we have used the conditional GAN, where
the generator networks $G_{PR}$ and $G_{FR}$ are conditioned on input profile
and frontal face images, respectively. We have used the conditional GAN loss
function [mirza_2014_conditional] to train the generators and the
corresponding discriminators in order to ensure that the discriminators cannot
distinguish the images reconstructed by the generators from the corresponding
ground truth images. Let $L_{PR}$ and $L_{FR}$ denote the conditional GAN loss
functions for the profile and the frontal GANs, respectively, where $L_{PR}$
and $L_{FR}$ are given as:

$L_{PR}=F_{cGAN}(D_{PR},G_{PR},y^{i},x^{i}_{PR}),$ (10)
$L_{FR}=F_{cGAN}(D_{FR},G_{FR},y^{j},x^{j}_{FR}),$ (11)

where function $F_{cGAN}$ is the conditional GAN objective defined in (4). The
term $x^{i}_{PR}$ denotes the profile image used as a condition for the
profile GAN, and $y^{i}$ denotes the real profile image. Note that the real
profile image $y^{i}$ and the network condition given by $x^{i}_{PR}$ are the
same. Similarly, $x^{j}_{FR}$ denotes the frontal image used as a condition
for the frontal GAN and $y^{j}$ denotes the real frontal image. Again, the
real frontal image $y^{j}$ and the network condition given by $x^{j}_{FR}$ are
the same. The total loss for the coupled conditional GAN is given by:

$L_{GAN}=L_{PR}+L_{FR}.$ (12)

###  4.3 $L_{2}$ Reconstruction Loss

We also consider the $L_{2}$ reconstruction loss for both the profile GAN and
frontal GAN. The $L_{2}$ reconstruction loss measures the reconstruction error
in terms of the Euclidean distance between the reconstructed image and the
corresponding real image. Let $L_{2_{PR}}$ denote the reconstruction loss for
the profile GAN and is defined as:

$L_{2_{PR}}=\left\lVert G_{PR}(z|x^{i}_{PR})-y^{i}\right\rVert^{2}_{2},$ (13)

where $y^{i}$ is the ground truth profile image, $G_{PR}(z|x^{i}_{PR})$ is the
output of the profile generator.

Similarly, Let $L_{2_{FR}}$ denote the reconstruction loss for the frontal
GAN:

$L_{2_{FR}}=\left\lVert G_{FR}(z|x^{j}_{FR})-y^{j}\right\rVert^{2}_{2},$ (14)

where $y^{j}$ is the ground truth profile image, $G_{FR}(z|x^{j}_{FR})$ is the
output of the frontal generator.

The total $L_{2}$ reconstruction loss function is given by:

$L_{2}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}(L_{2_{PR}}+L_{2_{FR}}).$
(15)

###  4.4 Perceptual Loss

In addition to the GAN loss and the reconstruction loss which are used to
guide the generators, we have also used perceptual loss, which was introduced
in [Johnson2016PerceptualLF] for style transfer and super-resolution. The
perceptual loss function is used to compare high level differences, like
content and style discrepancies, between images. The perceptual loss function
involves comparing two images based on high-level representations from a
pretrained CNN, such as VGG-16 [simonyan2014very]. The perceptual loss
function is a good alternative to solely using $L_{1}$ or $L_{2}$
reconstruction error, as it gives better and sharper high quality
reconstruction images [Johnson2016PerceptualLF].

In our proposed approach, perceptual loss is added to both the profile and the
frontal module using a pre-trained VGG-16 [simonyan2014very] network. We
extract the high-level features (ReLU3-3 layer) of VGG-16 for both the real
input image and the reconstructed output of the U-Net generator. The $L_{1}$
distance between these features of real and reconstructed images is used to
guide the generators $G_{PR}$ and $G_{FR}$ . The perceptual loss for profile
network is defined as:

$\begin{split}L_{P_{PR}}=&\frac{1}{C_{p}W_{p}H_{p}}\sum_{c=1}^{C_{p}}\sum_{w=1}^{W_{p}}\sum_{h=1}^{H_{p}}\\\
&\left\lVert
V(G_{PR}(z|x^{i}_{PR}))^{c,w,h}-V(y^{i})^{c,w,h}\right\rVert,\end{split}$ (16)

where $V(.)$ denotes a particular layer of the VGG-16, where the layer
dimensions are given by $C_{p}$, $W_{p}$, and $H_{p}$.

Likewise the perceptual loss for frontal network is:

$\begin{split}L_{P_{FR}}=&\frac{1}{C_{p}W_{p}H_{p}}\sum_{c=1}^{C_{p}}\sum_{w=1}^{W_{p}}\sum_{h=1}^{H_{p}}\\\
&\left\lVert
V(G_{FR}(z|x^{j}_{FR}))^{c,w,h}-V(y^{j})^{c,w,h}\right\rVert.\end{split}$ (17)

The total perceptual loss function is given by:

$L_{P}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}(L_{P_{PR}}+L_{P_{FR}}).$
(18)

###  4.5 Overall Objective Function

The overall objective function for learning the network parameters in the
proposed method is given as the sum of all the loss functions defined above:

$\begin{split}L_{tot}=L_{cpl}+\lambda_{1}L_{GAN}+\lambda_{2}L_{P}+\lambda_{3}L_{2},\end{split}$
(19)

where $L_{cpl}$ is the coupling loss, $L_{GAN}$ is the total generative
adversarial loss, $L_{P}$ is the total perceptual loss, and $L_{2}$ is the
total reconstruction error. Variables $\lambda_{1},\lambda_{2},and\lambda_{3}$
are the hyper-parameters to weigh the different loss terms.

##  5 Experiments

We initially describe our training setup and the datasets that we have used in
our experiments. We show the efficiency of our method for the task of frontal
to profile face verification by comparing its performance with state-of the-
art face verification methods across pose-variation. Moreover, we explore the
effect of face yaw in our algorithm. Finally, we conduct an ablation study to
investigate the effect of each term in our total training loss defined in
(19).

###  5.1 Experimental Details

Datasets: The Celebrities in Frontal-Profile (CFP) dataset [sengupta_wacv_cfp]
is a mixture of constrained (i.e., carefully collected under different pose,
illumination and expression conditions) and unconstrained (i.e., collected
images from the Internet) settings. CFP includes 500 celebrities, averaging
ten frontal and four profile face images per each celebrity. Following the
standard 10-fold protocol [sengupta_wacv_cfp], we divide the dataset into 10
folds, each of which consists of 350 same and 350 different pairs generated
from 50 subjects (i.e., 7 same and 7 different pairs for each subject).

The CMU Multi-PIE database [CMU-PIE] contains 750,000 images of 337 subjects.
Subjects were imaged from 15 viewing angles and 19 illumination conditions
while exhibiting a range of facial expressions. It is the largest database for
graded evaluation with respect to pose, illumination, and expression
variations. For fair comparison, the database setting was made consistent with
FNM [FNM], where 250 subjects from Multi-PIE have been used. The training set
and testing split is consistent with FNM.

The IARPA Janus Benchmark A (IJB-A) [IJB-A] is a challenging dataset collected
under complete unconstrained conditions covering full pose variation (yaw
angles $-90\degree$ to $+90\degree$). IJB-A contains 500 subjects with 5,712
images and 20,414 frames extracted from videos. Following the standard
protocol in [IJB-A], we evaluate our method on both verification and
identification. The IARPA Janus Benchmark B (IJB-B) dataset [IJB-B] builds on
the IJB-A by adding more 1345 subjects making it a total of 1845 subjects, and
a total of 21,798 still images and 55,026 frames from 7,011 videos. The IARPA
Janus Benchmark C (IJB-C) dataset [IJB-C] builds on IJB-A, and IJB-B datasets
and has a total of 31,334 images for a total number of 3,531 subjects. We have
also evaluated our method on IJB-A and IJB-C datasets.

Implementation Details: We have implemented the U-Net with ResNet-18
[He2015_Resnet] encoder pre-trained on ImageNet. We have added an additional
fully-connected layer after the average pooling layer for ResNet-18 for our
U-Net encoder. The U-Net decoder has the same number of layers as the encoder.
The entire framework has been implemented in Pytorch. For convergence,
$\lambda_{1}$ is set to 1, and $\lambda_{2}$, and $\lambda_{3}$ are both set
to 0.25. We used a batch size of 128 and an Adam optimizer [Kingma2015AdamAM]
with first-order momentum of 0.5, and learning rate of 0.0004. We have used
the ReLU activation function for the generator and Leaky ReLU with a slope of
0.3 for the discriminator.

For training, genuine and impostor pairs were required. The genuine/impostor
pairs are created by frontal and profile images of the same/different subject.
During the experiments, we ensure that the training set are balanced by using
the same number of genuine and impostor pairs.

Table 1: Performance comparison on CFP dataset. Mean Accuracy and equal error
rate (EER) with standard deviation over 10 folds

| Frontal-Profile | Frontal-Frontal
---|---|---
Algorithm | Accuracy | EER | Accuracy | EER
HoG+Sub-SML [sengupta_wacv_cfp] | 77.31(1.61) | 22.20(1.18) | 88.34(1.31) | 11.45(1.35)
LBP+Sub-SML [sengupta_wacv_cfp] | 70.02(2.14) | 29.60(2.11) | 83.54(2.40) | 16.00(1.74)
FV+Sub-SML [sengupta_wacv_cfp] | 80.63(2.12) | 19.28(1.60) | 91.30(0.85) | 8.85(0.74)
FV+DML [sengupta_wacv_cfp] | 58.47(3.51) | 38.54(1.59) | 91.18(1.34) | 8.62(1.19)
Deep Features [deep_features] | 84.91(1.82) | 14.97(1.98) | 96.40(0.69) | 3.48(0.67)
PR-REM [Cao2018PoseRobustFR] | 93.25(2.23) | 7.92(0.98) | 98.10(2.19) | 1.10(0.22)
PF-cpGAN | 93.78(2.46) | 7.21(0.65) | 98.88(1.56) | 0.93(0.14)

###  5.2 Evaluation on CFP with Frontal-Profile Setting

We first perform evaluation on the Celebrities in Frontal-Profile (CFP)
dataset[sengupta_wacv_cfp], a challenging dataset created to examine the
problem of frontal to profile face verification in the wild. We follow the
standard 10-fold protocol[sengupta_wacv_cfp] in our evaluation. The same
protocol is applied on both the Frontal-Profile and Frontal-Frontal settings.
For fair comparison and as given in [sengupta_wacv_cfp], we consider different
types of feature extraction techniques like HoG[hog], LBP[lbp], and Fisher
Vector[fisher_vector_dml] along with metric learning techniques like Sub-
SML[sub_sml], and Diagonal metric learning (DML) as reported in
[fisher_vector_dml]. We also compare against deep learning techniques,
including Deep Features[deep_features], and PR-REM[Cao2018PoseRobustFR]. The
results are summarized in Table 1.

We can observe from Table 1 that our proposed framework, PF-cpGAN, gives much
better performance than the methods that use standard hand-crafted features of
HoG, LBP, or FV, providing minimum of $13\%$ improvement in accuracy with a
$12\%$ decrease in EER for the profile-frontal setting. PF-cpGAN also improves
on the performance of the Deep Features by approximately $9\%$ with a $7.5\%$
decrease in EER for the profile-frontal setting. Finally, PF-cpGAN performs
on-par with the best deep learning method of PR-REM, and, in-fact, does
slightly better than PR-REM by $\approx 0.5\%$ improvement in accuracy with a
$0.7\%$ decrease in EER for the profile-frontal setting. This performance
improvement clearly shows that usage of a GAN framework for projecting the
profile and frontal images in the latent embedding subspace and maintaining
the sematic similarity in the latent space is better than some other deep
learning techniques such as Deep Features or PR-REM.

Table 2: Performance comparison on IJB-A benchmark. Results reported are the
’average$\pm$standard deviation’ over the 10 folds specified in the IJB-A
protocol. Symbol ’-’ indicates that the metric is not available for that
protocol

Method | Verification | Identification
---|---|---
GAR@ FAR$=0.01$ | GAR@ FAR$=0.001$ | @ Rank-1 | @ Rank-5
OPENBR [klare2015pushing] | $23.6\pm 0.9$ | $10.4\pm 1.4$ | $24.6\pm 1.1$ | $37.5\pm 0.8$
GOTS [klare2015pushing] | $40.6\pm 1.4$ | $19.8\pm 0.8$ | $43.3\pm 2.1$ | $59.5\pm 2.0$
PAM [Masi_pose_aware_2016] | $73.3\pm 1.8$ | $55.2\pm 3.2$ | $77.1\pm 1.6$ | $88.7\pm 0.9$
DCNN [chen2016unconstrained] | $78.7\pm 4.3$ | - | $85.2\pm 1.8$ | $93.7\pm 1.0$
DR-GAN [tran2017disentangled] | $77.4\pm 2.7$ | $53.9\pm 4.3$ | $85.5\pm 1.5$ | $94.7\pm 1.1$
FF-GAN [yin2017towards] | $85.2\pm 1.0$ | $66.3\pm 3.3$ | $90.2\pm 0.6$ | $95.4\pm 0.5$
FNM [FNM] | $93.4\pm 0.9$ | $83.8\pm 2.6$ | $96.0\pm 0.5$ | $98.6\pm 0.3$
PR-REM [Cao2018PoseRobustFR] | $94.4\pm 0.9$ | $86.8\pm 1.5$ | $94.6\pm 1.1$ | $96.8\pm 1.0$
PF-cpGAN | $95.8\pm 0.82$ | $91.2\pm 1.3$ | $97.6\pm 1.0$ | $98.8\pm 0.4$

(a) IJBA

(b) CMU Multi-PIE

(c) Ablation Study

Figure 3: ROC curve comparison against the baselines for different datasets is
shown in (a) and (b). In (c), we show the ROC curves showing the importance of
different loss functions for ablation study.

###  5.3 Evaluation on IJB-A and IJB-C

Figure 4: Reconstruction of frontal images at the output of the frontal U-Net
generator with profile images as input to the profile U-Net generator. Every
odd number column represent the input profile image and every even number
column represents the output frontal image. The input images belong to the
CMU-MultiPIE dataset.

Here, we focus on unconstrained face recognition on IJB-A dataset to quantify
the superiority of our PF-cpGAN for profile to frontal face recognition. Some
of the baselines for comparison on IJB-A are DR-GAN
[Tran_disentangled_cvpr_17], FNM [FNM], PR-REM [Cao2018PoseRobustFR],and FF-
GAN [Yin2017TowardsLF]. We have also compared them with other methods as
listed in [FNM] and shown in Table 2. As shown in Table 2, we perform better
than the state-of-the-art methods for both verification and identification.
Specifically, for verification, we improve the genuine accept rate (GAR) by at
least $1.4\%$ compared to other methods. For instance, at the false accept
rate (FAR) of 0.01, the best previously-used method is PR-REM, with an average
GAR of $94.4\%$. PF-cpGAN improves upon PR-REM and gives an average GAR of
$95.8\%$ at the same FAR. We also show improvement on identification.
Specifically, the rank-1 recognition rate shows an improvement of around
$1.6\%$ in comparison to the best state-of-the-art method, FNM [FNM].

Table 3: Performance comparison on IJB-C benchmark. Results reported are the
’average$\pm$standard deviation’ over the 10 folds specified in the IJB-C
protocol. Symbol ’-’ indicates that the metric is not available for that
protocol

Method | Verification | Identification
---|---|---
GAR@ FAR$=0.01$ | GAR@ FAR$=0.001$ | @ Rank-1 | @ Rank-5
GOTS [IJB-C] | $62.1\pm 1.1$ | $36.3\pm 1.2$ | $38.5\pm 1.6$ | $53.8\pm 1.8$
FaceNet [facenet] | $82.3\pm 1.18$ | $66.3\pm 1.3$ | $70.4\pm 1.2$ | $78.8\pm 2.3$
VGG-CNN [Vgg_CNN] | $87.2\pm 1.09$ | $74.3\pm 0.9$ | $79.6\pm 1.04$ | $87.8\pm 1.3$
FNM [FNM] | $91.2\pm 0.8$ | $80.4\pm 1.8$ | $84.6\pm 0.6$ | $93.7\pm 0.9$
PR-REM [Cao2018PoseRobustFR] | $92.1\pm 0.8$ | $83.4\pm 1.5$ | $83.1\pm 0.4$ | $92.6\pm 1.1$
PF-cpGAN | $93.8\pm 0.67$ | $86.1\pm 0.7$ | $88.3\pm 1.2$ | $94.8\pm 0.6$

We have also plotted receiver operating characteristic (ROC) curve and
compared with the baselines given above. The ROC curves for the IJB-A dataset
are given in Fig. 3(a). As we can clearly see from the curves, the proposed
PF-cpGAN method improves upon other methods and gives much better performance,
even at a FAR of 0.001.

We have also performed the task of verification and identification using the
IJB-C dataset according to the verification and the identification protocol
given in the dataset. The results are provided in Table 3, showing that PF-
cpGAN improve on the existing state-of-the-art methods for both verification
and identification. For instance, at the false accept rate (FAR) of 0.01, the
best previously-used method is PR-REM, with an average GAR of $92.1\%$. PF-
cpGAN improves upon PR-REM and gives an average GAR of $93.8\%$ at the same
FAR. We also observe that, for identification, specifically, rank-1
recognition, shows an improvement over the previous best state-of-the-art
method FNM [FNM] by about $1.1\%$.

Figure 5: Reconstruction of profile images at the output of the profile U-Net
generator with frontal images as input to the frontal U-Net generator. Every
odd number column represents the input frontal image, and every even number
column represents the output profile image. The input images belong to the
CMU-MultiPIE dataset.

###  5.4 A Further Analysis on Influences of Face Yaw

In addition to complete profile to frontal face recognition, we also perform a
more in-depth analysis on the influence of face yaw angle on the performance
of face recognition to better understand the effectiveness of the PF-cpGAN for
profile to frontal face recognition. We perform this experiment for the CMU
Multi-PIE dataset [CMU-PIE] under setting-1 for fair comparison with other
state-of-the-art methods. As shown in Table 4, we achieve comparable
performance with other state-of-the-art methods for different yaw angles.
Under extreme pose, PF-cpGAN achieves significant improvements (i.e., approx.
$77\%$ to $88\%$ under $\pm 90\degree$).

Table 4: Rank-1 recognition rates ($\%$) across poses and illuminations under
Multi-PIE Setting-1

Method | $\pm 90\degree$ | $\pm 75\degree$ | $\pm 60\degree$ | $\pm 45\degree$ | $\pm 30\degree$ | $\pm 15\degree$
---|---|---|---|---|---|---
HPN [ding2017pose] | $29.82$ | $47.57$ | $61.24$ | $72.77$ | $78.26$ | $84.23$
c-CNN [xiong2015conditional] | $47.26$ | $60.7$ | $74.4$ | $89.0$ | $94.1$ | $97.0$
TP-GAN [huang2017beyond] | $64.0$ | $84.1$ | $92.9$ | $98.6$ | $99.99$ | $99.8$
PIM [zhao2018towards] | $75.0$ | $91.2$ | $97.7$ | $98.3$ | $99.4$ | $99.8$
CAPG-GAN [hu2018pose] | $77.1$ | $87.4$ | $93.7$ | $98.3$ | $99.4$ | $99.99$
FNM$+$VGG-Face [FNM] | $41.1$ | $67.3$ | $83.6$ | $93.6$ | $97.2$ | $99.0$
FNM$+$Light CNN [FNM] | $55.8$ | $81.3$ | $93.7$ | $98.2$ | $99.5$ | $99.9$
PF-cpGAN | $88.1$ | $94.2$ | $97.6$ | $98.9$ | $99.9$ | $99.9$

For further testing on the Multi-PIE dataset under setting-1, we have also
plotted ROC curves and compared with other state-of-the-art methods. The ROC
curves for Multi-PIE dataset are given in Fig. 3(b). The curves clearly
indicate that the proposed method of PF-cpGAN improves upon other methods and
gives much better performance, even at FAR of 0.001.

###  5.5 Reconstruction of frontal and profile images

As noted in Sec. 1, the PF-cpGAN framework can also be used for reconstruction
of frontal images by using profile images as input and vice versa. The results
of reconstructing frontal images using the profile images as input are given
in Fig. 4, and the results of reconstructing profile images using the frontal
images as input is given in Fig. 5. The reconstruction procedure for frontal
images is given as follows: The profile image is given as input to the profile
U-Net generator and the feature vector generated at the bottleneck of the
profile generator (i.e., at the output of the encoder of the profile U-Net
generator) is passed through the decoder section of the frontal U-Net
generator to reconstruct the frontal image. Similarly the reconstruction
procedure for profile images is given as follows: The frontal image is given
as input to the frontal U-Net generator and the feature vector generated at
the bottleneck of the frontal generator (i.e., at the output of the encoder of
the frontal U-Net generator) is passed through the decoder section of the
profile U-Net generator to reconstruct the profile image. As we can see from
Fig. 4 and Fig. 5, the PF-cpGAN can preserve the identity and generate high-
fidelity faces from an unconstrained dataset such as CMU-MultiPIE. These
results show the robustness and effectiveness of PF-cpGAN for multiple use of
profile to frontal matching in the latent common embedding subspace, as well
as in the reconstruction of facial images.

###  5.6 Ablation Study

The objective function defined in (19) contains multiple loss functions:
coupling loss ($L_{cpl}$), perceptual loss ($L_{P}$), $L_{2}$ reconstruction
loss ($L_{2}$), and GAN loss ($L_{GAN}$). It is important to understand the
relative importance of different loss functions and the benefit of using them
in our proposed method. For this experiment, we use different variations of
PF-cpGAN and perform the evaluation using the IJB-A dataset. The variations
are: 1) PF-cpGAN with only coupling loss and $L_{2}$ reconstruction loss
($L_{cpl}+L_{2}$); 2) PF-cpGAN with coupling loss, $L_{2}$ reconstruction
loss, and GAN loss ($L_{cpl}+L_{2}+L_{GAN}$); 3) PF-cpGAN with all the loss
functions ($L_{cpl}+L_{2}+L_{GAN}+L_{P}$).

We use these three variations of our framework and plot the ROC for profile to
frontal face verification using the features from the common embedding
subspace. We can see from Fig. 3(c) that the generative adversarial loss helps
improve the profile to frontal verification performance, and adding the
perceptual loss (blue curve) results in an additional performance improvement.
The reason for this improvement is that using perceptual loss along with the
contrastive loss leads to a more discriminative embedding subspace leading to
a better face recognition performance.

##  6 Conclusion

We proposed a new framework which uses a coupled GAN for profile to frontal
face recognition. The coupled GAN contains two sub-networks which project the
profile and frontal images into a common embedding subspace, where the goal of
each sub-network is to maximize the pair-wise correlation between profile and
frontal images during the process of projection. We thoroughly evaluated our
model on several standard datasets and the results demonstrate that our model
notably outperforms other state-of-the-art algorithms for profile to frontal
face verification. Moreover, the improvement achieved by different losses in
our proposed algorithm has been studied in an ablation study.


# Asymptotic expansions of finite Hankel transforms and the surjectivity of
convolution operators††thanks: 2010 Mathematics Subject Classification:
primary 45E10; secondary 33C10, 44A15

Yasunori Okada and Hideshi Yamane The first author is supported by JSPS
KAKENHI Grant Number 21K03265.

###### Abstract

A compactly supported distribution is called invertible in the sense of
Ehrenpreis-Hörmander if the convolution with it induces a surjection from
$\mathcal{\mathcal{C}^{\infty}}(\mathbb{R}^{n})$ to itself. We give sufficient
conditions for radial functions to be invertible. Our analysis is based on the
asymptotic expansions of finite Hankel transforms. The dominant term may be
the contribution from the origin or from the boundary of the support of the
function. For the proof, we propose a new method to calculate the asymptotic
expansions of finite Hankel transforms of functions with singularities at a
point other than the origin.

The first author:
Graduate School of Science, Chiba University,
Yayoicho 1-33, Inage-ku, Chiba, 263-8522, Japan
E-mail address<EMAIL_ADDRESS>
The second author:
Department of Mathematical Sciences, Kwansei Gakuin University,
Uegahara, Gakuen, Sanda, Hyogo, 669-1330, Japan
E-mail address<EMAIL_ADDRESS>
##  1 Introduction

It is known that
$P(D)\colon\mathcal{C}^{\infty}(\mathbb{R}^{n})\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$
and
$P(D)\colon\mathcal{D}^{\prime}(\mathbb{R}^{n})\to\mathcal{D}^{\prime}(\mathbb{R}^{n})$
are both surjective, where $P(D)\neq 0$ is an arbitrary linear partial
differential operator with constant coefficients ([5, 6, 13]). Since these
mappings coincide with the convolution operator $P(D)\delta*$ on respective
spaces, a natural question is to characterize compactly supported
distributions that induce surjective convolution operators. Such a
distribution is called invertible. Notice that $\delta(x-a)$ induces
translation and hence is invertible. It was found that a compactly supported
distribution is invertible if and only if its Fourier transform, an entire
function, is slowly decreasing in a certain sense. This condition was found by
Ehrenpreis ([7]) and was further studied by Hörmander ([10]). A self-contained
account can be found in [11]. Invertible distributions and slowly decreasing
entire functions became and still are fundamental concepts in the research of
convolution equations. Perturbation of invertible distributions are studied in
[14], and a recent paper [4] investigates invertible distributions in an
abstract setting.

Slow decrease is a technical estimate from below and is not easy to grasp. The
present authors expect that the notions of slow decrease and of invertibility
will become less mysterious if many examples or sufficient conditions are
found. The delta function $\delta_{S(0,r)}$ supported on a sphere is
invertible ([12]). Its normal derivatives are invertible as well ([16]). In
both [12] and [16], the asymptotic behavior of Bessel functions were used. As
is well-known ([17]), the Fourier transform of a radial function can be
written in terms of an integral involving a Bessel function. The Fourier
transforms of $\delta_{S(0,r)}$ and its normal derivatives are limit cases.

In the present article, we prove the invertibility of some radial functions by
using two methods of calculating asymptotic expansions of finite Hankel
transforms $\int_{0}^{1}\varphi(s)J_{\nu}(rs)\,ds$. One is a result in Wong
[19], in which singularities at $s=0$ determine the asymptotic behaviors. The
other method, devised by the present authors, is useful to deal with
singularities at $s=1$. As far as they know, this is the first result about
the asymptotic behaviors of Hankel transforms of functions singular at a point
other than $s=0$.

##  2 Invertibility and slow decrease

In this article, we follow the convention

$\hat{f}(\xi)=\int_{\mathbb{R}^{n}}e^{-i\xi\cdot
x}f(x)\,dx\;\left(\xi\in\mathbb{R}^{n}\right)$

for the definition of the Fourier transform of a compactly supported
distribution $f\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$. When $f(x)\in
L^{1}(\mathbb{R}^{n})$ is a radial function, its Fourier transform can be
written in terms of an integral involving a Bessel function.

######  Theorem 2.1 ([17, p.155]).

Let $f_{0}(s),\,s>0$, be a function of a single variable. The Fourier
transform of $f(x)=f_{0}(|x|),\,x\in\mathbb{R}^{n},$ is written in terms of a
Hankel transform. More precisely,

(2.1)
$\displaystyle\hat{f}(\xi)=\frac{(2\pi)^{n/2}}{r^{n/2-1}}\int_{0}^{\infty}s^{n/2}f_{0}(s)J_{n/2-1}(rs)\,ds,\quad$
$\displaystyle r=|\xi|,\;\xi\in\mathbb{R}^{n}.$

Here $J_{n/2-1}(z)$ denotes the Bessel function of the first kind of order
$n/2-1$.

######  Remark 2.2.

Notice that $z^{-(n/2-1)}J_{n/2-1}(sz)$ is an even entire function in a single
variable $z$ for each $s>0$. If the support of $f_{0}(s)$ is bounded, then the
Fourier transform of $f(x)=f_{0}(|x|)$ is an even entire function which is
expressed by

$\hat{f}(\zeta)=(2\pi)^{n/2}\int_{0}^{\infty}s^{n/2}f_{0}(s)\left(\sqrt{\zeta^{2}}\right)^{-(n/2-1)}J_{n/2-1}\left(s\sqrt{\zeta^{2}}\right)\,ds,$

where $\zeta=(\zeta_{1},\dots,\zeta_{n})\in\mathbb{C}^{n}$,
$\zeta^{2}=\sum_{j=1}^{n}\zeta_{j}^{2}$.

We employ the notions of invertibility and slow decrease originated by
Ehrenpreis ([7]) and refined by Hörmander ([10]).

######  Theorem 2.3 ([7, Theorem 2.2, Proposition 2.7], [10, Definition 3.1,
Corollary 3.1], [11, Theorem 16.3.9, 16.3.10]).

For $u\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$, the following statements are
equivalent.

(i) There is a constant $A>0$ such that we have

(2.2)
$\sup\left\\{|\hat{u}(\zeta)|\,;\,\zeta\in\mathbb{C}^{n},|\zeta-\xi|<A\log(2+|\xi|)\right\\}>(A+|\xi|)^{-A}$

for any $\xi\in\mathbb{R}^{n}$.

(ii) There is a constant $A>0$ such that we have

(2.3)
$\sup\left\\{|\hat{u}(\eta)|\,;\,\eta\in\mathbb{R}^{n},|\eta-\xi|<A\log(2+|\xi|)\right\\}>(A+|\xi|)^{-A}$

for any $\xi\in\mathbb{R}^{n}$.

(iii) If $w\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and $\hat{w}/\hat{u}$ is
an holomorphic function, then $\hat{w}/\hat{u}$ is the Fourier transform of a
distribution in $\mathcal{E}^{\prime}(\mathbb{R}^{n})$.

(iv) If $v\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ satisfies
$u*v\in\mathcal{\mathcal{C}}^{\infty}(\mathbb{R}^{n})$, then
$v\in\mathcal{\mathcal{C}}^{\infty}(\mathbb{R}^{n})$.

(v) The mapping
$u*\colon\mathcal{\mathcal{C}}^{\infty}(\mathbb{R}^{n})\to\mathcal{\mathcal{C}}^{\infty}(\mathbb{R}^{n})$
is surjective.

(vi) The mapping
$u*\colon\mathcal{D}^{\prime}(\mathbb{R}^{n})\to\mathcal{D}^{\prime}(\mathbb{R}^{n})$
is surjective.

######  Remark 2.4.

In the present paper, we only deal with surjectivity on $\mathbb{R}^{n}$. If
one wants to study surjectivity on subsets of $\mathbb{R}^{n}$, one needs the
notions of $\mu$-convexity for supports or singular supports ([11]).

######  Definition 2.5 ([11, Definition 16.3.12]).

An element $u(x)$ of $\mathcal{E}^{\prime}(\mathbb{R}^{n})$ is called
invertible and its Fourier transform $\hat{u}(\zeta)$ is called slowly
decreasing if the equivalent conditions in Theorem 2.3 are fulfilled. See
Definition 2.8 below.

######  Remark 2.6.

A finitely supported non-zero distribution is invertible ([10, Theorem 4.4]).
The most important example is $P(D)\delta(x)$, where $P(D)\neq 0$ is a linear
partial differential operator with constant coefficients. It give rise to
$P(D)\delta(x)*$, which is nothing but $P(D)$. So
$P(D)\colon\mathcal{D}^{\prime}(\mathbb{R}^{n})\to\mathcal{D}^{\prime}(\mathbb{R}^{n})$
and
$P(D)\colon\mathcal{C}^{\infty}(\mathbb{R}^{n})\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$
are surjective.

Other classes of invertible distributions are discussed in [1]. Let $\mu$ be a
compactly supported measure with an atom,
$\nu\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ have singular support disjoint
from that of $\mu$ and $P(D)$ be a non-zero linear partial differential
operator with constant coefficients. Then $P(D)\mu+\nu$ is invertible.

Another useful fact in [1] is the following. Let
$\mu\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$,
$f\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$. If $f$ is real analytic in a
neighborhood of $\mathrm{singsupp}\,\mu$ and $f\mu$ is invertible, then $\mu$
is invertible. In [12] and [16], the spherical mean value operator and its
variants are discussed from the view point of invertibility.

######  Proposition 2.7.

(i) If $u,v\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ are invertible, then so is
$u*v$.

(ii) Let $\alpha\in\mathbb{R}\setminus\left\\{0\right\\}$. If
$u(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ is invertible, so is $u(\alpha
x)$.

(iii) Let $a\in\mathbb{R}^{n}$. If
$u(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ is invertible, so is $u(x-a)$.

(iv) If $u(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ is invertible, then so
is $\sum_{j=1}^{J}P_{j}(D)u(x-a_{j})$, where $P_{j}(D)$ is a non-zero linear
partial differential operator with constant coefficients and
$a_{j}\in\mathbb{R}^{n}$.

(v) Let $u\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and
$v\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{n})$. Then, $u+v$ is invertible if
and only if $u$ is.

(vi) If $u(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{m})$ and
$v(x^{\prime})\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ are invertible, then so
is $u(x)v(x^{\prime})\in\mathcal{E}^{\prime}(\mathbb{R}^{m+n})$***This can be
proved in an alternative manner using general facts about the tensor product
of Fréchet nuclear topological vector spaces. .

###### Proof.

(i) If $u,v\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ both satisfy (v) in
Theorem 2.3, then so does $v*u$.

(ii) For any $g(x)$, there exists $f(x)$ such that
$\int_{\mathbb{R}^{n}}u(x-y)f(y)\,dy=|\alpha|^{n}g(x/\alpha)$, hence
$\int_{\mathbb{R}^{n}}u(\alpha x-y)f(y)\,dy=|\alpha|^{n}g(x)$. Set $y=\alpha
z$. Then $\int_{\mathbb{R}^{n}}u(\alpha(x-z))f(\alpha z)\,dz=g(x)$. Therefore
$u(\alpha x)*$ is surjective.

(iii) We obtain $\int_{\mathbb{\mathbb{R}}^{n}}u(x-a-y)f(y)\,dy=g(x)$ from
$\int_{\mathbb{\mathbb{R}}^{n}}u(x-y)f(y)\,dy=g(x+a)$.

(iv) Since $\mu=\sum_{j=1}^{J}P_{j}(D)\delta(x-a_{j})$ is invertible, so is
$\mu*u$.

(v) Use the Paley-Wiener-Schwartz Theorem and Theorem 2.3 (ii).

(vi) For $\xi\in R,\,\xi^{\prime}\in\mathbb{R}^{n}$, set

$\displaystyle
S_{\xi,A}=\left\\{\eta\in\mathbb{R}^{m};\,|\eta-\xi|<A\log(2+|\xi|)\right\\},$
$\displaystyle
S_{\xi^{\prime},A}=\left\\{\eta^{\prime}\in\mathbb{R}^{n};\,|\eta^{\prime}-\xi^{\prime}|<A\log(2+|\xi^{\prime}|)\right\\},$
$\displaystyle
S_{\xi,\xi^{\prime},2A}=\left\\{(\eta,\eta^{\prime})\in\mathbb{R}^{m+n};\,|(\eta,\eta^{\prime})-(\xi,\xi^{\prime})|<2A\log(2+|(\xi,\xi^{\prime})|)\right\\}.$

We have $S_{\xi,A}\times S_{\xi^{\prime},A}\subset S_{\xi,\xi^{\prime},2A}$.

By assumption, there exists $A>0$ such that the supremum of
$\left|\hat{u}(\eta)\right|$ on $S_{\xi,A}$ and that of
$\left|\hat{v}(\eta^{\prime})\right|$ on $S_{\xi^{\prime},A}$ are estimated
from below by $(A+|\xi|)^{-A}$ and $(A+|\xi^{\prime}|)^{-A}$ respectively for
any ($\xi,\xi^{\prime})$. The supremum of
$\left|\hat{u}(\eta)\hat{v}(\eta^{\prime})\right|$ on $S_{\xi,A}\times
S_{\xi^{\prime},A}$ on $S_{\xi,\xi^{\prime},2A}$ is not smaller than that on
$S_{\xi,A}\times S_{\xi^{\prime},A}$ and is estimated from below by
$(A+|\xi|)^{-A}(A+|\xi^{\prime}|)^{-A}>(A+|(\xi,\xi^{\prime})|)^{-2A}>(2A+|(\xi,\xi^{\prime})|)^{-2A}$.
∎

######  Definition 2.8.

In the present paper, we extend the terminology of Definition 2.5 slightly: an
entire function $p(\zeta)$, not necessarily the Fourier transform of a
compactly supported distribution, is called slowly decreasing if there is a
constant $A>0$ such that we have

(2.4)
$\sup\left\\{|p(\eta)|\,;\,\eta\in\mathbb{R}^{n},|\eta-\xi|<A\log(2+|\xi|)\right\\}>(A+|\xi|)^{-A}$

for any $\xi\in\mathbb{R}^{n}$.

######  Proposition 2.9.

Let $p(\zeta)$ be an entire function. There are constants $A,B>0$ such that

(2.5)
$\sup\left\\{|p(\eta)|\,;\,\eta\in\mathbb{R}^{n},|\eta-\xi|<A\log(2+|\xi|)\right\\}>(A+|\xi|)^{-A}$

for any $\xi\in\mathbb{R}^{n}$ satisfying $|\xi|\geq B$, if and only if
$p(\zeta)$ is slowly decreasing in the sense of Definition 2.8.

###### Proof.

The “if” part is trivial. We prove the “only if” part. If $|\xi_{0}|=B$, we
set

$S(\xi_{0})=\left\\{\eta\in\mathbb{R}^{n};\,|\eta-\xi_{0}|<A\log(2+B)\right\\}.$

By (2.5),

(2.6) $\sup\left\\{|p(\eta)|;\,\eta\in S(\xi_{0})\right\\}>(A+B)^{-A}.$

On the other hand, we set

$S=\left\\{\eta\in\mathbb{R}^{n};\,|\eta|<B+A\log(2+B)\right\\}\supset
S(\xi_{0}).$

Here we may assume $A$ is so large that $A\log(2+B)>B.$ Then

(2.7) $\cup_{|\xi_{0}|=B}S(\xi_{0})=S.$

By (2.6) and (2.7), we have

(2.8) $\sup\left\\{|p(\eta)|;\,\eta\in S\right\\}>(A+B)^{-A}.$

Assume $|\xi|\leq B$ and set

$S_{\xi}=\left\\{\eta\in\mathbb{R}^{n};\,|\eta-\xi|\leq\frac{2B+A\log(2+B)}{\log
2}\log(2+|\xi|)\right\\}.$

Since

$\displaystyle S$
$\displaystyle\subset\left\\{\eta\in\mathbb{R}^{n};\,|\eta-\xi|<2B+A\log(2+B)\right\\}\subset
S_{\xi},$

(2.8) implies

$\sup\left\\{|p(\eta);\,\eta\in
S_{\xi}\right\\}>(A+B)^{-A}>(A+B+|\xi|)^{-(A+B)}.$

We set

$\tilde{A}=\max\left\\{\frac{2B+A\log(2+B)}{\log 2},\,A+B\right\\},$

then (2.4) holds if we replace $A$ with $\tilde{A}$. ∎

######  Proposition 2.10.

Let $p(z)$ be an even entire function of a single variable. If $p(z)$ is
slowly decreasing, then $p\left(\sqrt{\zeta^{2}}\right)$ is a slowly
decreasing function in $\zeta\in\mathbb{C}^{n}$.

###### Proof.

Notice that $p\left(\sqrt{\zeta^{2}}\right)$ is well-defined since $p(z)$ is
even. There are constants $A,B>0$ such that we have

(2.9)
$\sup\left\\{|p(y)|\,;\,y\in\mathbb{R},|y-x|<A\log(2+x)\right\\}>(A+x){}^{-A}$

for any $x\in\mathbb{R}$ satisfying $x\geq B$. For any $\xi\in\mathbb{R}^{n}$,
set $x=|\xi|$. The radial function
$p\left(|\eta|\right),\eta\in\mathbb{R}^{n},$ satisfies the condition in
Proposition 2.9, since the combination of

$\displaystyle\left\\{p\left(|\eta|\right)\,;\,\eta\in\mathbb{\mathbb{R}}^{n},|\eta-\xi|<A\log(2+|\xi|)\right\\}$
$\displaystyle=\left\\{p\left(|\eta|\right)\,;\,\eta\in\mathbb{R}^{n},\left||\eta|-|\xi|\right|<A\log(2+|\xi|)\right\\}$
$\displaystyle=\left\\{p\left(y\right)\,;\,y\in\mathbb{R},|y-x|<A\log(2+x)\right\\}$

and (2.9) implies

$\displaystyle\sup\left\\{p\left(|\eta|\right)\,;\,\eta\in\mathbb{\mathbb{R}}^{n},|\eta-\xi|<A\log(2+|\xi|)\right\\}$
$\displaystyle>(A+x)^{-A}=(A+|\xi|)^{-A}$

if $x=|\xi|\geq B$. ∎

######  Proposition 2.11.

Let $p(z)$ be an even entire function of a single variable. Set
$q(x)=x^{\alpha}p(x)$ for $x>0,\,\alpha\geq 0$. Assume there is a sufficiently
small constant $C>0$ and sufficiently large constants $A,B>0$ such that we
have

(2.10) $\sup\left\\{|q(y)|\,;\,y>0,|y-x|<B\right\\}>Cx{}^{-A}$

for any $x\geq B$. Then $p\left(\sqrt{\zeta^{2}}\right)$ is a slowly
decreasing entire function in
$\zeta=(\zeta_{1},\dots,\zeta_{n})\in\mathbb{C}^{n}$, where
$\zeta^{2}=\sum_{j=1}^{n}\zeta_{j}^{2}$.

###### Proof.

The assertion is trivial when $\alpha=0$ (Proposition 2.10). We have only to
prove the case $\alpha>0$. By choosing a larger $B$ if necessary, we may
assume that

(2.11) $2^{-\alpha}C>B^{-\alpha}.$

If $y>0,\,|y-x|<B$, we have $y^{\alpha}<(x+B)^{\alpha}<(2x)^{\alpha}$ and

(2.12) $|p(y)|=|y^{-\alpha}q(y)|>(2x)^{-\alpha}|q(y)|.$

By (2.10), (2.11) and (2.12), we have

$\displaystyle\sup\left\\{|p(y)|;\,y>0,\,|y-x|<B\right\\}$
$\displaystyle>(2x)^{-\alpha}Cx^{-A}>B^{-\alpha}x^{-(A+\alpha)}$
$\displaystyle>x^{-(A+2\alpha)}>(A+2\alpha+x)^{-(A+2\alpha)}.$

On the other hand, $B<A\log(2+x)$ holds if $B/\log 2\leq A$. We see that
$p(z)$ is slowly decreasing since (2.9) is valid if we adopt
$\max\left\\{A+2\alpha,B/\log 2\right\\}$ as a new value of $A$. Apply
Proposition 2.10 to complete the proof. ∎

##  3 Asymptotic expansion and invertibility

First, we utilize the results of [19] to consider the contribution of the
singularities at $s=0$ to Hankel transforms.

######  Proposition 3.1.

Let $\mu,\,\nu\in\mathbb{C}$ and $\varphi(s)$ be an infinitely differentiable
function in $(0,1)$. We make the following assumptions.

(3.1) $\displaystyle\mathrm{Re\,}(\mu+\nu)>-1,$ (3.2)
$\displaystyle{\displaystyle\varphi^{(j)}(s)\sim\sum_{k=0}^{\infty}c_{k}\frac{d^{j}}{ds^{j}}s^{\mu+k}}\quad(s\to+0;\,j=0,1,2,\dots),$
(3.3) $\displaystyle c_{0}\neq 0.$

Moreover, let $\chi_{0}(s)$ be an infinitely differentiable function such that
$\chi_{0}(s)=1$ in $0<s\leq\varepsilon$ and $\chi_{0}(s)=0$ in
$1-\varepsilon\leq s<1$, where $0<\varepsilon<1/3$. We define the set
$K=K\left(\mu,\nu,\left\\{c_{k}\right\\}_{k}\right)$ by

(3.4) $\displaystyle K$
$\displaystyle=K\left(\mu,\nu,\left\\{c_{k}\right\\}_{k}\right)$
$\displaystyle=\left\\{k\in\mathbb{N}_{0}\,;\,c_{k}\neq
0,\;\frac{1}{2}(\mu+k-\nu-1)\not\in\mathbb{N}_{0}\right\\},$

where $\mathbb{N}_{0}$ is the set of nonnegative integers. Then, we have the
following.

If $K\neq\emptyset$,

$\displaystyle\int_{0}^{1}\chi_{0}(s)\varphi(s)J_{\nu}(rs)\,ds$
$\displaystyle=c_{k_{0}}\frac{\Gamma\left(\frac{1}{2}(\mu+k_{0}+\nu+1)\right)2^{\mu+k_{0}}}{\Gamma\left(\frac{1}{2}(-\mu-
k_{0}+\nu+1)\right)r^{\mu+k_{0}+1}}+o(r^{-\mathrm{Re\,}(\mu+k_{0}+1)})$

as $r\to\infty$, where $k_{0}=\min K$.

If $K=\emptyset$,

(3.5) $\int_{0}^{1}\chi_{0}(s)\varphi(s)J_{\nu}(rs)\,ds=o(r^{-A})$

as $r\to\infty$, where $A$ is an arbitrarily large real number.

###### Proof.

We employ [19, (3.6), (4.2)]. Notice that the terms corresponding to $k\not\in
K$ vanish since $1/\Gamma(-n)=0,\,n\in\mathbb{N}_{0}$. ∎

######  Proposition 3.2.

Assume $\mathrm{Re\,}(2\nu)>-2$. If $\varphi(s)$ is an infinitely
differentiable function in $(0,\infty)$, its support is bounded and
$\varphi(s)=s^{\nu+1}(1-s^{2})^{\alpha},\;\alpha\in\mathbb{C}$ near $s=0$,
then

$\int_{0}^{\infty}\varphi(s)J_{\nu}(rs)\,ds=o(r^{-A}),$

where $A$ is an arbitrarily large real number.

###### Proof.

We employ [19, (3.6), (4.2)] again. ∎

Next we consider the contribution of the singularities at $s=1$ to finite
Hankel transforms.

######  Proposition 3.3.

Let $N$ be a non-negative integer and $\Lambda$ be a complex number with
$\mathrm{Re\,}\Lambda\geq N$. Assume that $\psi(t)$ is an infinitely
differentiable function in $0<t<1$ such that $\psi(t)=0$ in $0<t<\varepsilon$
and that $\psi^{(k)}(t)$ is integrable for $0\leq k\leq N$. Set
$\phi(t)=(1-t)^{\Lambda}\psi(t)$. Then $\phi(s^{2})$ is an infinitely
differentiable function in $0<s<1$ and we have

$\int_{0}^{1}s^{\nu+1}\phi(s^{2})J_{\nu}(rs)\,ds=o(r^{-(N+1/2)})$

as $r\to\infty$.

###### Proof.

We have

$\phi^{(k)}(t)=\sum_{j=0}^{k}\binom{k}{j}\Lambda(\Lambda-1)\cdots(\Lambda-j+1)(-1)^{j}(1-t)^{\Lambda-j}\psi^{(k-j)}(t).$

Assume $N\geq 1$. We have $\phi^{(k)}(1-0)=0$ if $k\leq N-1$. Set

$I_{k}=\int_{0}^{1}s^{\nu+k+1}\phi^{(k)}(s^{2})J_{\nu+k}(rs)\,ds\quad(k=0,1,2,\dots,N).$

The recurrence relation
$\left(z^{\nu+1}J_{\nu+1}(z)\right)^{\prime}=z^{\nu+1}J_{\nu}(z)$ (see for
example [15, 10.6.6]) yields

(3.6)
$\frac{d}{ds}\left\\{s^{\nu+1}J_{\nu+1}(rs)\right\\}=rs^{\nu+1}J_{\nu}(rs).$

Combining this formula and integration by parts, we get

(3.7) $\displaystyle I_{k}$
$\displaystyle=\frac{1}{r}\int_{0}^{1}\phi^{(k)}(s^{2})\cdot
rs^{\nu+k+1}J_{\nu+k}(rs)\,ds$
$\displaystyle=\frac{1}{r}\int_{0}^{1}\phi^{(k)}(s^{2})\frac{d}{ds}\left\\{s^{\nu+k+1}J_{\nu+k+1}(rs)\right\\}\,ds$
$\displaystyle=-\frac{2}{r}I_{k+1},$

if $k\leq N-1$. We obtain

(3.8) $I_{0}=\left(-\frac{2}{r}\right)^{N}I_{N}$

for $N\geq 1$. Notice that (3.8) holds for $N=0$ as well.

On the other hand, it is well known that

(3.9) $\displaystyle J_{\alpha}(z)$
$\displaystyle=\frac{2^{1/2}}{\pi^{1/2}}z^{-1/2}\cos\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)+O(z^{-3/2})$

as $z\to\infty$ (see for example [15, 10.17.3]). By (3.9) and the boundedness
of $J_{\nu+N}(z)$ as $z\to+0$, there exists a bounded function
$R(z)\;(0<z<\infty)$ such that

(3.10)
$J_{\nu+N}(z)=\frac{2^{1/2}}{\pi^{1/2}}z^{-1/2}\cos\left(z-\frac{(\nu+N)\pi}{2}-\frac{\pi}{4}\right)+z^{-3/2}R(z).$

We have

$\displaystyle I_{N}$
$\displaystyle=\frac{2^{1/2}}{\pi^{1/2}}r^{-1/2}\int_{0}^{1}\phi^{(N)}(s^{2})s^{\nu+N+1/2}\cos\left(rs-\frac{(\nu+N)\pi}{2}-\frac{\pi}{4}\right)\,ds$
$\displaystyle\quad+r^{-3/2}\int_{0}^{1}\phi^{(N)}(s^{2})s^{\nu+N-1/2}R(rs)\,ds.$

The first term in the right hand side is of order $o(r^{-1/2})$ as
$r\to\infty$ by the Riemann-Lebesgue lemma. The second term is of order
$O(r^{-3/2})$ by the boundedness of $R$. We have shown $I_{N}=o(r^{-1/2})$ and
the combination of it and (3.8) gives $I_{0}=o(r^{-(N+1/2)})$. ∎

######  Proposition 3.4.

Let $N$ be a non-negative integer and $\nu$, $\lambda_{0},\dots,\lambda_{m}$,
$\Lambda$, $a_{0},$ $\dots,$ $a_{m}$ be complex numbers. Assume
$\mathrm{Re\,}\nu>-1$,
$-1<\mathrm{Re\,}\lambda_{0}<\mathrm{Re\,}\lambda_{1}<\cdots<\mathrm{Re\,\lambda}_{m}<\mathrm{Re\,\Lambda}$,
$N\leq\mathrm{Re\,\Lambda}$, $\mathrm{Re\,\lambda}_{0}\leq N-1$. Assume that
the function $\phi(t)$ in $0<t<1$ satisfies

$\phi(t)=\sum_{k=0}^{m}a_{k}(1-t)^{\lambda_{k}}+(1-t)^{\Lambda}\psi(t)\quad(1-2\varepsilon<t<1),$

where $a_{0}\neq 0$ and $\psi(t)$ is an infinitely differentiable function in
$0<t<1$ such that $\psi^{(k)}(t)$ is integrable in $1-2\varepsilon<t<1$ for
$0\leq k\leq N$.

Let $\chi_{1}(t)$ be an infinitely differentiable function such that
$\chi_{1}(t)=0$ in $0<t\leq 1-2\varepsilon$ and $\chi_{1}(t)=1$ in
$1-\varepsilon\leq t<1$. Set†††Notice that the cutoff is incomplete in the
sense that $\chi_{1}(t)$ does not appear in the summation term.

$\tilde{\phi}(t)=\sum_{k=0}^{m}a_{k}(1-t)^{\lambda_{k}}+\chi_{1}(t)(1-t)^{\Lambda}\psi(t)\quad(0<t<1).$

Then as $r\to\infty$,

$\displaystyle\int_{0}^{1}s^{\nu+1}\tilde{\phi}(s^{2})J_{\nu}(rs)\,ds$
$\displaystyle=a_{0}\frac{2^{\lambda_{0}+1/2}}{\pi^{1/2}}\Gamma(\lambda_{0}+1)r^{-(\lambda_{0}+3/2)}\cos\left(r-\frac{\pi}{2}(\nu+\lambda_{0}+1)-\frac{\pi}{4}\right)$
$\displaystyle\quad+o\left(r^{-\mathrm{Re\,}(\lambda_{0}+3/2)}\right).$

###### Proof.

By [8, 26(33)a] or [9, 6.567.1], we have

(3.11)
$\displaystyle\int_{0}^{1}s^{\nu+1}\left(1-s^{2}\right)^{\alpha}J_{\nu}(rs)\,ds$
$\displaystyle=2^{\alpha}\Gamma(\alpha+1)r^{-(\alpha+1)}J_{\nu+\alpha+1}(r),$

if $r>0,\;\mathrm{Re\,}\nu>-1,\;\mathrm{Re\,}\alpha>-1$‡‡‡If we set
$s=\sin\theta$, we get Sonine’s first finite integral. Two methods of its
evaluation are given in [18, 12.11 (1)]). There is another proof of (3.11).
When $\alpha$ is a nonnegative integer, we can prove it following (3.7) in the
proof of Proposition 3.3. To generalize it to $\mathrm{Re\,}\alpha>-1$, we
divide the equality by $\Gamma(\alpha+1)$ and apply the Carlson’s theorem ([2,
p.110]). Boundedness is guaranteed by Poisson’s integral representation of
Bessel functions ([15, 10.9.4]).. We employ this formula and Proposition 3.3
to obtain

$\displaystyle\int_{0}^{1}s^{\nu+1}\tilde{\phi}(s^{2})J_{\nu}(rs)\,ds=$
$\displaystyle\sum_{k=0}^{m}a_{k}2^{\lambda_{k}}\Gamma(\lambda_{k}+1)r^{-(\lambda_{k}+1)}J_{\nu+\lambda_{k}+1}(r)$
$\displaystyle+o(r^{-(N+1/2)}).$

We finish the proof by using (3.9). ∎

Now we give our main results.

######  Theorem 3.5.

Let $n\geq 2$ be an integer. Let $\varphi(s)$ be an infinitely differentiable
function in the open interval $(0,1)$ and $\phi(t)\,(0<t<1)$ be the function
such that

$\varphi(s)=s^{n/2}\phi(s^{2})\quad(0<s<1).$

Set $\nu=n/2-1$. If $\mu$ satisfies (3.1), $\varphi(s)$ satisfies (3.2), (3.3)
and $\phi(t)$ satisfies the conditions of Proposition 3.4, then

(3.12)
$f(x)=|x|^{-n/2}\varphi(|x|)\chi_{[0,1]}(|x|)=\phi\left(|x|^{2}\right)\chi_{[0,1]}(|x|)\;(x\in\mathbb{R}^{n})$

is an invertible distribution. Here $\chi_{[0,1]}(\cdot)$ is the indicator
function of the interval $[0,1]$.

###### Proof.

Set $\nu=\nu(n)=n/2-1$ and

$\tilde{\varphi}(s)=\chi_{0}(s)\left\\{\varphi(s)-s^{n/2}\sum_{k=0}^{m}a_{k}(1-s^{2})^{\lambda_{k}}\right\\}.$

Notice that

${\displaystyle\tilde{\varphi}{}^{(j)}(s)\sim\sum_{k=0}^{\infty}c_{k}\frac{d^{j}}{ds^{j}}s^{\mu+k}+\sum_{\ell=0}^{\infty}A_{\ell}\frac{d^{j}}{ds^{j}}s^{n/2+2\ell}\quad(s\to+0)}$

for some $A_{\ell}$. The second sum, which corresponds to
$s^{n/2}\sum_{k=0}^{m}a_{k}(1-s^{2})^{\lambda_{k}}$, does not contribute to
the finite Hankel transform because of Proposition 3.2.

Let $K_{n}=K\left(\mu,\nu(n),\left\\{c_{k}\right\\}_{k}\right)$ be the set
defined by (3.4) with $\nu=\nu(n)$. If $K_{n}\neq\emptyset$, we have

(3.13) $\displaystyle H_{0}$
$\displaystyle:=\int_{0}^{1}\tilde{\varphi}(s)J_{\nu(n)}(rs)\,ds$
$\displaystyle=c_{k_{0}}\frac{\Gamma\left(\frac{1}{2}(\mu+k_{0}+\nu(n)+1)\right)2^{\mu+k_{0}}}{\Gamma\left(\frac{1}{2}(-\mu-
k_{0}+\nu(n)+1)\right)r^{\mu+k_{0}+1}}+o(r^{-\mathrm{Re\,}(\mu+k_{0}+1)}),$

and if $K_{n}=\emptyset$, $H_{0}$ decreases rapidly of arbitrary order by
(3.5). On the other hand, for $\tilde{\phi}$ defined in Proposition 3.4, we
have

$\displaystyle H_{1}:=$
$\displaystyle\int_{0}^{1}s^{\nu(n)+1}\tilde{\phi}(s^{2})J_{\nu(n)}(rs)\,ds$
$\displaystyle=$ $\displaystyle
a_{0}\frac{2^{\lambda_{0}+1/2}}{\pi^{1/2}}\Gamma(\lambda_{0}+1)r^{-(\lambda_{0}+3/2)}\cos\left(r-\frac{\pi}{2}(\nu(n)+\lambda_{0}+1)-\frac{\pi}{4}\right)$
$\displaystyle+o\left(r^{-\mathrm{Re}\,(\lambda_{0}+3/2)}\right).$

We can prove that $H_{0}+H_{1}$ satisfies (2.10) in the following way.

If $K_{n}=\emptyset$, $H_{1}$ is dominant. We assume $K_{n}\neq\emptyset$ from
now on. If $\mathrm{Re\,}\mu+k_{0}+1<\mathrm{Re\,\lambda_{0}+3/2}$, then
$H_{0}$ is dominant. If
$\mathrm{Re\,}\mu+k_{0}+1>\mathrm{Re\,\lambda_{0}+3/2}$, then $H_{1}$ is
dominant. If $\mathrm{Re\,}\mu+k_{0}+1=\mathrm{Re\,\lambda_{0}+3/2}$, we can
pick up those $r$’s for which the cosine factor vanishes and the contribution
of $H_{1}$ becomes irrelevant. In any case, $H_{0}+H_{1}$ satisfies (2.10). By
Theorem 2.1 and Proposition 2.11,
$|x|^{-n/2}\tilde{\varphi}(|x|)+\tilde{\phi}\left(|x|^{2}\right)$ is
invertible.

Notice that $\varphi(s)-\tilde{\varphi}(s)-s^{n/2}\tilde{\phi}(s^{2})$
vanishes near $s=0,1$ and

$\displaystyle
f(x)-|x|^{-n/2}\tilde{\varphi}(|x|)-\tilde{\phi}\left(|x|^{2}\right)\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{n}).$

By Proposition 2.7 (v), $f(x)$ is invertible. ∎

######  Example 3.6.

If $\lambda>-n/2,\,\rho>0$, then

$f(x)=|x|^{\lambda-n/2}\left(1-|x|^{2}\right)^{\rho-1}\chi_{[0,1]}(|x|)$

is an invertible distribution. This corresponds to the case of
$\varphi(s)=s^{\lambda}\left(1-s^{2}\right)^{\rho-1}$ and
$\phi(t)=t^{\lambda/2-n/4}\left(1-t\right)^{\rho-1}$.

The finite Hankel transform
$\int_{0}^{1}s^{\lambda}\left(1-s^{2}\right)^{\rho-1}J_{\nu}(rs)\,ds$ can be
written in terms of the generalized hypergeometric function ${}_{2}F_{3}$ by
[9, 6.569] and it is good enough to prove invertibility. The advantage of our
method is that it is stable under small perturbations and works even if no
closed form expression is available.

######  Remark 3.7.

In (3.2), $j$ can be arbitrarily large. This assumption can be relaxed in some
cases. We give such an example. Assume

$\varphi(s)\sim 0=\sum_{k=0}^{\infty}0\cdot s^{\mu+k}$

as $s\to+0$, where $-n/2<\mathrm{Re}\,\mu\leq-1$. We do not assume that term
by term differentiation is possible§§§e.g. $\varphi(s)=e^{-1/s}\sin e^{1/s}$.:
$m$ in [19] is 0. Moreover, we have removed the assumption $c_{0}\neq 0$ and
do not need the set $K_{n}$. We keep all the other assumptions of Theorem 3.5
unchanged. By [19, (3.6), (4.2)], we have $H_{0}=o(1)$. If
$\mathrm{Re}\,\lambda_{0}\leq-3/2$, $H_{1}$ is dominant and $f(x)$ defined by
(3.12) is invertible.

######  Theorem 3.8.

Let $n\geq 2$ be an integer and $N$ be a nonnegative integer. Let $\varphi(s)$
be an infinitely differentiable function in the open interval $(0,1)$ and
$\phi(t)\,(t<t<1)$ be the function such that $\varphi(s)=s^{n/2}\phi(s^{2})$.
Set $\nu=n/2-1$. Suppose that $\mu$ satisfies (3.1) and that $\varphi(s)$
satisfies (3.2), (3.3). Moreover suppose $\mathrm{Re\,}\Lambda\geq N$ and

$\phi(t)=(1-t)^{\Lambda}\psi(t)\quad(1-2\varepsilon<t<1),$

where $\psi(t)$ is an infinitely differentiable function in $0<t<1$ such that
$\psi^{(k)}(t)$ is integrable in $1-2\varepsilon<t<1$ for $0\leq k\leq N$.

If $K_{n}=K\left(\mu,n/2-1,\left\\{c_{k}\right\\}_{k}\right)\neq\emptyset$ and
$\mathrm{Re\,}(\mu+k_{0}+1/2)\leq N$, then

$f(x)=|x|^{-n/2}\varphi(|x|)\chi_{[0,1]}(|x|)=\phi\left(|x|^{2}\right)\chi_{[0,1]}(|x|)\;(x\in\mathbb{R}^{n})$

is an invertible distribution.

###### Proof.

We have

$\displaystyle H_{0}$
$\displaystyle=c_{k_{0}}\frac{\Gamma\left(\frac{1}{2}(\mu+k_{0}+\nu(n)+1)\right)2^{\mu+k_{0}}}{\Gamma\left(\frac{1}{2}(-\mu-
k_{0}+\nu(n)+1)\right)r^{\mu+k_{0}+1}}+o(r^{-\mathrm{Re\,}(\mu+k_{0}+1)}),$
$\displaystyle H_{1}$ $\displaystyle=o(r^{-(N+1/2)})$

by Propositions 3.1 and 3.3. Since $\mathrm{Re\,}(\mu+k_{0}+1)\leq N+1/2$,
$H_{0}$ is dominant and $H_{1}$ is negligible. ∎

######  Remark 3.9.

We can find a lot of invertible distributions by combining Theorems 3.5 and
3.8 with Remark 2.6 and Proposition 2.7.

######  Remark 3.10.

We can formulate an invertibility theorem by using [20] instead of [19]. In
[20], the function $\varphi(s)$ is allowed to have an asymptotic expansion
involving powers of logarithms.

##  4 Concluding remarks

The fundamental works by Ehrenpreis and Hörmander were based on Fourier
analysis and complex analysis. Our study of convolution equations involves
special functions and has a flavor of classical analysis, while some papers,
such as a recent one [4], are based on abstract methods of functional analysis
and homological algebra. All these works show that the study of convolution
equations can be a very rich field, where several branches of mathematics meet
and influence each other. On the other hand, convolution equations can be
formulated on symmetric spaces (e.g. [3]) and it is another reason for the
importance of the subject.

## References

  * [1] Abramczuk, W., A class of surjective convolution operators, Pacific J. Math., 110 (1984), no. 1, 1-7.
  * [2] Andrews, G. E., Askey R., and Roy R., Special functions, Cambridge University Press, 1999.
  * [3] Christensen, J., Gonzalez, F. and Kakehi, T., Surjectivity of mean value operators on noncompact symmetric space, J. Funct. Anal., 272 (2017), 3610-3646.
  * [4] Debrouwere, A. and Kalmes, T., Linear topological invariants for kernels of convolution and differential operators, J. Funct. Anal., 284 (2023), no. 10, Paper No. 109886, 20 pp.
  * [5] Ehrenpreis, L., Solution of some problems of division. I. Division by a polynomial of derivation, Amer. J. Math., 76 (1954), 883–903.
  * [6] Ehrenpreis, L., Solutions of some problems of division. III. Division in the spaces, $\mathcal{D}^{\prime},\mathcal{H},\mathcal{Q}_{A},\mathcal{O}$, Amer. J. Math., 78 (1956), 685–715.
  * [7] Ehrenpreis, L., Solution of some problems of division IV. Invertible and elliptic operators, Amer. J. Math., 82 (1960), 522-588.
  * [8] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Tables of integral transforms. Vol. II. Based, in part, on notes left by Harry Bateman., McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. xvi+451 pp.
  * [9] Gradshteyn, I. S. and Ryzhik, M., edited by Zwillinger, D. and Moll, V., Tables of Integrals, Series, and Products, 8th edition, Academic Press, 2015.
  * [10] Hörmander, L., On the range of convolution operators, Ann. of Math. (2), 76 (1962), 148-170.
  * [11] Hörmander, L., The analysis of linear partial differential operators II, Springer-Verlag, 1983.
  * [12] Lim, K-T., The spherical mean value operators on Euclidean and hyperbolic spaces, PhD thesis, Tufts University, Medford, MA, 2012\. ProQuest Dissertations Publishing, 2012. 3546361.
  * [13] Malgrange, B., Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble), 6 (1955/56), 271-355.
  * [14] Musin, I. Kh., Perturbation of a surjective convolution operator, Ufa Mathematical Journal, 8 (4) (2016), 123-130.
  * [15] NIST Digital Library of Mathematical Functions,
https://dlmf.nist.gov/

  * [16] Okada, Y. and Yamane, H., Generalized spherical mean value operators on Euclidean space, Tsukuba J. Math., 45 (1) (2021), 37-50.
  * [17] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ Press (1971).
  * [18] Watson, G. N., A treatise on the theory of Bessel functions, Cambridge University Press (1922).
  * [19] Wong, R., Error bounds for asymptotic expansions of Hankel transforms, SIAM J. Math. Anal., 7 (1976), no. 6, 799-808.
  * [20] Wong, R., Asymptotic expansions of Hankel transforms of functions with logarithmic singularities, Comput. Math. Appl., 3 (1977), no. 4, 271-286.
  * [21] Wong, R., Error bounds for asymptotic expansions of integrals, SIAM Rev., 22 (1980), no. 4, 401-435. (This paper contains a minor correction of [19]. )