Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

Forum for Information Retrieval Evaluation, December 13-17, 2021, India

<EMAIL_ADDRESS>url=https://seanbenhur.github.io/, ]

[email=<EMAIL_ADDRESS>]

# Pretrained Transformers for Offensive Language Identification in Tanglish

Sean Benhur  Kanchana Sivanraju  PSG College of Arts and Science, Civil
Aerodrome Post, Coimbatore, India

(2021)

###### Abstract

This paper describes the system submitted to Dravidian-Codemix-HASOC2021: Hate
Speech and Offensive Language Identification in Dravidian Languages (Tamil-
English and Malayalam-English). This task aims to identify offensive content
in code-mixed comments/posts in Dravidian Languages collected from social
media. Our approach utilizes pooling the last layers of pretrained transformer
multilingual BERT for this task which helped us achieve rank nine on the
leaderboard with a weighted average score of 0.61 for the Tamil-English
dataset in subtask B. After the task deadline, we sampled the dataset
uniformly and used the MuRIL pretrained model, which helped us achieve a
weighted average score of 0.67, the top score in the leaderboard. Furthermore,
our approach to utilizing the pretrained models helps reuse our models for the
same task with a different dataset. Our code and models are available in
GitHub 111https://github.com/seanbenhur/tanglish-offensive-language-
identification

###### keywords:

Hate Speech Offensive ContentBERT Transformer

##  1 Introduction

In the era of the internet, people from various age groups engage in social
media, it has become a one-stop shop for all activities from learning to
entertainment, but it is also filled with offensive and disturbing content,
which is potentially harmful to everyone [1]. To prevent this, an automated
system of identifying and flagging offensive content should be developed.
Though there is a substantial amount of work done on major languages like
English to identify offensive content [2], it is a challenging task to
identify and flag offensive content in low resource languages, since many
users tend to write their language in English script, which is called code-
switching or code-mixing [3, 4, 5]. Developing NLP systems on code-mixed text
is challenging since the number of datasets is scarce[6, 7, 8, 9] and there
are no clear patterns on these texts. The spelling and context might vary
depending upon the user.

Dravidian languages are under-resourced in natural language processing [10].
Dravidian name was derived from Tamil, Dravidian means Tamil [11], Dravidian
languages are Tamili languages [12]. Tamil is a language spoken by Tamils in
Tamil Nadu, India, Sri Lanka, and the Tamil diaspora worldwide, with official
recognition in India, Sri Lanka, and Singapore [13, 14, 15]. Current Tamil
script was developed from the Tamili script, the Vatteluttu alphabet, and the
Chola-Pallava script. There are 12 vowels, 18 consonants, and 1 ytam in this
word (voiceless velar fricative) [16, 17, 18, 19]. The Tamil script is also
used to write minority languages including Saurashtra, Badaga, Irula, and
Paniya. Tamil Eluttu "means" sound, letter, phoneme" in Tolkappiyam (about
5,320 BCE), and thus includes the sounds of the Tamil language, as well as how
they are created (phonology) [20, 21, 22, 23]. All the Tamili (Dravidian)
languages evolved from Tamil language [24, 25].

HASOC2021: Hate Speech and Offensive Content Identification is a competition
that helps increase research in offensive language identification in code
mixed languages such as Tamil-English and Malayalam-English [6]. The dataset
consists of comments/posts that were collected from Youtube and social media.
Each comment/post is annotated with an offensive language label at the
comment/post level. This dataset also has class imbalance problems depicting
real-world scenarios.

In this paper, we present our system developed for HASOC 2021; the rest of the
paper is organized as follows. Section 2 discusses the research work on
offensive language identification and natural language processing in under-
resourced languages. Following this, in section 3, we present the methodology
of our work, from preprocessing the dataset, our model architecture, and
training procedures. In section 4, we discuss our results. Finally, in section
6, we conclude with a summary and future work.

##  2 Related Work

Offensive Language identification has been widely made across many people in
multiple languages. Shared tasks like HASOC-19 [26] dealt with hate speech and
offensive language identification in Indo-European languages. HASOC-Dravidian-
CodeMix - FIRE 2020 [27][28] is the first shared task for identifying
offensive content in Tamili languages. Previous work on Tamili languages on
hope speech [29, 30], troll meme detection [31], multimodal sentiment analysis
[9] have paved the way to research in Tamili languages.

Researchers have used a wide variety of techniques for the identification of
offensive language. There have been previous work [32] in using classical
machine learning models with efficient feature generation. Other researchers
in [33] [34] have used an ULMFit model [35] and pretrained XLM-Roberta model
with translated and transliterated texts for this task.

##  3 Methodology

This section briefly describes our methodology for this task, including data
preparation, model architecture, and training strategies. For this HASOC 2021
competition, we only use the datasets that were provided for the HASOC task.
Table 1 shows the statistics of the train and dev distribution.

###  3.1 Dataset

The dataset given for subtask, Offensive Language Identification in Tamil-
English, consists of Youtube comments, present in code-mixed data containing
text written in both native and roman scripts in English.

For training our model, we concatenate both training and dev sets; we remove
the URLs, English stopwords, @username mentions, NAN values, emojis, and also
punctuations; this preprocessing method is applied to all the train, dev, and
test sets. After the task deadline, we sample the dataset uniformly to handle
the class imbalance problem in this dataset, which helps us improve our score.
Table 1 shows the statistics of the given dataset after preprocessing.

Table 1: Distribution of Tamil English Dataset Distribution | Data
---|---
Train | 4937
Test | 1000

###  3.2 Model Architecture

We use pretrained transformer models with custom pooled output for this task
of identifying offensive content. We have used mBERT and MuRIL pretrained
models from huggingface checkpoints. In this section, we describe our pooling
operations on the pretrained models and the pretrained models.

Attention Pooler: In this method, the attention operation described in the
below equation is applied to the last hidden state of the pretrained
transformer; empirically, this helps the model learn the contribution of
individual tokens. Finally, the returned pooled output from the transformer is
further passed to a linear layer for the prediction of the label.

$o=W^{T}_{h}softmax(qh^{T}_{CLS})h_{CLS}$ (1)

where $W^{T}_{h}$ and $q$ are learnable weights.

$y=softmax(W^{T}_{o}+bo)$ (2)

Mean Pooler: In this method, the average of the last layer of the pretrained
transformer is taken. This acts like a pooling layer in a convolutional neural
net. An alternative to this method is to use max pooling, but max-pooling
selects only the words with essential features rather than utilizing all the
words. Since our dataset is code-mixed and the spelling of the tokens are not
precise, we choose to go with mean pooling approach.

mBERT Multilingual models of BERT [36]. This model was pre-trained using the
same pretraining strategy that was employed to BERT, which is Masked Language
Modeling (MLM) and Next Sentence Prediction (NSP). It was pretrained on the
Wikipedia dump of top 104 languages. To account for the data imbalance due to
the size of Wikipedia for a given language, exponentially smoothed weighting
of data was performed during data creation and word piece vocabulary creation.
This results in high resource languages being under-sampled while low
resourced languages being over-sampled.

MuRIL MuRIL [37], pretrained model is trained on 16 different Indian
Languages; the model was pretrained on Masked Language Modeling(MLM) and
Translated Language Modelling(TLM). This model outperforms mBERT on all the
tasks in XTREME [38]

###  3.3 Training

Though finetuning transformers gives better results and is dominant across
leaderboards of various NLP competitions. Transformer models are still
unstable due to catastrophic forgetting [39]. For this offensive language
identification task, we carefully choose our hyperparameters for
experimentation. We finetune our custom models with binary-cross-entropy loss
and AdamW optimizer, which decouples the weight decay from the optimization
step. Linear scheduler for learning rate scheduling with 2e-5 as an initial
step is used with this training strategy. The training hyperparameters are
listed in Table 2.

Table 2: Hyperparameters used across experiments Hyperparameters | Values |
---|---|---
Learning Rate | 2e-5 |
Maximim Sequence Length | 512 |
Batch Size | 8 |
Epochs | 5 |
Weight Decay | 0.01 |
Dropout | 0.5 |
AdamW epsilon | 1e-06 |

##  4 Results and Discussion

In this HASOC 2021 competition, the teams were ranked by the weighted F1-score
of their classification system. This section discusses our experimental
results; since we have used both training and dev sets for training, the train
set in the dataset distribution refers to the concatenated given train and dev
sets. The W-Precision, W-Recall, and W-F1-Score refer to the Weighted
precision, weighted recall, and weighted F1-Score. Table 3 shows our results
obtained before the task deadline using Attention Pooler and mBERT without
sampling the dataset. After the task deadline, we uniformly sample our dataset
and run or experiments on MuRIL and mBERT with Attention Pooling and Mean
Pooling. The results are provided in Table 4 and Table 5. The W-precision,
W-Recall and W-F1 Score stands for Weighted Precision, Weighted Recall and
Weighted F1-Score.

Table 3: Results before task deadline Dataset Distribution | W-Precision | W-Recall | W-F1 Score
---|---|---|---
Train | 0.90 | 0.91 | 0.92
Test | 0.61 | 0.60 | 0.61
Table 4: Results on Train dataset Model | W-Precision | W-Recall | W-F1 Score
---|---|---|---
mBERT with AttentionPooler | 0.93 | 0.90 | 0.93
mBERT with MeanPooler | 0.90 | 0.92 | 0.91
MuRIL with AttentionPooler | 0.88 | 0.88 | 0.88
MuRIL with MeanPooler | 0.93 | 0.93 | 0.93
Table 5: Results on Test data Model | W-Precision | W-Recall | W-F1 Score
---|---|---|---
mBERT with AttentionPooler | 0.65 | 0.65 | 0.65
mBERT with MeanPooler | 0.61 | 0.61 | 0.61
MuRIL with AttentionPooler | 0.63 | 0.63 | 0.63
MuRIL with MeanPooler | 0.67 | 0.67 | 0.67

From the above results, we conclude that the pretrained model MuRIL with
MeanPooler performs best than others. Also, one can infer that the difference
between training and test scores shows that the model is suffering from
overfitting, and also sampling the dataset uniformly is a crucial step to
increasing the score.

##  5 Conclusion

In this paper, we have presented our solution for the Offensive Language
Identification system, which uses pretrained transformers mBERT and MuRIL. As
a result, we achieve Rank 9 on the leaderboard and a 0.67 f1-score after the
task deadline. For future research, we will consider improving the results by
using any external dataset and other pretrained models and reducing the
generalization error of the model.

## References

  * Hegde et al. [2021] S. U. Hegde, A. Hande, R. Priyadharshini, S. Thavareesan, R. Sakuntharaj, S. Thangasamy, B. Bharathi, B. R. Chakravarthi,  Do Images really do the Talking? Analysing the significance of Images in Tamil Troll meme classification,  arXiv preprint arXiv:2108.03886 (2021).
  * Chakravarthi et al. [2021] B. R. Chakravarthi, R. Priyadharshini, R. Ponnusamy, P. K. Kumaresan, K. Sampath, D. Thenmozhi, S. Thangasamy, R. Nallathambi, J. P. McCrae,  Dataset for Identification of Homophobia and Transophobia in Multilingual YouTube Comments,  arXiv preprint arXiv:2109.00227 (2021).
  * Suryawanshi et al. [2021] S. Suryawanshi, B. R. Chakravarthi, M. Arcan, S. Little, P. Buitelaar,  TrollsWithOpinion: A Dataset for Predicting Domain-specific Opinion Manipulation in Troll Memes,  arXiv preprint arXiv:2109.03571 (2021).
  * Hande et al. [2021a] A. Hande, K. Puranik, K. Yasaswini, R. Priyadharshini, S. Thavareesan, A. Sampath, K. Shanmugavadivel, D. Thenmozhi, B. R. Chakravarthi,  Offensive Language Identification in Low-resourced Code-mixed Dravidian languages using Pseudo-labeling,  arXiv preprint arXiv:2108.12177 (2021a).
  * Hande et al. [2021b] A. Hande, S. U. Hegde, R. Priyadharshini, R. Ponnusamy, P. K. Kumaresan, S. Thavareesan, B. R. Chakravarthi,  Benchmarking multi-task learning for sentiment analysis and offensive language identification in under-resourced Dravidian languages,  arXiv preprint arXiv:2108.03867 (2021b).
  * Chakravarthi et al. [2021] B. R. Chakravarthi, P. K. Kumaresan, R. Sakuntharaj, A. K. Madasamy, S. Thavareesan, P. B, S. Chinnaudayar Navaneethakrishnan, J. P. McCrae, T. Mandl,  Overview of the HASOC-DravidianCodeMix Shared Task on Offensive Language Detection in Tamil and Malayalam,  in: Working Notes of FIRE 2021 - Forum for Information Retrieval Evaluation, CEUR, 2021\.
  * Priyadharshini et al. [2021] R. Priyadharshini, B. R. Chakravarthi, S. Thavareesan, D. Chinnappa, T. Durairaj, E. Sherly,  Overview of the DravidianCodeMix 2021 Shared Task on Sentiment Detection in Tamil, Malayalam, and Kannada,  in: Forum for Information Retrieval Evaluation, FIRE 2021, Association for Computing Machinery, 2021\.
  * Chakravarthi et al. [2021a] B. R. Chakravarthi, R. Priyadharshini, V. Muralidaran, N. Jose, S. Suryawanshi, E. Sherly, J. P. McCrae,  DravidianCodeMix: Sentiment Analysis and Offensive Language Identification Dataset for Dravidian Languages in Code-Mixed Text,  arXiv preprint arXiv:2106.09460 (2021a).
  * Chakravarthi et al. [2021b] B. R. Chakravarthi, K. Soman, R. Ponnusamy, P. K. Kumaresan, K. P. Thamburaj, J. P. McCrae, et al.,  DravidianMultiModality: A Dataset for Multi-modal Sentiment Analysis in Tamil and Malayalam,  arXiv preprint arXiv:2106.04853 (2021b).
  * Chakravarthi [2020] B. R. Chakravarthi, Leveraging orthographic information to improve machine translation of under-resourced languages, Ph.D. thesis, NUI Galway, 2020.
  * Chinnappa and Dhandapani [2021] D. Chinnappa, P. Dhandapani,  Tamil lyrics corpus: Analysis and experiments,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021, pp. 1–9. URL: https://aclanthology.org/2021.dravidianlangtech-1.1.
  * Andrew [2021] J. J. Andrew,  JudithJeyafreedaAndrew@DravidianLangTech-EACL2021:offensive language detection for Dravidian code-mixed YouTube comments,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021, pp. 169–174. URL: https://aclanthology.org/2021.dravidianlangtech-1.22.
  * Sarveswaran et al. [2021] K. Sarveswaran, G. Dias, M. Butt,  ThamizhiMorph: A morphological parser for the Tamil language,  Machine Translation 35 (2021) 37–70.
  * Bharathi and A [2021a] B. Bharathi, A. S. A,  SSNCSE_NLP@DravidianLangTech-EACL2021: Offensive language identification on multilingual code mixing text,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021a, pp. 313–318. URL: https://aclanthology.org/2021.dravidianlangtech-1.45.
  * Bharathi and A [2021b] B. Bharathi, A. S. A,  SSNCSE_NLP@DravidianLangTech-EACL2021: Meme classification for Tamil using machine learning approach,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021b, pp. 336–339. URL: https://aclanthology.org/2021.dravidianlangtech-1.49.
  * Thavareesan and Mahesan [2020a] S. Thavareesan, S. Mahesan,  Word embedding-based Part of Speech tagging in Tamil texts,  in: 2020 IEEE 15th International Conference on Industrial and Information Systems (ICIIS), 2020a, pp. 478–482. doi:10.1109/ICIIS51140.2020.9342640.
  * Thavareesan and Mahesan [2020b] S. Thavareesan, S. Mahesan,  Sentiment Lexicon Expansion using Word2vec and fastText for Sentiment Prediction in Tamil texts,  in: 2020 Moratuwa Engineering Research Conference (MERCon), 2020b, pp. 272–276. doi:10.1109/MERCon50084.2020.9185369.
  * Thavareesan and Mahesan [2019] S. Thavareesan, S. Mahesan,  Sentiment Analysis in Tamil Texts: A Study on Machine Learning Techniques and Feature Representation,  in: 2019 14th Conference on Industrial and Information Systems (ICIIS), 2019, pp. 320–325. doi:10.1109/ICIIS47346.2019.9063341.
  * Thenmozhi and Aravindan [2018] D. Thenmozhi, C. Aravindan,  Ontology-based Tamil–English cross-lingual information retrieval system,  Sādhanā 43 (2018) 1–14.
  * Sakuntharaj and Mahesan [2016] R. Sakuntharaj, S. Mahesan,  A novel hybrid approach to detect and correct spelling in Tamil text,  in: 2016 IEEE International Conference on Information and Automation for Sustainability (ICIAfS), IEEE, 2016, pp. 1–6.
  * Sakuntharaj and Mahesan [2017] R. Sakuntharaj, S. Mahesan,  Use of a novel hash-table for speeding-up suggestions for misspelt Tamil words,  in: 2017 IEEE International Conference on Industrial and Information Systems (ICIIS), IEEE, 2017, pp. 1–5.
  * Sakuntharaj and Mahesan [2018a] R. Sakuntharaj, S. Mahesan,  Detecting and correcting real-word errors in Tamil sentences,  Ruhuna Journal of Science 9 (2018a).
  * Sakuntharaj and Mahesan [2018b] R. Sakuntharaj, S. Mahesan,  A refined pos tag sequence finder for Tamil sentences,  in: 2018 IEEE International Conference on Information and Automation for Sustainability (ICIAfS), IEEE, 2018b, pp. 1–6.
  * R and C N [2019] A. R, S. C N,  Building discourse parser for Thirukkural,  in: Proceedings of the 16th International Conference on Natural Language Processing, NLP Association of India, International Institute of Information Technology, Hyderabad, India, 2019, pp. 18–25. URL: https://aclanthology.org/2019.icon-1.3.
  * Subalalitha [2019] C. Subalalitha,  Information extraction framework for Kurunthogai,  Sādhanā 44 (2019) 1–6.
  * Majumder et al. [2019] P. Majumder, D. Patel, S. Modha, T. Mandl,  Overview of the hasoc track at fire 2019: Hate speech and offensive content identification in indo-european languages,  2019\. doi:10.1145/3368567.3368584.
  * Chakravarthi et al. [2020] B. R. Chakravarthi, R. Priyadharshini, V. Muralidaran, S. Suryawanshi, N. Jose, E. Sherly, J. P. McCrae,  Overview of the track on sentiment analysis for dravidian languages in code-mixed text,  in: Forum for Information Retrieval Evaluation, 2020, pp. 21–24.
  * Mandl et al. [2020] T. Mandl, S. Modha, A. Kumar M, B. R. Chakravarthi,  Overview of the hasoc track at fire 2020: Hate speech and offensive language identification in tamil, malayalam, hindi, english and german,  in: Forum for Information Retrieval Evaluation, 2020, pp. 29–32.
  * Chakravarthi [2020] B. R. Chakravarthi,  HopeEDI: A multilingual hope speech detection dataset for equality, diversity, and inclusion,  in: Proceedings of the Third Workshop on Computational Modeling of People’s Opinions, Personality, and Emotion’s in Social Media, Association for Computational Linguistics, Barcelona, Spain (Online), 2020, pp. 41–53. URL: https://aclanthology.org/2020.peoples-1.5.
  * Chakravarthi and Muralidaran [2021] B. R. Chakravarthi, V. Muralidaran,  Findings of the shared task on hope speech detection for equality, diversity, and inclusion,  in: Proceedings of the First Workshop on Language Technology for Equality, Diversity and Inclusion, Association for Computational Linguistics, Kyiv, 2021, pp. 61–72. URL: https://aclanthology.org/2021.ltedi-1.8.
  * Suryawanshi and Chakravarthi [2021] S. Suryawanshi, B. R. Chakravarthi,  Findings of the shared task on troll meme classification in Tamil,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021, pp. 126–132. URL: https://aclanthology.org/2021.dravidianlangtech-1.16.
  * Lakshmanan and Ravindranath [2020] B. Lakshmanan, S. K. Ravindranath, Theedhum nandrum@dravidian-codemix-fire2020: A sentiment polarity classifier for youtube comments with code-switching between tamil, malayalam and english, 2020. arXiv:2010.03189.
  * Vasantharajan and Thayasivam [2021] C. Vasantharajan, U. Thayasivam,  Hypers@DravidianLangTech-EACL2021: Offensive language identification in Dravidian code-mixed YouTube comments and posts,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021, pp. 195–202. URL: https://aclanthology.org/2021.dravidianlangtech-1.26.
  * Sai and Sharma [2021] S. Sai, Y. Sharma,  Towards offensive language identification for Dravidian languages,  in: Proceedings of the First Workshop on Speech and Language Technologies for Dravidian Languages, Association for Computational Linguistics, Kyiv, 2021, pp. 18–27. URL: https://aclanthology.org/2021.dravidianlangtech-1.3.
  * Howard and Ruder [2018] J. Howard, S. Ruder,  Fine-tuned language models for text classification,  CoRR abs/1801.06146 (2018). URL: http://arxiv.org/abs/1801.06146. arXiv:1801.06146.
  * Devlin et al. [2019] J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Bert: Pre-training of deep bidirectional transformers for language understanding, 2019. arXiv:1810.04805.
  * Khanuja et al. [2021] S. Khanuja, D. Bansal, S. Mehtani, S. Khosla, A. Dey, B. Gopalan, D. K. Margam, P. Aggarwal, R. T. Nagipogu, S. Dave, S. Gupta, S. C. B. Gali, V. Subramanian, P. Talukdar, Muril: Multilingual representations for indian languages, 2021\. arXiv:2103.10730.
  * Hu et al. [2020] J. Hu, S. Ruder, A. Siddhant, G. Neubig, O. Firat, M. Johnson, Xtreme: A massively multilingual multi-task benchmark for evaluating cross-lingual generalization, 2020. arXiv:2003.11080.
  * Mosbach et al. [2021] M. Mosbach, M. Andriushchenko, D. Klakow, On the stability of fine-tuning bert: Misconceptions, explanations, and strong baselines, 2021\. arXiv:2006.04884.


# Effects of Small-Scale User Mobility on Highly Directional XR Communications

Asad Ali${}^{1,2}\href https://orcid.org/0000-0001-6293-4771$, Olga Galinina1*
—, Jiri Hosek2* —, and Sergey Andreev1,2* —
1Tampere University, Tampere, Finland
2Brno University of Technology, Brno, Czech Republic
{asad.ali, olga.galinina<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The development of next-generation communication systems promises to enable
extended reality (XR) applications, such as XR gaming with ultra-realistic
content and human-grade sensory feedback. These demanding applications impose
stringent performance requirements on the underlying wireless communication
infrastructure. To meet the expected Quality of Experience (QoE) for XR
applications, high-capacity connections are necessary, which can be achieved
by using millimeter-wave (mmWave) frequency bands and employing highly
directional beams. However, these narrow beams are susceptible to even minor
misalignments caused by small-scale user mobility, such as changes in the
orientation of the XR head-mounted device (HMD) or minor shifts in user body
position. This article explores the impact of small-scale user mobility on
mmWave connectivity for XR and reviews approaches to resolve the challenges
arising due to small-scale mobility. To deepen our understanding of small-
scale mobility during XR usage, we prepared a dataset of user mobility during
XR gaming. We use this dataset to study the effects of user mobility on highly
directional communication, identifying specific aspects of user mobility that
significantly affect the performance of narrow-beam wireless communication
systems. Our results confirm the substantial influence of small-scale mobility
on beam misalignment, highlighting the need for enhanced mechanisms to
effectively manage the consequences of small-scale mobility.

##  I Introduction

Extended reality (XR) headsets are envisioned to be as lightweight and compact
as standard eyeglasses. The compact design introduces hardware constraints
such as limited processing power, memory, and battery capacity. To meet the
expected Quality of Experience (QoE) for XR applications, XR headsets might
need to offload computation tasks to the network. This calls for wireless
connectivity that offers exceptional reliability, ultra-high data rates, and
low latency. By utilizing millimeter-wave (mmWave) frequency bands, 5G and
beyond networks open up possibilities for XR due to their immense available
bandwidth [1]. However, wireless communication at these frequencies requires
effective beam management, which involves identifying and maintaining the best
beam pairs for communication [2]. Unfortunately, these highly directional beam
pairs are prone to misalignment when the user equipment (UE) is moving.

Traditionally, research on wireless communication systems predominantly
focused on large-scale mobility, where the movement of UE is based on the
general trajectory of the user. This form of mobility has been extensively
studied within the context of wireless communication systems, often
overlooking small-scale mobility, which refers to the lateral shifts and
rotations of the UE that occur closer to the main trajectory of the user, or
movement experienced by the UE when the user is within a confined space.
Unlike large-scale mobility, which involves significant changes in the UE’s
location such as another geographical location, small-scale mobility
encompasses comparatively smaller movements due to posture adjustments or head
turns.

For XR applications that result in high user mobility, such as gaming, the
frequent movement of UE leads to rapid and abrupt changes in the wireless
channel negatively impacting link quality. The challenge is further
exacerbated by blockages, deafness, and propagation loss from factors like
object and human body shadowing. In the context of XR, antenna arrays
integrated into head-mounted devices (HMDs) worn by users form highly
directional beams directed at the access node. These beams change direction as
the user turns their head. Even minor misalignments can result in significant
link degradation or outages [3], primarily due to the delay in beam
realignment. This highlights the importance of considering small-scale user
mobility in scenarios involving highly directional communication. To gain
deeper insights into the effects of small-scale mobility, it is crucial to
observe and analyze real user mobility data. Leveraging real-world data
provides a more comprehensive understanding of the aspects of mobility that
significantly affect wireless communication such as movement pattern, type of
movement, and speed of movement.

In [4], the authors perform statistical analysis and modeling by considering
micro-mobility (i.e., more subtle movements as compared to small-scale
mobility) of a user standing still under different applications. This is
achieved by adopting a direct measurements methodology involving the usage of
a laser attached to the UE and a camera. In [5], the authors investigate the
impact of using phased antenna arrays for UE under small-scale mobility and
compare the results to a reference horn antenna measurement. The authors used
angle of arrival (AoA) and angle of departure (AoD) measurements to assess the
achievable data rates, directional link opportunities, and robustness. The
results show considerable losses in data rate due to small beam misalignments
with significant and irregular variations. The study in [6], explores mmWave
beam misalignment effects on small- and large-scale user mobility with the
help of channel and mobility measurements in an outdoor setting. The beam
misalignment due to small-scale mobility strongly depends on the mobile device
use case.

The main objective of this article is to investigate the impact of small-scale
user mobility on wireless communication in XR applications. In what follows,
we provide an overview of the challenges posed by small-scale mobility during
wireless communication with highly directional beams in the mmWave frequency
band, specifically for XR applications. We also outline potential approaches
to resolving the challenges associated with small-scale mobility. Further, to
advance our understanding of small-scale mobility and its implications on
highly directional communication, we collect and present a dataset of user
mobility during VR gaming sessions. This dataset is subsequently employed to
evaluate the effects of user mobility on highly directional communication
during XR usage and inform on the aspects of mobility that negatively impact
wireless communication.

##  II Small-Scale Mobility and Highly Directional Communication

###  II-A Small-Scale Mobility in XR

Throughout an immersive XR session, the UE experiences continuous small-scale
mobility due to factors such as user behavior, content attributes, and
interactions with the physical and digital environment. For example, a seated
XR user viewing 360-degree videos tends to exhibit a higher degree of
rotational movements [7]. On the other hand, a walking user typically performs
synchronized cyclic head movements with each step [8], supplemented by
rotational movements to explore their surroundings.

In a fully mobile XR HMD setup, various sensory data are transmitted to a
remote XR server. These data include information about user head motion, eye
tracking, simultaneous localization and mapping (SLAM), light detection and
ranging (LiDAR), and six degrees of freedom (6DoF) sensors, as well as user
input. The server renders and encodes frames based on the reported motion data
and sends them back to the XR device to be decoded and shown on the XR display
panel. As the XR user moves in the real world, whether laterally or by turning
their head, their field of view adapts to mirror these changes in the digital
world. This real-time adjustment ensures that the displayed digital content
aligns with the user’s perspective thereby preserving the immersive
experience.

As the user interacts with the digital world, the position and orientation of
the antennas on the XR HMD change relative to the access node. Depending on
the user’s behavior and reaction to the digital content, the small-scale
mobility experienced by the UE can be slow and continuous or sudden and
abrupt. As the UE’s antenna array position and orientation change, the beams
adapt to maintain seamless communication with the access node as the user’s
perspective alters in response to the dynamic digital content.

As users turn their heads to navigate within immersive XR environments, small-
scale mobility manifests in faster and more abrupt changes in orientation as
compared to lateral movement [7]. These sudden and abrupt movements initiated
by the user to alter their perspective during scene transitions can
potentially influence their reactions. The high capacity and reliable
connectivity facilitated by highly directional beam is crucial for the closed-
loop delay-sensitive interaction between user actions (input and sensor data)
and the rendered frames displayed to the user.

Figure 1: Illustration of directional XR communication and beam misalignment
due to small-scale mobility.

###  II-B Wireless Challenges due to Small-Scale XR Mobility

From a wireless communications perspective in XR applications, it is crucial
to maintain synchronization between the render rate and the display rate. This
helps prevent the effects such as frame judder, which can result in issues
including double images, ghosting, and blurring. Mismatches between the
displayed frame and the sensory expectations degrade the QoE of an XR session.
Therefore, minimizing frame judder is essential for delivering a seamless and
immersive XR experience, one that minimizes the likelihood of motion sickness.

In scenarios with high mobility, the performance of the connection formed by
highly directional beams is adversely affected by changes in the environment
between the transmit and the receive antennas. The AoA and AoD change rapidly,
requiring frequent beam alignment, which incurs significant overhead. Highly
directional beams are more susceptible to degradation or disconnection due to
even minor misalignments resulting in significant signal-to-noise ratio (SNR)
degradation and causing short-term outages [9].

In the context of mobility, short-term outages refer to temporary
interruptions in wireless connection that occur when beams become misaligned
and last until beam management procedures realign the beams. In high-mobility
scenarios, the time provisioned for a beam to realign based on the channel
conditions during the initial misalignment may not be sufficient. Even after
switching beams, the newly selected beam pairs can become misaligned due to
the rapid movement of the device.

Small-scale mobility introduces beam misalignment through both lateral and
angular movements, which alters AoA and AoD at both the access node and the UE
sides. However, the most pronounced changes occur at the UE side. Slight
lateral movements during communication have the risk to result in outages due
to signal obstruction or stepping out of the coverage area, requiring the
selection of a new beam. Consequently, by causing momentary connectivity loss,
these outages disrupt the immersive perception for the user and greatly
degrade the QoE.

Furthermore, the speed of the user’s head movement directly impacts the data
size transmitted from the access node to the UE [10] to ensure a smooth and
immersive experience. As the user’s head motion prompts changes in the
displayed view, bursts of new frames are needed to accommodate these changes.
Consequently, greater user mobility results in increased data transmission,
while at the same time, higher mobility also demands more capacity and
reliability. This disparity between the user’s movements and the link
performance in terms of reliability and capacity highlights the need for
efficient handling of user mobility, especially small-scale mobility.

The small-scale mobility experienced by UEs is further aggravated by blockers,
making it even more challenging to maintain reliable communication links. The
highly directional beams in the mmWave frequency bands are susceptible to
disruptions caused by blockages. Primarily, three types of blockages commonly
need to be considered: (i) static blockers, such as buildings and trees; (ii)
dynamic blockers, including neighboring users; and (iii) self-blockages caused
by user body parts like the head or arms. While static blockages are
effectively modeled and not explicitly considered in indoor mmWave networks,
accurately predicting dynamic and self-blockages relies on understanding human
behavior, which varies depending on the environment.

Employing real-world mobility data allows for a more accurate assessment of
the challenges posed by small-scale mobility and aids in the identification of
specific mobility aspects that significantly influence wireless communication
performance. The intricate nature of small-scale mobility introduces
complexities that disrupt the connections between access nodes and XR formed
by highly directional beams. Small-scale mobility experienced by UE during XR
sessions is considerably different from the typical large-scale mobility and
requires special consideration grounded in realistic scenarios. Unlike large-
scale mobility, which involves more gradual and predictable movements, small-
scale mobility can lead to frequent and less expected beam misalignments and
disruptions in signal transmission. For this reason, it is essential to
consider real mobility data during an XR session to quantify the impact on
wireless communication, notably for highly directional beams.

###  II-C Addressing Challenges of Small-Scale Mobility

To ensure seamless and immersive XR experiences, side-information and
learning-driven beamforming can be employed to mitigate the impact of small-
scale mobility during highly directional communication for XR. Beam management
is a central aspect in ensuring seamless and immersive XR experience,
particularly in addressing the challenges posed by small-scale mobility.
Efficient beam switching in response to user movement is essential to counter
the performance degradation caused by such mobility.

XR HMDs are equipped with an inertial measurement unit (IMU) that enables
real-time tracking of HMD orientation, including the antenna array. Leveraging
IMU data improves beam management, resulting in faster and more agile beam
switching. By predicting the AoA and AoD using IMU data, XR communication
systems should be able to reduce communication outages caused by abrupt
changes in user orientation. This integration of motion sensor data enhances
the responsiveness of XR communication systems, hence providing users with a
smoother and more immersive experience. Statistical analysis of user movements
based on actual XR device data can offer valuable insights into user mobility
and help develop intuition to improve XR beamforming.

Machine learning (ML) algorithms can analyze user’s historical movement
patterns and real-time sensor data from the headset’s IMU. ML algorithms may
also be used to predict user’s future movement trajectory and anticipate
potential changes in the AoA and AoD of the communication beams. As the user
starts to turn their head or move, a suitable ML algorithm can proactively
suggest the preferred beam direction to maintain a stable link with the access
node. By employing ML-aided beam switching at both the UE and the access node,
XR devices may dynamically adapt and respond to user movements with faster
beam selection. This translates into improved reliability and capacity of the
wireless link formed by the highly directional beam pair and the XR QoE.

Another opportunity is using an ML-based approach for on-demand beam alignment
(in contrast to a periodic beam alignment case), where the beam alignment
procedure runs every time when the UE experiences an outage. With an ML-based
approach, the UE can rely on IMU and historical channel data to anticipate an
outage before it occurs and proactively perform beam realignment.

In summary, the aforementioned approaches offer promising solutions to address
the challenges arising from small-scale mobility in XR communications. In the
following section, we present our numerical results derived from real-world
mobility data collected during XR gaming sessions, which illustrate the
challenges stemming from user small-scale mobility. Additionally, we identify
specific aspects of small-scale mobility that impact highly directional
communication during XR gaming scenarios.

##  III Evaluation of Small-Scale Mobility Effects on XR Communications

In this section, we experimentally evaluate the impact of user small-scale
mobility on highly directional communication by using mobility data collected
from participants during VR gaming sessions. We further describe the user
mobility dataset and simulation setup, present our selected numerical results,
and analyze the impact of user mobility. This study aims to identify and
quantify the specific aspects of small-scale mobility that significantly
affect highly directional communication for XR applications, with a particular
focus on VR gaming.

###  III-A User Mobility Dataset

We use real-world mobility data obtained during VR gaming sessions to assess
the impact of small-scale mobility on the performance of highly directional
beams. Our approach involves gathering user mobility data from VR gaming
scenarios, providing valuable practical insights into small-scale mobility in
XR, which are then utilized to evaluate its impact on wireless performance.

We collected mobility data from 40 participants engaged in VR gameplay using
an HTC Vive Pro headset. This headset was equipped with 6DoF IMU sensors to
accurately track body and head movements. Additionally, user movement was
captured by two infrared (IR) devices mounted on tripod stands. The combined
data from the headset’s IMU sensors and the IR stations provided precise
position and orientation information. The VR headset was connected to a high-
performance PC workstation featuring an Intel Xeon 6-core CPU, 16 GB of RAM,
and a GeForce GTX 1650 GPU. Using OpenTrack, an open-source head-tracking
software [11] running on the workstation, we recorded the headset’s position
and orientation. The participants enjoyed a 5m × 5m play area. These
measurements were obtained indoors while participants played a game and
allowed for real-time collection of orientation and position data.

The data was collected for 5 minutes from each participant, with the mean
sampling rate of approximately 250 Hz. The data was stored in .csv file
format, containing 7 fields: recording time, displacement of the HMD position
in meters (x, y, z), and angular displacement of the HMD (yaw, pitch, roll).
The yaw and roll angles ranged from -180 to 180 degrees, while the pitch angle
ranged from -90 to 90 degrees. To foster broader community engagement and
facilitate further research and benchmarking, this user mobility dataset was
made publicly available as part of this publication [12].

###  III-B Evaluation Methodology and Results

To investigate the impact of user mobility on directional communication, we
conducted additional simulations that focused on the interactions between an
access node and a wireless VR headset. These simulations employed realistic
antenna movement patterns based on the collected dataset we described earlier.
The simulations were carried out within a 3D environment, where both the
access node and the VR headset were equipped with antenna arrays and adhered
to IEEE 802.11ad/ay communication standards. The key system parameters are
detailed in Table I. Utilizing the dataset described earlier in our
simulations, we proceed to present our findings.

Table I: System parameters Parameter | Value
---|---
Area of interest | 5$\times$5 m2
Height of access node | 5 m
Channel model | mmAgic InH
Antenna model | 3GPP
Traffic model | Full buffer
Antenna model | 3GPP antenna array[13]
Access node antenna arrays | 64 $\times$ 64
UE antenna arrays | var.
Beam management | Hierarchical codebook-based
Beacon interval | 100 ms
Sector-level sweep duration | 16 $\mu$s
Beam refinement duration | 1 $\mu$s
Carrier frequency | 60 GHz
Carrier bandwidth | 2.16 GHz
Receiver sensitivity | -78 dBm
Transmit power | 0 dBm
Noise figure | 9 dB
Power spectral density of noise | 174 dBm/Hz
Maximum EIRP | 40 dBm
Minimum SNR threshold | -10.3 dB (MCS 0) [14]

####  III-B1 Impact of Lateral Movements vs. Angular Movements

To assess the impact of mobility on highly directional beams, we focus on beam
misalignment, which is the difference between intended and actual beam
directions. In Fig. 2, we present cumulative distributions of beam
misalignment for all 40 participants, separately angular and lateral movements
for different antenna array sizes. To analyze the effects of lateral movement,
we maintain constant orientation angles and focus solely on the x, y, and z
displacements of the UE to quantify their impact on misalignment. Conversely,
to assess the influence of angular movement, we assume a constant lateral
position while examining changes in UE orientation. Our study investigates
beam misalignment for varying numbers of antenna elements. In codebook-based
beamforming, the number of beams aimed toward specific directions depend on
the antenna array size. During each Beacon Interval (BI), the beam with the
strongest signal is selected for the remainder of the BI, which may not be
perfectly aligned. Smaller antenna arrays form wider beams, resulting in
increased mean misalignment angles as compared to larger antenna arrays.

Our results highlight the more pronounced influence of angular movement on
beam misalignment. Furthermore, we observe that broader beams formed with
fewer antenna elements exhibit higher misalignment degrees. After the the
Sector Level Sweep (SLS) and the Beam Refinement Protocol (BRP), a new beam is
chosen for each BI by prioritizing the highest directivity gain. Importantly,
this selected beam is not necessarily perfectly aligned but is chosen for the
highest gain even when mostly misaligned. Wider beams demonstrate greater
tolerance for misalignment, which allows them to maintain sufficient gain even
under considerable misalignment. In contrast, narrower beams have lower
tolerance for misalignment, thereby requiring a new beam selection for even
minor misalignments.

The size of the antenna array is closely linked to the specific type of
immersive device being used. Smaller arrays are a practical choice for AR
glasses given their limited available space, while larger arrays are better
suited for VR goggles with more surface area. Additionally, continuous
technological advancements (such as smaller element spacing and on-lens
antennas) should enable the reduction of space requirements for these arrays
in the future, hence making them even more compact. It is important to
emphasize that, for all UE antenna arrays, the effective isotropic radiated
power (EIRP) has consistently remained below the maximum EIRP limit of 40 dBm,
in compliance with FCC and ETSI regulations. These outcomes highlight the
substantial effect of user orientation changes on XR communication system
performance.

Figure 2: Misalignment due to angular and lateral movement.

####  III-B2 Impact of Angular Speed on Beam Misalignment

Fig. 3 presents side-by-side comparisons of angular speeds and misalignment
angles concerning the headset orientation for all participants. Notably, the
central point of interest in VR gaming occurs when both pitch and yaw angles
are close to 0. Our observations reveal that the angular speed at this central
focus point is relatively slow, and as the user’s gaze moves away from the
center, the angular speed tends to increase. This phenomenon has potential
implications for link utilization. The speed at which the user’s head moves
directly affects the frame size in XR applications, particularly during higher
small-scale mobility characterized by the rapid angular speed of user head
rotation [10].

The gradual shift in gaze around the central focus point can be due to the
user’s focus changing more gradually when gazing towards the center,
potentially due to compensatory eye movements [15]. In contrast, challenges
arise when higher angular speeds manifest as the user explores the digital
environment or shifts focus to the peripheries, leading to increased data
transmission requirements. This exposes the headset to the risks of capacity
degradation and outages, particularly when higher data rates and reliability
are demanded.

Additionally, the right side of Fig. 3 displays the beam misalignment
corresponding to the angular speed on the left side. The results reveal a
clear relationship, indicating that higher angular speeds proportionally
correspond to greater angles of beam misalignment. Such misalignment can lead
to significant reductions in SNR hence directly impacting the QoE during
immersive VR sessions by introducing buffering, lower resolution, and
increased motion-to-photon latency. These findings highlight the importance of
monitoring and accounting for user angular speed to optimize the wireless
connection performance in VR scenarios, especially for delivering seamless and
immersive experiences. This highlights the need for responsive beam management
strategies, where beams can swiftly switch to compensate for misalignment.

Figure 3: Beam misalignment during VR gaming.

####  III-B3 Small-Scale Mobility Challenges in XR Communication

To analyze the effects of angular speed on system performance, we employ
synthetic user mobility traces, which capture angular speeds in a controlled
manner. Synthetic data enable testing wireless performance across a wide range
of parameters range, giving us control over the angular speed and facilitating
the calculation of the mean performance values.

As illustrated in Fig. 4, it is evident that narrower beams perform better at
lower angular speeds in terms of SNR. However, as the angular speed increases,
the performance of the system employing narrower beams starts to deteriorate,
leading to a decline in the mean SNR. This decrease can be attributed to the
growing need for more frequent beam-switching at higher speeds, which causes
substantial SNR drops during these transitions and potentially leads to
connection disruptions or outages. Conversely, smaller antenna arrays forming
wider beams offer broader coverage per beam, thereby reducing the necessity
for frequent beam switches and associated performance decline. The wider beams
benefit from better mean SNR at higher angular speeds, which suggests their
suitability for scenarios involving faster user movement, while employing
narrower beams during slower user movement.

In Fig. 5, we present the outage probability for varying numbers of antenna
elements at different angular speeds. Outage probability, which is a crucial
parameter for evaluating wireless system performance, represents the
proportion of time when the SNR falls below a minimum threshold, hence leading
to connection disruptions. Influenced by factors such as propagation
conditions, receiver sensitivity, transmit power, and antenna arrays at both
ends, outage probability increases with both higher numbers of antenna
elements and angular speeds. This rise can be attributed to the growing
frequency of beam switch events at higher speeds, where SNR drops drastically
during transitions, by leading to outages and temporary connection losses.
Additionally, transitioning from line-of-sight to non-line-of-sight beams as
the user head turns away from the access node may trigger outages, which are
more likely at higher speeds. For XR applications, this can cause frame
judder, hence leading to double images, ghosting, and blurring, thereby
disrupting the immersive experience.

Overall, our findings from the simulations highlight the significant impact of
user small-scale mobility, thereby particularly the angular speed, on mmWave
link performance between the UE and the access node in XR applications. Wider
beams perform better during faster user movement, while narrower beams yield
better performance during slower user movement. This necessitates dynamic
beamwidth adjustments according to the user movement. In summary, our findings
emphasize the importance of optimizing beam management strategies and
introducing redundancy in wireless systems to accommodate user mobility,
ensure reliability, and enhance the performance of directional connections in
immersive XR applications.

Figure 4: SNR w.r.t. angular speed.
Figure 5: Outage w.r.t. antenna elements.

##  IV Key Outcomes and Conclusions

Due to the inherent nature of human behavior during an XR session, the UE
experiences a significant degree of small-scale mobility. This fact, coupled
with the stringent communication requirements of XR applications that demand
highly directional communication, poses a formidable challenge in consistently
delivering a resilient connection without outages. As XR services become
increasingly prevalent, it is of significant importance for network
architectures and protocols to evolve to meet the XR wireless connectivity
requirements by providing reliable connectivity.

In this article, we investigate the impact of small-scale user mobility on
wireless XR communications, with a particular focus on its significance in the
context of highly directional transmission. We provide a comprehensive
overview of the small-scale mobility phenomenon during XR device usage and its
influence on wireless communication using highly directional beams.
Additionally, we discuss innovative approaches to address the challenges posed
by small-scale mobility in XR applications. To deepen our understanding of
small-scale mobility during consumer VR gaming, we collect a dataset
encompassing the movements of 40 participants.

Utilizing this dataset, we further conduct extensive simulations to study the
effects of small-scale mobility on highly directional communication and
analyze the output to identify the aspects of small-scale mobility that impact
performance. Our dataset and results provide valuable insights for researchers
and industry practitioners aiming to enhance wireless communication for XR
applications. The dataset can serve as a benchmark for evaluating and
comparing different mobility models and communication protocols to develop new
beam management mechanisms.

The key learning from our study is the notable influence of angular movements
on beam misalignment as compared to lateral movements. We further investigate
the relationship between angular speed and beam misalignment during VR gaming,
by revealing that higher speeds lead to greater misalignment angles and
potential SNR reductions, ultimately causing outages. Therefore, a careful
consideration of the trade-off between directivity gain and link reliability
is essential for the development of next-generation XR systems.

## References

  * [1] C.-X. Wang, X. You, X. Gao, X. Zhu, Z. Li, C. Zhang, H. Wang, Y. Huang, Y. Chen, H. Haas, J. S. Thompson, E. G. Larsson, M. D. Renzo, W. Tong, P. Zhu, X. Shen, H. V. Poor, and L. Hanzo, “On the Road to 6G: Visions, Requirements, Key Technologies, and Testbeds,” _IEEE Communications Surveys & Tutorials_, vol. 25, no. 2, pp. 905–974, 2023.
  * [2] Y. Heng, J. G. Andrews, J. Mo, V. Va, A. Ali, B. L. Ng, and J. C. Zhang, “Six Key Challenges for Beam Management in 5.5G and 6G Systems,” _IEEE Communications Magazine_ , vol. 59, no. 7, pp. 74–79, 2021.
  * [3] E. Khayrov and Y. Koucheryavy, “Packet Level Performance of 5G NR System Under Blockage and Micromobility Impairments,” _IEEE Access_ , vol. 11, pp. 90 383–90 395, 2023.
  * [4] N. Stepanov, D. Moltchanov, V. Begishev, A. Turlikov, and Y. Koucheryavy, “Statistical Analysis and Modeling of User Micromobility for THz Cellular Communications,” _IEEE Transactions on Vehicular Technology_ , vol. 71, no. 1, pp. 725–738, 2022.
  * [5] L. Grannemann, A. Ichkov, P. Mähönen, and L. Simić, “Urban Outdoor Measurement Study of Phased Antenna Array Impact on Millimeter-Wave Link Opportunities and Beam Misalignment,” _IEEE Transactions on Wireless Communications_ , vol. 20, no. 3, pp. 1727–1741, 2020.
  * [6] A. Ichkov, I. Gehring, P. Mähönen, and L. Simić, “Millimeter-Wave Beam Misalignment Effects of Small- and Large-Scale User Mobility Based on Urban Measurements,” in _Proceedings of the 5th ACM Workshop on Millimeter-Wave and Terahertz Networks and Sensing Systems_ , 2021, pp. 13–18.
  * [7] R. M. Magnani, S. M. Bruijn, J. H. van Dieen, and M. F. Vieira, “Head Orientation and Gait Stability in Young Adults, Dancers and Older Adults,” _Gait & Posture_, vol. 80, pp. 68–73, 2020.
  * [8] H. Hicheur, S. Vieilledent, and A. Berthoz, “Head Motion in Humans Alternating between Straight and Curved Walking Path: Combination of Stabilizing and Anticipatory Orienting Mechanisms,” _Neuroscience Letters_ , vol. 383, no. 1-2, pp. 87–92, 2005.
  * [9] C. Liu, M. Li, S. V. Hanly, I. B. Collings, and P. Whiting, “Millimeter Wave Beam Alignment: Large Deviations Analysis and Design Insights,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 7, pp. 1619–1631, 2017\.
  * [10] S. Kim and J.-H. Yun, “Motion-Aware Interplay between WiGig and WiFi for Wireless Virtual Reality,” _Sensors_ , vol. 20, no. 23, p. 6782, 2020.
  * [11] Opentrack tracking. Accessed on 08.07.2024. [Online]. Available: https://github.com/opentrack/opentrack
  * [12] A. Ali. Accessed on 08.07.2024. [Online]. Available: https://github.com/asadaliTUNI/VR-Gaming-Mobility-Dataset
  * [13] 3GPP SA TR 37.840, “Study of Radio Frequency (RF) and Electromagnetic Compatibility (EMC) requirements for Active Antenna Array System (AAS) Base Station,” 2014.
  * [14] Q. Chen, P. H. Tan, Z. Lin, S. Sun, and W. Zhao, “Design and Optimization of IEEE 802.11ad-Based Dense Network in Cabin Environment,” in _GLOBECOM 2017 IEEE Global Communications Conference_ , 2017, pp. 1–6.
  * [15] L. Sidenmark and H. Gellersen, “Eye, Head and Torso Coordination during Gaze Shifts in Virtual Reality,” _ACM Transactions on Computer-Human Interaction (TOCHI)_ , vol. 27, no. 1, pp. 1–40, 2019.

## Acknowledgements

This work was supported by the European Union’s Horizon 2020 Research and
Innovation Programme under the Marie Skłodowska-Curie grant agreement No.
813278 (A-WEAR; http://www.a-wear.eu/). This work was also supported by the
Research Council of Finland (Projects RADIANT, ECO-NEWS, and SOLID).

## Biography

Asad Ali<EMAIL_ADDRESS>is a joint Ph.D. student at Tampere University,
Finland, and Brno University of Technology (BUT), Czech Republic. He holds a
B.Sc. in Electrical Engineering from COMSATS University, Pakistan, and an
M.Sc. in Electrical Engineering from the University of Rostock, Germany. His
research focuses on 5G/6G wireless networks and extended reality.

Olga Galinina<EMAIL_ADDRESS>is a Senior Research Fellow at Tampere
University, Finland. She earned her B.Sc. and M.Sc. degrees from Peter the
Great St. Petersburg Polytechnic University and her Ph.D. from Tampere
University of Technology in 2015. Her research interests include applied
mathematical modeling and analysis of wireless networks and statistical
machine learning.

Jiri Hosek<EMAIL_ADDRESS>received his M.S. and Ph.D. in electrical
engineering from BUT in 2007 and 2011, respectively. He is currently an
associate professor (2016), deputy vice-head of R$\&$D and International
Relations in the Department of Telecommunications (2018), and coordinates the
WISLAB research group (since 2012) at the Department of Telecommunications,
BUT. He has (co-)authored 130+ research works in networking, wireless
communications, quality of service, quality of experience, and IoT
applications.

Sergey Andreev<EMAIL_ADDRESS>is a Professor of wireless
communications and an Academy Research Fellow at Tampere University, Finland.
He is also a Research Specialist with Brno University of Technology, Brno,
Czech Republic. He received his Ph.D. (2012) from TUT as well as his
Specialist (2006), Cand.Sc. (2009), and Dr.Habil. (2019) degrees from SUAI.
Sergey (co-)authored more than 300 published research works on intelligent
IoT, mobile communications, and heterogeneous networking.


# Fischer decompositions for entire functions of sufficiently low order

J.M. Aldaz and H. Render  H. Render: School of Mathematical Sciences,
University College Dublin, Dublin 4, Ireland<EMAIL_ADDRESS>J.M.
Aldaz: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento
de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco 28049, Madrid,
Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The existence of decompositions of the form $f=P\cdot q+r$ with
$P_{k}^{\ast}\left(D\right)r=0$, where $f$ is entire, $P$ a polynomial and
$P^{\ast}_{k}$ the principal part of $P$ with its coefficients conjugated, was
achieved in [1] under certain restrictions on the order of $f$. Here we prove
uniqueness, thereby obtaining Fischer decompositions, under conditions that
sometimes match those required for existence, and sometimes are more
restrictive, depending on the parameters involved.

2020 Mathematics Subject Classification: _Primary: 32A05_ , _Secondary: 35A01_

Key words and phrases: _Fischer decomposition, entire function of finite
order_

The first named author was partially supported by Grant PID2019-106870GB-I00
of the MICINN of Spain, by ICMAT Severo Ochoa project CEX2019-000904-S
(MICINN), and by V PRICIT (Comunidad de Madrid - Spain).

##  1\. Introduction

Let $\mathcal{P}\left(\mathbb{C}^{d}\right)=\mathbb{C}[z_{1},\dots,z_{d}]$ be
the set of all polynomials in
$z=\left(z_{1},\dots,z_{d}\right)\in\mathbb{C}^{d}$, and let
$P\in\mathcal{P}\left(\mathbb{C}^{d}\right)$. By $P^{\ast}\left(z\right)$ we
denote the polynomial obtained from $P\left(z\right)$ by conjugating its
coefficients, and by $P\left(D\right)$, the linear differential operator
obtained by replacing each appearance of the variable $z_{i}$ with
$\partial/\partial{z_{i}}$, for $i=1,\dots,d$. Fischer’s decomposition theorem
states that for every _homogeneous_ polynomial $P$ and every polynomial $f$
there exist _unique_ _polynomials_ $q$ and $r$ such that

$f=P\cdot q+r\text{ and }P^{\ast}\left(D\right)r=0.$

According to D. J. Newman and H. S. Shapiro, cf. [8, pg. 971], Fischer’s
result “underlies various formal schemes for exhibiting a basic set of
polynomial solutions of a partial differential equation”. H. S. Shapiro and
several of his collaborators studied Fischer decompositions in a wider setting
(cf. for instance [9], [7], [5], [6]) going beyond the case of polynomials to
more general function spaces, in particular, to the space
$E\left(\mathbb{C}^{d}\right)$ of all entire functions
$f:\mathbb{C}^{d}\rightarrow\mathbb{C}$. We recall a notion introduced in [9,
p. 522]: given a vector space $E$ of infinitely differentiable functions
$f:G\rightarrow\mathbb{C}$ (defined on an open subset $G$ of $\mathbb{C}^{d}$)
that is a module over $\mathcal{P}\left(\mathbb{C}^{d}\right)$, a polynomial
$P$ and a differential operator $Q\left(D\right)$ are said to form a _Fischer
pair for the space_ $E$, if for each $f\in E$ there exist _unique_ elements
$q\in E$ and $r\in E$ such that

(1) $f=P\cdot q+r\text{ and }Q\left(D\right)r=0.$

The question of uniqueness can be rephrased as follows: suppose that $f=P\cdot
q_{j}+r_{j}$ and $P_{k}^{\ast}\left(D\right)r_{j}=0$ for $j=1,2.$ Then with
$q:=q_{2}-q_{1}$ and $r=r_{2}-r_{1}$ we obtain the equation $0=Pq+r.$ Applying
the operator $P^{\ast}\left(D\right)$ we see that

(2) $P_{k}^{\ast}\left(D\right)\left(Pq\right)=0.$

Thus it is sufficient for uniqueness to show from (2) that $q=0.$

For simplicity in the exposition we indicate next a special case
(corresponding to the parameter $\tau=0$) of our main theorem, which will be
stated only after we recall the definition of the apolar inner product.

######  Theorem 1.

Let $0\leq\beta\leq k-1$, let $P$ be a non-homogeneous polynomial of degree
$k\geq 2$ with homogenous expansion $P=P_{0}+\cdots+P_{\beta}+P_{k}$, and let
$\varphi:\mathbb{C}^{d}\rightarrow\mathbb{C}$ be an entire function which
satisfies $P_{k}^{\ast}\left(D\right)\left(P\varphi\right)=0$ and has order
$\rho<2k^{-1}\left(k-\beta\right)$. Then $\varphi=0.$

In terms of the Fischer operator $F_{QP},$ defined via
$F_{QP}(\varphi):=Q(D)(P\varphi),$ the conclusion of Theorem 1 states that
$F_{P_{k}^{*}P}$ is injective on the entire functions of sufficiently low
order.

Note that we only deal with the case of a non-homogeneous polynomial $P$. The
bijectivity of the Fischer operator on the space of all entire functions when
$P$ is homogeneous, was proved by H. S. Shapiro in [9, Theorem 1].

We speak of _weak Fischer pairs_ when the decomposition $f=P\cdot q+r$ is not
required to be unique. In reference [1, Theorem 2], the authors proved the
existence of weak Fischer decompositions, under the bound
$\rho(k-\tau)<2(k-\beta)$ on the order $\rho$ of $f$. Our main result, Theorem
3 3 below, partially complements this existence result, yielding the
injectivity of the Fischer operator under sometimes more strict assumptions on
$\rho$, depending on the parameters.

##  2\. A lemma on sequences of nonnegative numbers

######  Lemma 2.

Let $\left\\{a_{m}\right\\}_{m\in\mathbb{N}}$ be a sequence of non-negative
numbers. Suppose there is a finite non-empty subset
$E\subset\mathbb{N}\setminus\\{0\\}$ satisfying the following two conditions:

(i) There exist constants $A\geq 1$, $D\geq 0$ and $\alpha\in\mathbb{R}$ such
that for all $m\in\mathbb{N}$,

$a_{m}\leq A\left(m+D\right)^{\alpha}\max_{j\in E}a_{m+j}.$

(ii) There exist constants $A_{0},b_{0}>0$ and $D_{0},\alpha_{0}\geq 0$ and
$\sigma\neq 0$ such that for all $m\in\mathbb{N}\setminus\\{0\\}$,

$a_{m}\leq A_{0}\ \frac{\left(m+D_{0}\right)^{\alpha_{0}}}{b_{0}^{m}}\
m^{-\frac{m}{\sigma}}\text{ }.$

Let $\beta_{\ast}:=\min E$ and $\beta^{\ast}=\max E.$ If

$0\leq\alpha<\frac{\beta_{\ast}}{\sigma},\text{ \quad or }\quad\alpha<0\text{
and }\alpha<\frac{\beta^{\ast}}{\sigma},$

then for all $m\in\mathbb{N}$ we have $a_{m}=0$. In the case
$\alpha=\frac{\beta_{\ast}}{\sigma}>0$ then the same conclusion holds under
the additional hypothesis that $A^{\beta_{\ast}}<b_{0}.$

###### Proof.

Assume that $a_{m}\neq 0.$ We derive a contradiction. Since

(3) $a_{m}\leq A\left(m+D\right)^{\alpha}\max_{j\in E}a_{m+j}$

there exists an $l_{1}\in E$ with

$a_{m}\leq A\left(m+D\right)^{\alpha}a_{m+l_{1}}.$

Applying inequality (3) to $a_{m+l_{1}}$ instead of $a_{m}$, we see that there
exists an $l_{2}\in E$ such that

$a_{m}\leq
A^{2}\left(m+D\right)^{\alpha}\left(m+l_{1}+D\right)^{\alpha}a_{m+l_{1}+l_{2}}.$

Now proceed inductively, to conclude that there exist $l_{1},\dots,l_{j-1}\in
E$ satisfying

$a_{m}\leq
A^{j-1}\left(m+D\right)^{\alpha}\left(m+l_{1}+D\right)^{\alpha}\cdots\left(m+D+l_{1}+\cdots+l_{j-1}\right)^{\alpha}a_{m+l_{1}+\cdots+l_{j}}.$

Define for $j\geq 1$,

$p_{m,j}:=\left(m+D\right)\left(m+l_{1}+D\right)\cdots\left(m+D+l_{1}+\cdots+l_{j-1}\right),$

and $k_{j}:=m+l_{1}+\cdots+l_{j}.$ Our assumption $a_{m}\leq
A_{0}\frac{\left(m+D_{0}\right)^{\alpha_{0}}}{b_{0}^{m}}\frac{1}{m^{\frac{m}{\sigma}}}$
implies that

$a_{k_{j}}\leq
A_{0}\frac{\left(k_{j}+D_{0}\right)^{\alpha_{0}}}{b_{0}^{k_{j}}}k_{j}^{-\frac{k_{j}}{\sigma}},\
$

so we have

$a_{m}\leq
A_{0}\frac{\left(k_{j}+D_{0}\right)^{\alpha_{0}}}{b_{0}^{k_{j}}}A^{j-1}p_{m,j}^{\alpha}k_{j}^{-\frac{k_{j}}{\sigma}}.$

Taking $k_{j}$-th roots in the last inequality shows that

(4) $a_{m}^{\frac{1}{k_{j}}}\leq
A_{0}^{\frac{1}{k_{j}}}\frac{\left(k_{j}+D_{0}\right)^{\frac{\alpha_{0}}{k_{j}}}}{b_{0}}A^{\frac{j-1}{k_{j}}}p_{m,j}^{\frac{\alpha}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}.$

Since $k_{j}=m+l_{1}+\cdots+l_{j}\geq m+\beta_{\ast}j\geq\beta_{\ast}j$ we see
that as $j\to\infty$, $k_{j}\rightarrow\infty$, so
$a_{m}^{\frac{1}{k_{j}}}\rightarrow 1$, $A_{0}^{\frac{1}{k_{j}}}\rightarrow
1$, $\left(k_{j}+D_{0}\right)^{\frac{\alpha_{0}}{k_{j}}}\rightarrow 1$, and
$A^{\frac{j-1}{k_{j}}}\leq A^{\frac{1}{\beta^{\ast}}}$. Thus, it suffices to
prove that $p_{m,j}^{\frac{\alpha}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}\rightarrow
0$ in order to obtain a contradiction, since then the left hand side of
inequality (4) converges to $1$ and the right hand side to $0.$

If $0\leq\alpha<\beta_{\ast}/\sigma$, then $m+D+l_{1}+\cdots+l_{s}\leq
D+k_{j}$ for $s=0,\dots,j-1$ and $p_{m,j}\leq\left(D+k_{j}\right)^{j},$ so

$p^{\frac{\alpha}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}\leq\left(D+k_{j}\right)^{\alpha\frac{j}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}\leq\left(D+k_{j}\right)^{\frac{\alpha}{\beta^{\ast}}}k_{j}^{-\frac{1}{\sigma}}=\left(\frac{D+k_{j}}{k_{j}}\right)^{\frac{\alpha}{\beta^{\ast}}}k_{j}^{\left(\frac{\alpha}{\beta^{\ast}}-\frac{1}{\sigma}\right)}.$

Since $\frac{\alpha}{\beta^{\ast}}-\frac{1}{\sigma}<0$ the term on the right
hand side converges to $0$. If $0<\alpha=\beta_{\ast}/\sigma$, then

$1=\lim\sup_{j\rightarrow\infty}a_{m}^{\frac{1}{k_{j}}}\leq\lim\sup_{j\rightarrow\infty}A_{0}^{\frac{1}{k_{j}}}\frac{\left(k_{j}+D_{0}\right)^{\frac{\alpha_{0}}{k_{j}}}}{b_{0}}A^{\frac{j-1}{k_{j}}}\left(\frac{D+k_{j}}{k_{j}}\right)^{\frac{\alpha}{\beta^{\ast}}}\leq\frac{1}{b_{0}}A^{\frac{1}{\beta^{\ast}}}<1.$

This finishes the proof when $\alpha\geq 0.$

Let us discuss next the case $\alpha<0.$ Since $k_{j}\geq\beta_{\ast}j\ $for
$j\geq 1$, we have

$p_{m,j}=\left(m+D\right)\left(m+l_{1}+D\right)\cdots\left(m+D+l_{1}+\cdots+l_{j-1}\right)\geq\beta_{\ast}^{j-1}\left(j-1\right)!.$

From $\alpha<0$ we conclude that

$p_{m,j}^{\frac{\alpha}{k_{j}}}\leq\left(\beta_{\ast}\right)^{\alpha\frac{j-1}{k_{j}}}\left[\left(j-1\right)!\right]^{\frac{\alpha}{k_{j}}}.$

Define $\beta^{\ast}=\max\left\\{l:l\in E\right\\}.$ Since
$k_{j}=m+l_{1}+\cdots+l_{j}$, it follows that

$\beta_{\ast}j\leq k_{j}\leq m+\beta^{\ast}j.$

Thus $\left(\beta_{\ast}\right)^{\alpha\frac{j-1}{k_{j}}}$ is bounded above.
Furthermore, a version of Stirling’s formula with explicit bounds for $m\geq
0$ tells us that

$m!\geq\left(2\pi
m\right)^{\frac{1}{2}}\left(\frac{m}{e}\right)^{m}e^{\frac{1}{12m+1}}.$

Using $\alpha<0$ and the preceding formula with $m=j-1\geq 1$ we obtain

$\left[\left(j-1\right)!\right]^{\frac{\alpha}{k_{j}}}\leq\left(2\pi\left(j-1\right)\right)^{\frac{1}{2}\frac{\alpha}{k_{j}}}\left(\frac{j-1}{e}\right)^{\alpha\frac{j-1}{k_{j}}}e^{\frac{\alpha}{12\left(j-1\right)+1}}.$

Note that

$\left(\frac{1}{e}\right)^{\alpha\frac{j-1}{k_{j}}}=e^{-\alpha\frac{j-1}{k_{j}}}\leq
e^{-\frac{\alpha}{\beta_{\ast}}},$

so

$\lim\sup_{j\rightarrow\infty}p^{\frac{\alpha}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}\leq\lim\sup_{j\rightarrow\infty}\left(\beta_{\ast}\right)^{\alpha\frac{j-1}{k_{j}}}e^{-\frac{\alpha}{\beta_{\ast}}}\left(j-1\right)^{\alpha\frac{j-1}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}$

Since $k_{j}\leq m+\beta^{\ast}j$ we have

$\frac{k_{j}}{j-1}\leq\frac{m}{j-1}+\beta^{\ast}+\frac{1}{j-1}\beta^{\ast}.$

By assumption $\alpha<\beta^{\ast}/\sigma$, where $\alpha<0$ (and we can have
either $\sigma<0$ or $\sigma>0$). Let $\varepsilon>0$ be such that
$\alpha<(1+\varepsilon)\beta^{\ast}/\sigma$. Now there exists a
$j_{\varepsilon}$ such that for all $j\geq j_{\varepsilon}$,

$\frac{k_{j}}{j-1}\leq\left(1+\varepsilon\right)\beta^{\ast}.$

Then whenever $j\geq j_{\varepsilon}$,

$j-1\geq k_{j}\left(1+\varepsilon\right)^{-1}\left(\beta^{\ast}\right)^{-1}.$

Using again that $\alpha<0$ we get

$\displaystyle\left(j-1\right)^{\alpha\frac{j-1}{k_{j}}}$ $\displaystyle\leq
k_{j}^{\alpha\frac{j-1}{k_{j}}}\left(1+\varepsilon\right)^{-\alpha\frac{j-1}{k_{j}}}\left(\beta^{\ast}\right)^{-\alpha\frac{j-1}{k_{j}}}$
$\displaystyle\leq
k_{j}^{\alpha\frac{j-1}{k_{j}}}\left(1+\varepsilon\right)^{-\alpha\frac{1}{\beta_{\ast}}}\left(\beta^{\ast}\right)^{-\alpha\frac{1}{\beta_{\ast}}}.$

Now

$\lim\sup_{j\rightarrow\infty}p^{\frac{\alpha}{k_{j}}}k_{j}^{-\frac{1}{\sigma}}\leq
e^{-\frac{\alpha}{\beta_{\ast}}}\left(1+\varepsilon\right)^{-\alpha\frac{1}{\beta_{\ast}}}\left(\beta^{\ast}\right)^{-\alpha\frac{1}{\beta_{\ast}}}\lim\sup_{j\rightarrow\infty}\left(\beta_{\ast}\right)^{\alpha\frac{j-1}{k_{j}}}k_{j}^{\alpha\frac{j-1}{k_{j}}}k_{j}^{-\frac{1}{\sigma}},$

and this limit equals $0$ since

$\alpha\frac{j-1}{k_{j}}\leq\frac{\alpha}{\left(1+\varepsilon\right)\beta^{\ast}}<\frac{1}{\sigma}.$

∎

##  3\. The apolar inner product, and statement of the main result

The _apolar inner product_ $\langle\cdot,\cdot\rangle_{a}$ on
$\mathbb{C}[z_{1},\dots,z_{d}]$ is defined by

(5) $\left\langle
P,Q\right\rangle_{a}:=\left(Q^{*}\left(D\right)P\right)\;(0)=\sum_{\alpha\in\mathbb{N}^{d}}\alpha!c_{\alpha}\overline{d_{\alpha}}.$

Note that for monomials $z^{\alpha}$ and $z^{\beta}$, definition (5) means
that $\left\langle z^{\alpha},z^{\beta}\right\rangle_{a}=0$ if
$\alpha\neq\beta$, and $\left\langle
z^{\alpha},z^{\alpha}\right\rangle_{a}=\alpha!$. Furthermore, from these
relations definition (5) is recovered by sesquilinearity.

A fundamental property of the apolar inner product, easily obtained from the
definition (alternatively, see [1, Proposition 3]) is that given polynomials
$Q,f$ and $g$, we have

(6) $\left\langle Q^{\ast}\left(D\right)f,g\right\rangle_{a}=\left\langle
f,Q\cdot g\right\rangle_{a}.$

Freom the apolar inner product we obtain the associated _apolar norm_ in the
usual way: $\left\|f\right\|_{a}:=\sqrt{\left\langle f,f\right\rangle_{a}}.$

We are now ready to state our main result. Regarding the notation used below
on the non-homogeneous polynomial
$P=P_{\beta_{1}}+\cdots+P_{\beta_{2}}+P_{k}$, we suppose not only that
$P_{k}\neq 0$, but also that $P_{\beta_{1}}\neq 0$ and $P_{\beta_{2}}\neq 0$;
however, it may happen that $\beta_{1}=\beta_{2}$.

######  Theorem 3.

Let $0\leq\beta_{1}\leq\beta_{2}\leq k-1$ and let $P$ be a non-homogeneous
polynomial of degree $k\geq 2$ with homogenous expansion
$P=P_{\beta_{1}}+\cdots+P_{\beta_{2}}+P_{k}$, where $P_{\beta_{1}}\neq 0$ and
$P_{\beta_{2}}\neq 0$. Suppose there exist $\tau\geq 0$ and $C>0$ such that
for all natural numbers $m\geq 0$ and all homogeneous polynomials $g_{m}$ of
degree $m$, we have

(7) $\left\|P_{k}g_{m}\right\|_{a}\geq
C\left(m+1\right)^{\tau/2}\left\|g_{m}\right\|_{a}.$

Let $\rho<2\left(k-\beta_{2}\right)/(k-\tau)$ if $\beta_{2}-\tau\geq 0$, and
let $\rho<2(k-\beta_{1})/(k+\beta_{2}-\beta_{1}-\tau)$ when
$\beta_{2}-\tau<0$. If $\varphi:\mathbb{C}^{d}\rightarrow\mathbb{C}$ is an
entire function of order $\rho$ satisfying
$P_{k}^{\ast}\left(D\right)\left(P\varphi\right)=0$, then $\varphi=0.$

Note that (7) always holds when $\tau=0$, as a consequence of Bombieri’s
inequality, cf. [3]. When $\tau\geq 1$, we call the inequalities (7)
_Khavinson-Shapiro bounds_ , cf. [1]. A more detailed discussion of condition
(7) can be found in [2]. If $k=1$, the Fischer operator $F_{P_{k}^{*}P}$ is
always bijective on the space of entire functions, by [1, Theorem 8].
Regarding the bounds on the order, we mention that if $\beta_{2}-\tau\geq 0$,
we have

$\frac{2}{k}\leq\frac{2(k-\beta_{2})}{k-\tau}\leq 2,$

while if $\beta_{2}-\tau<0$, then

$2<\frac{2(k-\beta_{1})}{k+\beta_{2}-\beta_{1}-\tau}\leq 2k.$

With respect to the existence result of [1, Theorem 2], there the assumption
on the order is

$\rho<\frac{2(k-\beta_{2})}{k-\tau},$

which coincides with the uniqueness bound for $\rho$ when $\beta_{2}\geq\tau$.
But if $\beta_{2}<\tau$, then

$\frac{2(k-\beta_{1})}{k+\beta_{2}-\beta_{1}-\tau}\leq\frac{2(k-\beta_{2})}{k-\tau},$

so the restriction on the order is at least as strong for the proof of
uniqueness as for existence.

##  4\. Entire functions of finite order

For $z\in\mathbb{C}^{d}$ we write
$\left|z\right|=\sqrt{\left|z_{1}\right|^{2}+\cdots+\left|z_{d}\right|^{2}}.$
The polynomial $P\left(z\right)$ is _homogeneous_ of degree $m$ if
$P\left(tz\right)=t^{m}P\left(z\right)$ for all $t>0$ and for all $z$. In
particular, the constant zero polynomial is homogeneous of every degree. We
use the standard notations for multi-indices
$\alpha=\left(\alpha_{1},\dots,\alpha_{d}\right)\in\mathbb{N}^{d}$, namely
$\alpha!=\alpha_{1}!\cdots\alpha_{d}!$,
$\left|\alpha\right|=\alpha_{1}+\cdots+\alpha_{d}$, and
$z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{d}^{\alpha_{d}}.$ We also write
$\mathbb{S}^{2d-1}=\left\\{z\in\mathbb{C}^{d}:\left|z\right|=1\right\\}.$

Suppose that $f:\mathbb{C}^{d}\rightarrow\mathbb{C}$ is an entire function.
Let

(8)
$M_{\mathbb{C}^{d}}\left(f,r\right):=\sup\left\\{\left|f\left(z\right)\right|:z\in\mathbb{C}^{d},\left|z\right|=r\right\\}.$

The order $\rho_{\mathbb{C}^{d}}\left(f\right)$ of $f$ is then defined as

$\rho_{\mathbb{C}^{d}}\left(f\right)=\lim_{r\rightarrow\infty}\sup\frac{\log\log
M_{\mathbb{C}^{d}}\left(f,r\right)}{\log r}.$

It is easy to see that $f$ has finite order $\leq\rho$ if and only if for all
$\varepsilon>0$ there exists a positive number $r_{\varepsilon}$ such that

$M_{\mathbb{C}^{d}}\left(f,r\right)\leq e^{r^{\rho+\varepsilon}}\text{ for all
}r\geq r_{\varepsilon}.$

Given $\rho\geq 0$, it is well known that
$\rho_{\mathbb{C}^{d}}\left(f\right)\leq\rho$ if and only if for every
$\varepsilon>0$, there exists an $m_{0}\geq 0$ such that for every $m\geq
m_{0}$ the following bounds hold:

(9)
$\max_{\theta\in\mathbb{S}^{2d-1}}\left|f_{m}\left(\theta\right)\right|\leq\frac{1}{m^{m/\left(\rho+\varepsilon\right)}}.$

Using the Taylor expansion one can write

$f\left(z\right)=\sum_{m=0}^{\infty}f_{m}\left(z\right)\text{ where
}f_{m}\left(z\right)=\sum_{\left|\alpha\right|=m}c_{m,\alpha}z^{\alpha},$

so each $f_{m}\left(z\right)$ is a homogeneous polynomial of degree $m$. As in
the complex case, for $x=\left(x_{1},\dots,x_{d}\right)\in\mathbb{R}^{d}$ the
euclidean distance $\left|x\right|$ is given by
$\left|x\right|^{2}=x_{1}^{2}+\cdots+x_{d}^{2}$. Let
$B_{R}:=\left\\{x\in\mathbb{R}^{n}:\left|x\right|<R\right\\}$ be the open ball
in $\mathbb{R}^{n}$ with center $0$ and radius $0<R\leq\infty.$

The following result is well known, see for instance [1, Lemma 17].

######  Lemma 4.

Let $f_{m}$ be a homogeneous polynomial of degree $m$, and denote by
$\mathbb{S}^{2d-1}$ the unit sphere in $\mathbb{R}^{2n}$. There is a
dimensional constant $C_{d}>0$ such that, identifying $\mathbb{C}^{n}$ with
$\mathbb{R}^{2n}$ as a measure space, we have

$\left\|f_{m}\right\|_{a}\leq
C_{d}\sqrt{\left(m+d-1\right)!}\max_{\theta\in\mathbb{S}^{2d-1}}\left|f_{m}\left(\eta\right)\right|.$

######  Proposition 5.

Assume that $f:\mathbb{C}^{d}\rightarrow\mathbb{C}$ is entire and has order
$\leq\rho.$ Then for every $m\geq 1$ and every $\varepsilon>0$, we have
$\left\|f_{m}\right\|_{a}\leq 2\sqrt{\pi}\ C_{d}\ e^{-m/2}\
\left(m+d-1\right)^{d/2}m^{m\left(\frac{1}{2}-\frac{1}{\rho+\varepsilon}\right)}.$

###### Proof.

By Lemma 4 we know that

$\left\|f_{m}\right\|_{a}\leq
C_{d}\sqrt{\left(m+d-1\right)!}\frac{1}{m^{m/\left(\rho+\varepsilon\right)}}.$

Furthermore $\left(m+d-1\right)!\leq m!\left(m+d-1\right)^{d}$. A standard
version of Stirling’s formula with explicit bounds for every $m\geq 1$, yields

$m!<\sqrt{2\pi m}\left(\frac{m}{e}\right)^{m}e^{\frac{1}{12m}}<2\sqrt{\pi
m}\left(\frac{m}{e}\right)^{m}$

since $e^{\frac{1}{12m}}<\sqrt{2}$. It follows that

$\displaystyle\left\|f_{m}\right\|_{a}$ $\displaystyle\leq
2C_{d}\sqrt{\pi}\left(m+d-1\right)^{d/2}\left(\frac{m}{e}\right)^{m/2}\frac{1}{m^{m/\left(\rho+\varepsilon\right)}}$
$\displaystyle=2C_{d}\sqrt{\pi}\ \frac{\left(m+d-1\right)^{d/2}}{e^{m/2}\
}\frac{1}{m^{m\left(\frac{1}{\rho+\varepsilon}-\frac{1}{2}\right)}}.$

∎

With different normalizations, the following result is essentially due to B.
Beauzamy, cf. [4, Formula (6)]. The exact statement appearing below is [1,
Lemma 13].

######  Lemma 6.

If
$P\left(z\right)={\displaystyle\sum_{\left|a\right|=k}}c_{\alpha}z^{\alpha}$
is a homogeneous polynomial of degree $k$, then

$\left\|Pf_{m}\right\|_{a}\leq\left\|f_{m}\right\|_{a}\left(1+m\right)^{\frac{k}{2}}{\displaystyle\sum_{\left|\alpha\right|=k}}\left|c_{\alpha}\right|\sqrt{\alpha!}.$

##  5\. Proof of Uniqueness

We are now ready to prove Theorem 3.

###### Proof.

Let $\varphi:\mathbb{C}^{d}\rightarrow\mathbb{C}$ be an entire function of
order $\rho<\frac{2\left(k-\beta_{2}\right)}{k-\tau}$, and let $P$ be a
polynomial of degree $k\geq 1$ with homogenous expansion
$P=P_{\beta_{1}}+\cdots+P_{\beta_{2}}+P_{k}$, where
$0\leq\beta_{1}\leq\beta_{2}\leq k-1.$ Let us write
$\varphi=\sum_{m=0}^{\infty}\varphi_{m}$, with each $\varphi_{m}$ a
homogeneous polynomial of degree $m$. Then $u:=P\varphi$ can be expressed as
$u=\sum_{m=0}^{\infty}u_{m},$ where the homogeneous polynomials $u_{m}$ of
degree $m$ satisfy

(10) $u_{m+k}=P_{k}\varphi_{m}+....+P_{0}\varphi_{m+k}.$

Note that $P_{k}^{\ast}\left(D\right)u=0$ implies that
$P_{k}^{\ast}\left(D\right)u_{m+k}=0$ for every $m\geq 0$, by uniqueness of
the representation of $u$ as a series of homogeneous polynomials. Taking the
inner product with $P_{k}\varphi_{m}$ on both sides of (10) and using
$P_{k}^{\ast}\left(D\right)u_{m+k}=0$, we obtain

$\left\|P_{k}\varphi_{m}\right\|_{a}^{2}+\sum_{j=1}^{k}\left\langle
P_{k-j}\varphi_{m+j},P_{k}\varphi_{m}\right\rangle_{a}=\left\langle
u_{m+k},P_{k}\varphi_{m}\right\rangle_{a}=\left\langle
P_{k}^{\ast}\left(D\right)u_{m+k},\varphi_{m}\right\rangle_{a}=0,$

so

$\left\|P_{k}\varphi_{m}\right\|_{a}^{2}\leq\sum_{j=1}^{k}\left|\left\langle
P_{k-j}\varphi_{m+j},P_{k}\varphi_{m}\right\rangle_{a}\right|\leq\sum_{j=1}^{k}\left\|P_{k-j}\varphi_{m+j}\right\|_{a}\left\|P_{k}\varphi_{m}\right\|_{a}$

by the Cauchy-Schwarz inequality. Thus
$\left\|P_{k}\varphi_{m}\right\|_{a}\leq\sum_{j=1}^{k}\left\|P_{k-j}\varphi_{m+j}\right\|_{a}.$
By Lemma 6, for each $j\in\\{1,\dots,k\\}$ there exists a constant $D_{k-j}$,
which depends only on $P_{k-j},$ and satisfies

$\left\|P_{k-j}\varphi_{m+j}\right\|_{a}\leq
D_{k-j}\left\|\varphi_{m+j}\right\|_{a}\left(1+m\right)^{\left(k-j\right)/2}.$

Defining

$E:=\left\\{j\in\left\\{1,\dots,k\right\\}:P_{k-j}\neq
0\right\\}\subset\left\\{k-\beta_{2},\dots,k-\beta_{1}\right\\},$

we conclude that

$\displaystyle\left\|P_{k}\varphi_{m}\right\|_{a}$
$\displaystyle\leq\sum_{j\in
E}D_{k-j}\left\|\varphi_{m+j}\right\|_{a}\left(1+m\right)^{\left(k-j\right)/2}$
$\displaystyle\leq\max_{s\in E}\left\|\varphi_{m+s}\right\|_{a}\sum_{j\in
E}D_{k-j}\left(1+m\right)^{\left(k-j\right)/2}.$

Note also that from the assumptions $P_{\beta_{1}}\neq 0$ and
$P_{\beta_{2}}\neq 0$ we obtain $\beta_{*}=k-\beta_{2}$ and
$\beta^{*}=k-\beta_{1}$. The change of indices $n:=k-j$ leads to

$\displaystyle\sum_{j\in E}D_{k-j}\left(1+m\right)^{\left(k-j\right)/2}$
$\displaystyle=\sum_{n=\beta_{1}}^{\beta_{2}}D_{n}\left(1+m\right)^{n/2}$
$\displaystyle\leq\left(1+m\right)^{\beta_{2}/2}\left(\sum_{n=\beta_{1}}^{\beta_{2}}\frac{D_{n}}{\left(1+m\right)^{\left(\beta_{2}-n\right)/2}}\right).$

Next we fix $0<\delta\ll 1$ and take $m_{0}=m_{0}(\delta)\gg 1$ so that

$\sum_{n=\beta_{1}}^{\beta_{2}}\frac{D_{n}}{\left(1+m_{0}\right)^{\left(\beta_{2}-n\right)/2}}\leq
D_{\beta_{2}}+\delta.$

By hypothesis there exist $\tau\geq 0$ and $C>0$ such that for all $m\geq
m_{0}$,

$C\left(m+1\right)^{\tau/2}\left\|\varphi_{m}\right\|_{a}\leq\left\|P_{k}\varphi_{m}\right\|_{a}.$

Hence

$\left\|\varphi_{m}\right\|_{a}\leq
C^{-1}\left(D_{\beta_{2}}+\delta\right)\frac{\left(1+m\right)^{\beta_{2}/2}}{\left(1+m\right)^{\tau/2}}\max_{j\in
E}\left\|\varphi_{m+j}\right\|_{a}\ .$

Let us write $a_{m}:=\left\|\varphi_{m}\right\|_{a}$,
$\alpha:=\left(\beta_{2}-\tau\right)/2$, and
$A:=C^{-1}\left(D_{\beta_{2}}+\delta\right)$. Then

$a_{m}\leq A\left(m+1\right)^{\alpha}\max_{j\in E}a_{m+j}$

for all $m\in\mathbb{N}$, so the first condition in Lemma 2 is satisfied. To
check the second condition we use Proposition 5, which tells us that for every
$\varepsilon>0$ the following inequality holds:

$a_{m}\leq
2\sqrt{\pi}C_{d}\frac{\left(m+d-1\right)^{d/2}}{e^{m/2}}\frac{1}{m^{m\left(\frac{1}{\rho+\varepsilon}-\frac{1}{2}\right)}}.$

so it holds for
$\sigma=\sigma_{\varepsilon}:=\frac{1}{\rho+\varepsilon}-\frac{1}{2}.$

Assume that $\beta_{2}-\tau\geq 0,$ so $\alpha\geq 0$ and
$\rho<2\frac{k-\beta_{2}}{k-\tau}.$ Choose $\varepsilon>0$ so that

(11) $\rho+\varepsilon<2\frac{k-\beta_{2}}{k-\tau}\leq 2.$

Since $\frac{1}{\sigma}=\frac{1}{\rho+\varepsilon}-\frac{1}{2}$, by Lemma 2 we
have that $a_{m}=0$ for every $m\geq 0$, provided
$\frac{1}{\sigma}>\frac{\alpha}{\beta_{\ast}}.$ This condition means that

$\frac{1}{\rho+\varepsilon}-\frac{1}{2}>\frac{1}{2}\frac{\beta_{2}-\tau}{k-\beta_{2}},$

which is equivalent to the first inequality in (11).

Now assume that $\beta_{2}-\tau<0$, and thus $\alpha<0$. By the bound on
$\rho$ there exists an $\varepsilon>0$ such that

$2\neq\rho+\varepsilon<2\frac{k-\beta_{1}}{k+\beta_{2}-\beta_{1}-\tau}.$

Setting $\sigma:=\frac{1}{\rho+\varepsilon}-\frac{1}{2}$, we conclude that

$\frac{\alpha}{\beta^{\ast}}=\frac{1}{2}\frac{\left(\beta_{2}-\tau\right)}{k-\beta_{1}}<\frac{1}{\sigma},$

so once again the hypotheses of Lemma 2 are satisfied. ∎

## References

  * [1] Aldaz, J.M., Render H.: _A Fischer type decomposition theorem from the apolar inner product._ Anal. Math. Phys. 13, 91, 29 pp. (2023).
  * [2] Aldaz, J.M., Render H.: _Asympotic bounds for Bombieri’s inequality on products of homogeneous polynomials._ Available at the Mathematics ArXiv.
  * [3] Beauzamy, B., Bombieri, E., Enflo, P., Montgomery, H.L.: _Products of Polynomials in Many Variables._ J. Number Theory 36 (1990) 219–245.
  * [4] Beauzamy, B.: _Extremal products in Bombieri’s norm._ Rend. Istit. Mat. Univ. Trieste, Suppl. Vol. XXVIII (1997), 73–89.
  * [5] Ebenfelt, P., Shapiro, H.S.: _The Cauchy–Kowaleskaya theorem and Generalizations._ Commun. Partial Diffential Equations, 20 (1995), 939–960.
  * [6] Ebenfelt, P., Shapiro, H.S.: _The mixed Cauchy problem for holomorphic partial differential equations._ J. D’Analyse Math. 65 (1996) 237–295.
  * [7] Khavinson, D., Shapiro, H.S.: _Dirichlet’s problem when the data is an entire function._ Bull. London Math. Soc. 24, 456–468 (1992)
  * [8] Newman, D.J., Shapiro, H.S.: _Certain Hilbert spaces of entire functions._ Bull. Amer. Math. Soc.  72, 971–977 (1966)
  * [9] Shapiro, H.S.: _An algebraic theorem of E. Fischer and the Holomorphic Goursat Problem._ Bull. London Math. Soc. 21, (1989) 513–537.