# LIME - a gas TPC prototype for directional Dark Matter search for the CYGNO
experiment

Fernando Domingues Amaro  LIBPhys; Department of Physics; University of
Coimbra; 3004-516 Coimbra; Portugal  Elisabetta Baracchini  Gran Sasso Science
Institute; 67100; L’Aquila; Italy  Istituto Nazionale di Fisica Nucleare;
Laboratori Nazionali del Gran Sasso; 67100; Assergi; Italy  Luigi Benussi
Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di Frascati;
00044; Frascati; Italy  Stefano Bianco  Istituto Nazionale di Fisica Nucleare;
Laboratori Nazionali di Frascati; 00044; Frascati; Italy  Cesidio Capoccia
Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di Frascati;
00044; Frascati; Italy  Michele Caponero  Istituto Nazionale di Fisica
Nucleare; Laboratori Nazionali di Frascati; 00044; Frascati; Italy  ENEA
Centro Ricerche Frascati; 00044; Frascati; Italy  Danilo Santos Cardoso
Centro Brasileiro de Pesquisas Físicas; Rio de Janeiro 22290-180; RJ; Brazil
Gianluca Cavoto  Dipartimento di Fisica; Università La Sapienza di Roma;
00185; Roma; Italy  André Cortez  Gran Sasso Science Institute; 67100;
L’Aquila; Italy  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali
del Gran Sasso; 67100; Assergi; Italy  Igor Abritta Costa  Dipartimento di
Matematica e Fisica; Università Roma TRE; 00146; Roma; Italy  Istituto
Nazionale di Fisica Nucleare; Sezione di Roma Tre; 00146; Rome; Italy
Emiliano Dané  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di
Frascati; 00044; Frascati; Italy  Giorgio Dho  Gran Sasso Science Institute;
67100; L’Aquila; Italy  Istituto Nazionale di Fisica Nucleare; Laboratori
Nazionali del Gran Sasso; 67100; Assergi; Italy  Flaminia Di Giambattista Gran
Sasso Science Institute; 67100; L’Aquila; Italy  Istituto Nazionale di Fisica
Nucleare; Laboratori Nazionali del Gran Sasso; 67100; Assergi; Italy  Emanuele
Di Marco  Istituto Nazionale di Fisica Nucleare; Sezione di Roma; 00185; Rome;
Italy  Giulia D’Imperio  Istituto Nazionale di Fisica Nucleare; Sezione di
Roma; 00185; Rome; Italy  Francesco Iacoangeli  Istituto Nazionale di Fisica
Nucleare; Sezione di Roma; 00185; Rome; Italy  Herman Pessoa Lima Júnior
Centro Brasileiro de Pesquisas Físicas; Rio de Janeiro 22290-180; RJ; Brazil
Guilherme Sebastiao Pinheiro Lopes  Universidade Federal de Juiz de Fora;
Faculdade de Engenharia; 36036-900; Juiz de Fora; MG; Brasil  Giovanni
Maccarrone  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di
Frascati; 00044; Frascati; Italy  Rui Daniel Passos Mano  LIBPhys; Department
of Physics; University of Coimbra; 3004-516 Coimbra; Portugal  Robert Renz
Marcelo Gregorio  Department of Physics and Astronomy; University of
Sheffield; Sheffield; S3 7RH; UK  David José Gaspar Marques  Gran Sasso
Science Institute; 67100; L’Aquila; Italy  Istituto Nazionale di Fisica
Nucleare; Laboratori Nazionali del Gran Sasso; 67100; Assergi; Italy  Giovanni
Mazzitelli  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di
Frascati; 00044; Frascati; Italy  Alasdair Gregor McLean  Department of
Physics and Astronomy; University of Sheffield; Sheffield; S3 7RH; UK  Andrea
Messina  Dipartimento di Fisica; Università La Sapienza di Roma; 00185; Roma;
Italy  Istituto Nazionale di Fisica Nucleare; Sezione di Roma; 00185; Rome;
Italy  Cristina Maria Bernardes Monteiro  LIBPhys; Department of Physics;
University of Coimbra; 3004-516 Coimbra; Portugal  Rafael Antunes Nobrega
Universidade Federal de Juiz de Fora; Faculdade de Engenharia; 36036-900; Juiz
de Fora; MG; Brasil  Igor Fonseca Pains  Universidade Federal de Juiz de Fora;
Faculdade de Engenharia; 36036-900; Juiz de Fora; MG; Brasil  Emiloano
Paoletti  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di
Frascati; 00044; Frascati; Italy  Luciano Passamonti  Istituto Nazionale di
Fisica Nucleare; Laboratori Nazionali di Frascati; 00044; Frascati; Italy
Sandro Pelosi  Istituto Nazionale di Fisica Nucleare; Sezione di Roma; 00185;
Rome; Italy  Fabrizio Petrucci  Dipartimento di Matematica e Fisica;
Università Roma TRE; 00146; Roma; Italy  Istituto Nazionale di Fisica
Nucleare; Sezione di Roma Tre; 00146; Rome; Italy  Stefano Piacentini
Dipartimento di Fisica; Università La Sapienza di Roma; 00185; Roma; Italy
Istituto Nazionale di Fisica Nucleare; Sezione di Roma; 00185; Rome; Italy
Davide Piccolo  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di
Frascati; 00044; Frascati; Italy  Daniele Pierluigi  Istituto Nazionale di
Fisica Nucleare; Laboratori Nazionali di Frascati; 00044; Frascati; Italy
Davide Pinci  Istituto Nazionale di Fisica Nucleare; Sezione di Roma; 00185;
Rome; Italy  Atul Prajapati  Gran Sasso Science Institute; 67100; L’Aquila;
Italy  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali del Gran
Sasso; 67100; Assergi; Italy  Francesco Renga  Istituto Nazionale di Fisica
Nucleare; Sezione di Roma; 00185; Rome; Italy  Rita Joanna da Cruz Roque
LIBPhys; Department of Physics; University of Coimbra; 3004-516 Coimbra;
Portugal  Filippo Rosatelli  Istituto Nazionale di Fisica Nucleare; Laboratori
Nazionali di Frascati; 00044; Frascati; Italy  Alessandro Russo  Istituto
Nazionale di Fisica Nucleare; Laboratori Nazionali di Frascati; 00044;
Frascati; Italy  Joaquim Marques Ferreira dos Santos  LIBPhys; Department of
Physics; University of Coimbra; 3004-516 Coimbra; Portugal  Giovanna Saviano
Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di Frascati;
00044; Frascati; Italy  Dipartimento di Ingegneria Chimica; Materiali e
Ambiente; Sapienza Università di Roma; 00185; Roma; Italy  Neil John Curwen
Spooner  Department of Physics and Astronomy; University of Sheffield;
Sheffield; S3 7RH; UK  Roberto Tesauro  Istituto Nazionale di Fisica Nucleare;
Laboratori Nazionali di Frascati; 00044; Frascati; Italy  Sandro Tommasini
Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali di Frascati;
00044; Frascati; Italy  Samuele Torelli  Gran Sasso Science Institute; 67100;
L’Aquila; Italy  Istituto Nazionale di Fisica Nucleare; Laboratori Nazionali
del Gran Sasso; 67100; Assergi; Italy

###### Abstract

The CYGNO experiment aims at the development of a large gaseous TPC with GEM-
based amplification and an optical readout by means of PMTs and scientific
CMOS cameras for 3D tracking down to O(keV) energies, for the directional
detection of rare events such as low mass Dark Matter and solar neutrino
interactions. The largest prototype built so far towards the realisation of
the CYGNO experiment demonstrator is the 50 L active volume LIME, with 4 PMTs
and a single sCMOS imaging a 33$\times$33 cm2 area for 50 cm drift, that has
been installed in underground Laboratori Nazionali del Gran Sasso in February
2022. We will illustrate LIME performances as evaluated overground in
Laboratori Nazionali di Frascati by means of radioactive X-ray sources, and in
particular the detector stability, energy response and energy resolution. We
will discuss the MC simulation developed to reproduce the detector response
and show the comparison with actual data. We will furthermore examine the
background simulation worked out for LIME underground data taking and
illustrate the foreseen expected measurement and results in terms of natural
and materials intrinsic radioactivity characterisation and measurement of the
LNGS underground natural neutron flux. The results that will be obtained by
underground LIME installation will be paramount in the optimisation of the
CYGNO demonstrator, since this is foreseen to be composed by multiple modules
with the same LIME dimensions and characteristics.

##  1 The CYGNO experiment

The identification of the nature of Dark Matter (DM) is one of the most
compelling tasks for fundamental physics today. A well motivated DM candidate
are weakly interacting massive particles (WIMPs), which could interact with
the ordinary matter of a target on Earth producing a nuclear recoil (NR). A
promising approach is to look for the directional signature of low energy
(less than 100 keV) NRs, whose angular distribution should point to the
direction of the Cygnus constellation [5]. The goal of the CYGNO experiment is
the direct detection of DM, with the use of a gaseous time projection chamber
(TPC) with an optical readout equipped with a triple-GEM (Gas Electron
Multiplier) amplification stage, operated at atmospheric pressure [2]. The
ionization electrons produced by the interaction of charged particles within
the gas volume are drifted by an electric field towards the triple GEM stack,
where they are amplified. The secondary scintillation light produced in the
electron avalanche is detected by scientific CMOS-based cameras (sCMOS), which
capture the light emission projected on the GEM plane, and Photo Multiplier
Tubes (PMTs) to determine the component along the drift direction. The He:CF4
gas mixture in 60/40 proportion provides a low energy threshold and a high
scintillation yield. Several prototypes were built and are being used for R&D
studies towards the final goal of a O(1 m3) CYGNO demonstrator with a modular
design, which will be installed underground at Laboratori Nazionali del Gran
Sasso (LNGS) of INFN to prove the feasibility of the approach.

##  2 The LIME prototype

The 50 L LIME detector is the largest prototype we built so far, and it
matches the dimension of one basic module of the future CYGNO demonstrator.
Triple thin GEMs with an area of 33$\times$33 cm2 amplify the charge produced
in the 50 cm long drift region. The high granularity Hamamatsu ORCA Fusion
sCMOS and 4 PMTs located at the readout area’s corners detect the secondary
scintillation light. The camera has 2304$\times$2304 pixels, each imaging an
area of about 160$\times$160 µm2, with a low noise of about 1 photon per
pixel, and a large quantum efficiency of 80% at 600 nm, which nicely matches
the spectral emission of our mixture. The employment of the camera and PMTs
together for 3D track reconstruction not only makes the detector sensitive to
the direction of the events, but also allows for the fiducialization of the
sensitive volume, lowering the impact of the background on the detector’s
sensitivity reach even further. LIME was tested using multiple X-ray sources
at the INFN Laboratori Nazionali di Frascati (LNF). For a month, the detector
ran in stable conditions, with a measured rate of current spikes less that 2.7
per hour, confirming the possibility of underground continuous operation. The
energy resolution was measured to be around 14% across the 50 cm drift length
using a 55Fe source generating 5.9 keV X-rays positioned on top of the
sensitive volume at varying distances from the GEM plane. An energy threshold
of 0.5 keV was set to ensure a maximum rate of 10 fake 55Fe events per year
due to noise. A multivariate regression analysis, taking the position of the
track and various shape parameters as inputs, is under development in order to
improve the energy resolution. The preliminary results show an energy
resolution better than 10% at 5.9 keV. To evaluate the detector’s response,
multiple radioactive X-ray sources with energies ranging from 3.7 keV (from
Ca) to 47 keV (from Tb) were employed, and a linear behavior was observed
across the whole energy range.

##  3 Monte Carlo simulation of the tracks

To reproduce the electronic recoil (ER) and NR tracks seen by the sCMOS, a
Monte Carlo (MC) simulation of the detector’s response is being performed.
First, a MC simulation of the ionization energy deposition as a function of 3D
position is done using GEANT4 [1] for the ERs and SRIM [6] for the NRs.
Various processes (such as ionization e- yield, diffusion along the drift
region and inside the GEMs, charge amplification, photon yield and collection
efficiency, absorption in the gas, gain saturation) are considered, and an
image is produced, which is then analyzed using an intensity-based DBSCAN
clustering algorithm [3], which we also use for the analysis of the real
images. The light integral and the track dimension of the simulated 5.9 keV
ERs was compared to 55Fe data, showing a good agreement across different GEM
gain values. In the linearity study, a preliminary comparison of X-ray data
and the MC simulation reveals an agreement within 10%. The study of the
discrimination between NRs and ERs is ongoing in order to improve the
background rejection capabilities of our approach, which was previously
measured with a smaller 7 L prototype [4].

##  4 Outlook

In February 2022, LIME was installed underground at LNGS to be tested under
the conditions of the future CYGNO experiment. These measurements will allow
us to characterize the performance of the detector in low radioactivity and
low pile-up configurations, measure the actual background, and validate the MC
simulation, as well as test the gas system and all the data analysis tools
developed for 3D track reconstruction and background rejection. The expected
background was simulated, taking into account both the intrinsic radioactivity
of the detector materials and the natural ambient gamma and neutron flux. The
application of fiducial cuts to our sensitive volume showed that the
radioactivity induced background events can be reduced by 96%. The shielding
design was optimized through dedicated MC simulations for different phases of
the upcoming data collection underground, and it comprises the use of copper
to shield the detector from gammas and water tanks to shield it from neutrons.
Before the installation of the water shielding, a spectral measurement of the
fast neutron flux underground will be done, which will provide valuable
information for all rare events search experiments running underground at
LNGS. The measurement will be done with LIME, which can detect the NRs
produced by the elastic scattering of fast environmental neutrons, whose
spectrum can be retrieved by deconvolving the measured NR spectrum with the
response of the detector. LIME is a critical step toward the development of
the CYGNO demonstrator, which will demonstrate the scalability of this
approach to large volumes.

## Acknowledgments

This project has received fundings under the European Union’s Horizon 2020
research and innovation programme from the European Research Council (ERC)
grant agreement No 818744. This project is supported by the Italian Ministry
of Education, University and Research through the project PRIN: Progetti di
Ricerca di Rilevante Interesse Nazionale ”Zero Radioactivity in Future
experiment” (Prot. 2017T54J9J). We want to thank General Services and
Mechanical Workshops of Laboratori Nazionali di Frascati (LNF) and Laboratori
Nazionali del Gran Sasso (LNGS) for their precious work and L. Leonzi (LNGS)
for technical support.

## References

  * [1] S. Agostinelli et al. GEANT4–a simulation toolkit. _Nucl. Instrum. Meth. A_ , 506:250–303, 2003
  * [2] Fernando Domingues Amaro et al. The CYGNO experiment. _Instruments_ , 6(1), 2022
  * [3] E. Baracchini et al. A density-based clustering algorithm for the CYGNO data analysis. _JINST_ , 15(12):T12003, 2020
  * [4] E. Baracchini et al. Identification of low energy nuclear recoils in a gas time projection chamber with optical readout. _Measurement Science and Technology_ , 32(2):025902, 2021
  * [5] F. Mayet et al. A review of the discovery reach of directional dark matter detection. _Physics Reports_ , 627, 03 2016
  * [6] James F. Ziegler, M.D. Ziegler, and J.P. Biersack. Srim – the stopping and range of ions in matter (2010). _Nucl. Instrum. Meth. B_ , 268(11):1818–1823, 2010.


155620.23+521520.0 | 0.227 | $1.17\pm 0.16$ | 1997-05-04 | $0.30\pm 0.02$ | 2010-12-10 | $0.58\pm 0.17$ | 2017-11-11 | $0.27\pm 0.01$ | 2019-08-13 | RQ | $-0.7$
162750.54+473623.5 | 0.262 | $1.26\pm 0.11$ | 1997-03-31 | $0.30\pm 0.02$ | 2010-12-7 | $1.06\pm 0.20$ | 2019-05-04 | $0.44\pm 0.01$ | 2019-09-20 | RQ | $-0.7$
115558.97+593129.2 | 0.241 | $0.63\pm 0.11$ | 2002-7-15 | $0.29\pm 0.01$ | 2010-12-22 | $0.62\pm 0.17$ | 2017-11-27 | $0.24\pm 0.02$ | 2019-08-04 | RQ | $-0.7$
085640.78+105755.8 | 0.274 | $1.35\pm 0.14$ | 2000-01-15 | $0.28\pm 0.01$ | 2010-12 | $0.72\pm 0.18$ | 2017-11-21 | $0.51\pm 0.10$ | 2019-08 | RQ | $-0.7$
123532.83+410445.1 | 0.212 | $<0.51$ | 1994-09-02 | $0.27\pm 0.02$ | 2010-12-27 | $0.45\pm 0.15$ | 2019-04-15 | $0.55\pm 0.07$ | 2019-08-13 | RQ | $-0.7$
161711.42+063833.4 | 0.229 | $<0.63$ | 2000-02-15 | $0.27\pm 0.01$ | 2010-12-23 | $<0.65$ | 2019-03-18 | $0.47\pm 0.01$ | 2019-08-20 | RQ | $-0.7$
130456.91+395111.4 | 0.271 | $0.62\pm 0.11$ | 1994-08-19 | $0.25\pm 0.02$ | 2010-12-16 | $0.59\pm 0.17$ | 2017-10-28 | $0.22\pm 0.01$ | 2019-08-17 | RQ | $-0.7$
125236.15+140213.9 | 0.266 | $0.63\pm 0.12$ | 2000-01-15 | $0.24\pm 0.01$ | 2010-12-17 | $<0.41$ | 2019-04-25 | $0.11\pm 0.01$ | 2019-08-17 | RQ | 0.0
100033.88+104723.7 | 0.226 | $1.78\pm 0.16$ | 2000-01-15 | $0.23\pm 0.01$ | 2010-12-5 | $0.87\pm 0.19$ | 2017-11-26 | $0.55\pm 0.01$ | 2019-08-04 | RQ | $-0.7$
082205.24+455349.1 | 0.300 | $0.43\pm 0.13$ | 1997-03-22 | $0.22\pm 0.01$ | 2010-12-20 | $<0.41$ | 2019-05-24 | $0.53\pm 0.03$ | 2019-08-13 | RQ | $-0.7$
130416.99+020537.0 | 0.229 | $0.87\pm 0.10$ | 1998-07-18 | $0.21\pm 0.01$ | 2010-12-18 | $<0.45$ | 2019-04-21 | $0.23\pm 0.04$ | 2019-08-20 | RQ | $-0.7$
104541.76+520235.5 | 0.284 | $0.49\pm 0.15$ | 1997-05-04 | $0.20\pm 0.01$ | 2010-12-19 | $<0.50$ | 2017-09-26 | $0.22\pm 0.03$ | 2019-08-04 | RQ | $-0.7$
104528.30+293344.7 | 0.293 | $0.60\pm 0.10$ | 1993-05-04 | $0.18\pm 0.01$ | 2010-12-12 | $<0.39$ | 2019-04-13 | $0.32\pm 0.01$ | 2019-08-04 | RQ | $-0.7$
145824.46+363119.5 | 0.246 | $<0.49$ | 1994-07-17 | $0.17\pm 0.01$ | 2010-12-25 | $0.51\pm 0.16$ | 2017-11-10 | $0.21\pm 0.03$ | 2019-09-20 | RQ | $-0.7$
142522.37+080327.1 | 0.230 | $<0.37$ | 2000-01-15 | $0.14\pm 0.01$ | 2010-12-29 | $<0.47$ | 2019-05-08 | $0.22\pm 0.01$ | 2019-08-17 | RQ | $-0.7$
153539.25+564406.5 | 0.207 | $0.76\pm 0.16$ | 1997-05-15 | $0.13\pm 0.02$ | 2010-12-24 | $<0.43$ | 2017-10-28 | $0.12\pm 0.01$ | 2019-08-13 | RQ | $-0.7$

Note. — RQQs whose 2019 A-configuration ($S_{\mathrm{A}}$) and 2010–2011
C-configuration ($S_{\mathrm{C}}$) flux densities differ by $>3\sigma$ (see
§4.3). Sources are listed by morphology R (Resolved) or U (Unresolved) (see
§4.1). “Inc.” (increase) indicates $S_{\mathrm{A}}>S_{\mathrm{C}}$); “Dec.”
(decrease) indicates $S_{\mathrm{A}}<S_{\mathrm{C}}$.

Note. — Data are from FIRST (Becker et al., 1995), Kimball et al. (2011b),
VLASS (Lacy et al., 2020), and the observations presented in this paper.
Columns 3 and 4 give each observation’s corresponding VLA configuration (and
fiducial angular resolution) and largest angular scale (LAS) detectable.

Note. — Number counts and luminosity-based definitions of RL/RI/RQ QSOs (as
introduced in Kimball et al., 2011b; Kellermann et al., 2016) presented in
§5.2.


# On integration with respect to filter

Dmytro Seliutin  School of Mathematics and Informatics V.N. Karazin Kharkiv
National University, 61022 Kharkiv, Ukraine<EMAIL_ADDRESS>
###### Abstract.

Using a concept of filter we propose one generalization of Riemann integral,
that is integration with respect to filter. We study this problem, demonstrate
different properties and phenomena of filter integration.

###### Key words and phrases:

integral, filter, ideal

##  1\. Introduction

Let us remind main concepts which we use in this paper. Throughout this
article $\Omega$ stand for a non-empty set. Non-empty family of subsets
$\mathfrak{F}\subset 2^{\Omega}$ is called filter on $\Omega$, if
$\mathfrak{F}$ satisfies the following axioms:

  1. (1)

$\emptyset\notin\mathfrak{F}$;

  2. (2)

if $A,\ B\in\mathfrak{F}$ then $A\cap B\in\mathfrak{F}$;

  3. (3)

if $A\in\mathfrak{F}$ and $D\supset A$ then $D\in\mathfrak{F}$.

Also very useful for us is a concept of filter base. Non-empty family of
subsets $\mathfrak{B}\subset 2^{\Omega}$ is called filter base on $\Omega$, if
$\emptyset\notin\mathfrak{B}$ and for every $A,\ B\in\mathfrak{B}$ there
exists $C\in\mathfrak{B}$ such that $C\subset A\cap B$. We say that filter
base generates filter $\mathfrak{F}$ if and only if for each
$A\in\mathfrak{F}$ there is $B\in\mathfrak{B}$ such that $B\subset A$.

Let $X$ be a topological vector space, $f:X\rightarrow{\mathbb{R}}$ be a
function. For $t\in X$ denote $\mathcal{O}(t)$ the family of all
neighbourhoods of $t$. Let $\mathfrak{F}$ be a filter on $X$,
$y\in{\mathbb{R}}$. Function $f$ is said to be convergent to $y$ over filter
$\mathfrak{F}$ (denote $y=\lim\limits_{\mathfrak{F}}f$), if for each
$U\in\mathcal{O}(y)$ there exists $A\in\mathfrak{F}$ such that for each $t\in
A$ the following holds true: $f(t)\in U$. We refers, for example, to [1] for
more information about filter and related concepts.

The concept of filter is a very powerful tool for studying different
properties of general topological vector spaces. For example, in [3] authors
study convergence over ideal, generated by the modular function. Ideal is a
concept dual to filter. In [2] we study completeness and its generalization
using filters.

In this this article we refer our attention to classical Riemann integral. Let
us remind how we can construct this object. Let $[a,b]\subset{\mathbb{R}}$,
let $f:[a,b]\rightarrow{\mathbb{R}}$ be a continuous function. Denote
$\Pi=\\{a\leqslant\xi_{1}\leqslant\xi_{2}\leqslant...\leqslant\xi_{n}=b\\}$
the partition of $[a,b]$, in other words,
$\overset{n}{\underset{k=1}{\cup}}[\xi_{k-1},\xi_{k}]=[a,b]$. Consider also
the set $T=\\{t_{1},t_{2},...,t_{n}\\}$ such that for each $k=1,2,...,n$
$t_{k}\in[\xi_{k-1},\xi_{k}]$. Let us call the pair $(\Pi,T)$ by the tagged
partition on the segment. Denote $d(\Pi)$ the diameter of the $\Pi$ – maximum
length of $[\xi_{k-1},\xi_{k}]$, where $k=1,2,...,n$. Let us recall that
function $f$ is said to be Riemann integrable if there exist the limit
$I=\lim\limits_{d(\Pi)\rightarrow
0}\sum\limits_{k=1}^{n}f(t_{k})\cdot|\xi_{k}-\xi_{k-1}|$, and we call this
limit the Riemann integral of the function $f$, and write
$I=\int\limits_{a}^{b}f(t)dt$. We know many different properties of this
integral, for example linearity, integration on subsegment of $[a,b]$ etc.

If we look at the definition of Riemann integral more attentively, we realize
that, in fact we can use one special filter and obtain desirable result. In
next section we are going to develop this idea.

##  2\. Integration with respect to filter

Just for simplicity we are going to consider functions, defined on $[0,1]$.
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a function. As above, denote
$\Pi=\\{a\leqslant\xi_{1}\leqslant\xi_{2}\leqslant...\leqslant\xi_{n}=b\\}$
the partition of $[0,1]$, in other words,
$\overset{n}{\underset{k=1}{\cup}}[\xi_{k-1},\xi_{k}]=[0,1]$. Consider also
the set $T=\\{t_{1},t_{2},...,t_{n}\\}$ such that for each $k=1,2,...,n$
$t_{k}\in[\xi_{k-1},\xi_{k}]$. For $k=1,2,...,n$ denote
$\Delta_{k}:=|\xi_{k}-\xi_{k-1}|$. Denote also $\text{TP}[0,1]$ the set of all
tagged partition of $[0,1]$. For a tagged partition $(\Pi,T)\in\text{TP}[0,1]$
denote

$S(f,\Pi,T)=\sum_{k=1}^{n}f(t_{k})\Delta_{k}.$

Now we are going to introduce the central definition of this paper

######  Definition 1.

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a function, $\mathfrak{F}$ be a filter
on $\text{TP}[0,1]$. We say that $f$ is integrable over filter $\mathfrak{F}$
($\mathfrak{F}$-integrable for short), if there exists $I\in{\mathbb{R}}$ such
that $I=\lim\limits_{\mathfrak{F}}S(f,\Pi,T)$. The number $I$ is called the
$\mathfrak{F}$-integral of the $f$ (denote
$I=\int\limits_{0}^{1}fd\mathfrak{F}$).

######  Remark 2.

The fact that $f$ is $\mathfrak{F}$-integrable we will write as follows:

$f\in\text{Int}(\mathfrak{F}).$

######  Remark 3.

Using Definition 1 we can construct the Riemann integral as follows. Let
$\delta>0$ be a real positive number. Denote

$P_{<\delta}=\\{(\Pi,T)\in\text{TP}[0,1]:\ d(\Pi)<\delta\\},$

where $d(\Pi)$ stands for diameter of $\Pi$. Consider now

$\mathfrak{B}_{<\delta}=\\{P_{<\delta}:\delta>0\\}.$

It is easy to check that $\mathfrak{B}_{<\delta}$ is a filter base. Denote
$\mathfrak{F}_{<\delta}$ filter generate by $\mathfrak{B}_{<\delta}$. Let
$f:[0,1]\rightarrow{\mathbb{R}}$ be a function. Then $f$ is integrable by
Riemann if there exists the limit
$\lim\limits_{\mathfrak{F}_{<\delta}}S(f,\Pi,T)$.

Bellow we study different properties of filter integration.

######  Definition 4.

Let $X$ be a non-empty set, $f:X\rightarrow{\mathbb{R}}$ be a function, and
$\mathfrak{F}$ be a filter on $X$. We say that $f$ is bounded with respect to
$\mathfrak{F}$ ($\mathfrak{F}$-bounded for short), if there is $C>0$ such that
there exists $A\in\mathfrak{F}$ such that for every $t\in A$ $|f(t)|<C$.

The following lemma is very simple, but for readers convenient we present its
proof.

######  Lemma 5.

Let $X$ be a non-empty set, $f:X\rightarrow{\mathbb{R}}$ be a function, and
$\mathfrak{F}$ be a filter on $X$. Suppose that there exists
$I\in{\mathbb{R}}$, $I=\lim\limits_{\mathfrak{F}}f$. Then $f$ is
$\mathfrak{F}$-bounded.

###### Proof.

We know that $I=\lim\limits_{\mathfrak{F}}f$. It means that for every
$\varepsilon>0$ there exists $A\in\mathfrak{F}$ such that for all $t\in A$
$|f(t)-I|<\varepsilon$. Consider

$|f(t)|-|I|\leqslant|f(t)-I|<\varepsilon.$

In other words, $|f(t)|\leqslant|I|+\varepsilon$. Then just put
$C:=|I|+\varepsilon$. ∎

The next theorem generalizes well-know fact about Riemann integral: if
function in integrable by Riemann then it’s bounded.

######  Theorem 6.

Let $\mathfrak{F}$ be a filter on $\text{TP}[0,1]$,
$f:[0,1]\rightarrow{\mathbb{R}}$ be a function, and
$f\in\text{Int}(\mathfrak{F})$. Then $S(f,\Pi,T)$ is $\mathfrak{F}$-bounded.

###### Proof.

Just use Lemma 5. ∎

Let us formulate well-known fact about Riemann integral, using filters.

######  Theorem 7.

Let $f:[0,1]\rightarrow{\mathbb{R}}$, there exists
$\lim\limits_{\mathfrak{F}_{<\delta}}S(f,\Pi,T)$. Then $f$ is bounded, in
other words, there is $C>0$ such that for all $t\in[0,1]$ $|f(t)|\leqslant C$.

The next theorem is natural generalization of the Theorem 7.

######  Theorem 8.

Let $f:[0,1]\rightarrow{\mathbb{R}}$, let $\mathfrak{F}$ be a filter on
$TP[0,1]$ such that for every $A\in\mathfrak{F}$ there exists
$B\in\mathfrak{F}_{<\delta}$ such that $B\subset A$ and let there exists
$I\in{\mathbb{R}}$ such that $I=\lim\limits_{\mathfrak{F}}S(f,\Pi,T)$. Then
$C>0$ such that for each $t\in[0,1]$ we have $|f(t)|<C$.

###### Proof.

There exists $I\in{\mathbb{R}}$ such that
$I=\lim\limits_{\mathfrak{F}}S(f,\Pi,T)$ $\Leftrightarrow$ for all
$\varepsilon>0$ there exists $A\in\mathfrak{F}$ such that for all $(\Pi,T)\in
A$ $|S(f,\Pi,T)-I|<\varepsilon$. We know that for $A\in\mathfrak{F}$ there is
$B\in\mathfrak{F}_{<\delta}$ such that $B\subset A$, then, particularly, for
all $\varepsilon>0$ there exists $A\in\mathfrak{F}$ there is
$B\in\mathfrak{F}_{<\delta}$ such that $B\subset A$ such that for all
$(\Pi,T)\in B$ $|S(f,\Pi,T)-I|<\varepsilon$ $\Rightarrow$ for all
$\varepsilon>0$ there exists $B\in\mathfrak{F}_{<\delta}$ such that for all
$(\Pi,T)\in B$ $|S(f,\Pi,T)-I|<\varepsilon$ $\overset{\text{Theorem
\ref{bound}}}{\Rightarrow}$ $C>0$ such that for each $t\in[0,1]$ we have
$|f(t)|<C$, in other words, $f$ is bounded. ∎

Now we are going to demonstrate that filter integration has additive property.
To demonstrate this we proof next easy two lemmas.

######  Lemma 9.

Let $X$ be a non-empty set, $f,\ g:X\rightarrow{\mathbb{R}}$ be a functions,
and $\mathfrak{F}$ be a filter on $X$. Let $x=\lim\limits_{\mathfrak{F}}f$,
$y=\lim\limits_{\mathfrak{F}}g$. Then $\lim\limits_{\mathfrak{F}}(f+g)=x+y$.

###### Proof.

We know that $x=\lim\limits_{\mathfrak{F}}f$, so for each $U\in\mathcal{O}(x)$
there is $A\in\mathfrak{F}$ such that $f(A)\subset U$. Analogically,
$y=\lim\limits_{\mathfrak{F}}f$, it means that for each $V\in\mathcal{O}(x)$
there is $B\in\mathfrak{F}$ such that $f(B)\subset V$. We have to demonstrate
that for each $W\in\mathcal{O}(x+y)$ there exists $C\in\mathfrak{F}$ such that
$(f+g)(C)\subset W$. Let fix $W\in\mathcal{O}(x+y)$. Then there exist
$W_{1}\in\mathcal{O}(x)$ and $W_{2}\in\mathcal{O}(y)$ such that $W\supset
W_{1}+W_{2}$. Then there are $C_{1},\ C_{2}\in\mathfrak{F}$ such that
$f(C_{1})\subset W_{1}$ and $f(C_{2})\subset W_{2}$. Denote $C:=C_{1}\cap
C_{2}$. Clearly that $C\in\mathfrak{F}$. So

$(f+g)(C)=f(C)+g(C)\subset W_{1}+W_{2}\subset W.$

∎

######  Lemma 10.

Let $X$ be a non-empty set, $f:X\rightarrow{\mathbb{R}}$ be a function,
$\mathfrak{F}$ be a filter on $X$, and $\alpha\in{\mathbb{R}}$. Let
$x=\lim\limits_{\mathfrak{F}}f$. Then $\lim\limits_{\mathfrak{F}}\alpha
f=\alpha x$.

###### Proof.

$x=\lim\limits_{\mathfrak{F}}f$, it means that for each $U\in\mathcal{O}(x)$
there is $A\in\mathfrak{F}$ such that $f(A)\subset U$. We have to demonstrate
that for all $V\in\mathcal{O}(\alpha x)$ there is $B\in\mathfrak{F}$ such that
$(\alpha f)(B)\subset V$. Suppose that $\alpha\neq 0$. The case $\alpha=0$ is
obvious. Remark that if $W\in\mathcal{O}(x)$ then $\alpha
W\in\mathcal{O}(\alpha x)$. So just put $B:=A$. Then $(\alpha f)(B)=\alpha
f(B)\subset\alpha U\in\mathcal{O}(\alpha x)$. ∎

######  Theorem 11.

Let $\mathfrak{F}$ be a filter on $\text{TP}[0,1]$,
$f,g:[0,1]\rightarrow{\mathbb{R}}$ be a functions,
$f\in\text{Int}(\mathfrak{F})$, $\alpha,\ \beta\in{\mathbb{R}}$, and
$f\in\text{Int}(\mathfrak{F})$ and $g\in\text{Int}(\mathfrak{F})$. Then
$(\alpha f+\beta g)\in\text{Int}(\mathfrak{F})$

###### Proof.

Just use Lemmas 9 and 10. ∎

##  3\. Integration with respect to different filters

In the previous section we’ve studied arithmetic properties of integral over
filter and problems deals with boundedness. This section is devoted to
integration over different filters and its relations.

For $(\Pi,T)\in\text{TP}[0,1]$ and $t\in T$ we denote $\Delta(t)$ length of
the element of partition of $\Pi$ which covers $t$.

Let $(\Pi_{1},T_{1}),(\Pi_{2},T_{2})$ be partitions of $[0,1]$. Consider

$\begin{split}\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=\\\ \sum_{t\in T_{1}\cap
T_{2}}|\Delta_{1}(t)-\Delta_{2}(t)|+\sum_{T_{1}\setminus
T_{2}}\Delta_{1}(t)+\sum_{T_{2}\setminus T_{1}}\Delta_{2}(t).\end{split}$
(3.1)

For easy using of concept defined in Equation 3.1 consider
${\mathbb{F}}:[0,1]\rightarrow l_{1}[0,1]$, such that ${\mathbb{F}}(t)=e_{t}$,
where

$e_{t}(\tau)=\begin{cases}1,\text{if }\tau=t;\\\
0,\text{otherwise}.\end{cases}$

It is clearly then that

$\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=||S({\mathbb{F}},\Pi_{1},T_{1})-S({\mathbb{F}},\Pi_{2},T_{2})||.$

Now we are going to demonstrate that the mapping $\rho$, defined above, is a
metric, or distance between two tagged partitions.

######  Proposition 1.

Consider $\rho:\text{TP}[0,1]\times\text{TP}[0,1]\rightarrow{\mathbb{R}}$,
$\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=||S({\mathbb{F}},\Pi_{1},T_{1})-S({\mathbb{F}},\Pi_{2},T_{2})||$.
Then $\rho$ satisfies all metric axioms.

###### Proof.

  1. (1)

let $(\Pi_{1},T_{1})=(\Pi_{2},T_{2})$.

It is clear that in this case $\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=0$;

  2. (2)

let $\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=0$.

Then $\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=\sum\limits_{t\in T_{1}\cap
T_{2}}|\Delta_{1}(t)-\Delta_{2}(t)|+\sum\limits_{T_{1}\setminus
T_{2}}\Delta_{1}(t)+\sum\limits_{T_{2}\setminus T_{1}}\Delta_{2}(t)=0$. We
have a sum of non-negative numbers equals to 0. This means that

     * •

$\forall t\in T_{1}\cap T_{2}$ $|\Delta_{1}(t)-\Delta_{2}(t)|=0$ $\Rightarrow$
$\forall t\in T_{1}\cap T_{2}$ $\Delta_{1}(t)=\Delta_{2}(t)$;

     * •

$\forall t\in T_{1}\setminus T_{2}$ $\Delta_{1}(t)=0$;

     * •

$\forall t\in T_{2}\setminus T_{1}$ $\Delta_{2}(t)=0$;

$\Rightarrow$ $(\Pi_{1},T_{1})=(\Pi_{2},T_{2})$.

  3. (3)

consider $(\Pi_{1},T_{1}),(\Pi_{2},T_{2}),(\Pi_{3},T_{3})$. Then

$\begin{split}\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))=\\\
||S({\mathbb{F}},\Pi_{1},T_{1})-S({\mathbb{F}},\Pi_{2},T_{2})+S({\mathbb{F}},\Pi_{3},T_{3})-S({\mathbb{F}},\Pi_{3},T_{3})||\leqslant\\\
||S({\mathbb{F}},\Pi_{1},T_{1})-S({\mathbb{F}},\Pi_{3},T_{3})||+||S({\mathbb{F}},\Pi_{3},T_{3})-S({\mathbb{F}},\Pi_{2},T_{2})||=\\\
\rho((\Pi_{1},T_{1}),(\Pi_{3},T_{3}))+\rho((\Pi_{3},T_{3}),(\Pi_{2},T_{2}))\end{split}$

∎

Now we introduce very important concept.

######  Definition 12.

Let $\mathfrak{F}_{1},\mathfrak{F}_{2}$ be filters on $TP[0,1]$. We say that
$\mathfrak{F}_{2}$ $\rho$-dominates filter $\mathfrak{F}_{1}$
($\mathfrak{F}_{2}\succ_{\rho}\mathfrak{F}_{1}$), if for every $\varepsilon<0$
and for each $A_{1}\in\mathfrak{F}_{1}$ there exists
$A_{2}\in\mathfrak{F}_{2}$ such that for all $(\Pi_{2},T_{2})\in A_{2}$ there
is $(\Pi_{1},T_{1})\in A_{1}$ such that
$\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))<\varepsilon$.

######  Proposition 2.

Let $\mathfrak{F}_{2}\supset\mathfrak{F}_{1}$. Then $\mathfrak{F}_{2}$
$\rho$-dominates $\mathfrak{F}_{1}$.

###### Proof.

As $\mathfrak{F}_{2}\supset\mathfrak{F}_{1}$ we obtain that if
$A\in\mathfrak{F}_{1}$ then $A\in\mathfrak{F}_{2}$. Consider an arbitrary
$\varepsilon>0$. Then for every $A_{1}\in\mathfrak{F}_{1}$ there is
$A_{2}\in\mathfrak{F}_{2}$, $A_{2}:=A_{1}$ such that for each
$(\Pi_{2},T_{2})\in A_{2}$ there exists $(\Pi_{1},T_{1})\in A_{1}$,
$(\Pi_{1},T_{1}):=(\Pi_{2},T_{2})$ such that
$\rho\left((\Pi_{1},T_{1}),(\Pi_{2},T_{2})\right)=\rho\left((\Pi_{2},T_{2}),(\Pi_{2},T_{2})\right)=0<\varepsilon$.
∎

Previous proposition shows us that $\rho$-dominance generates some relation of
order on $\text{TP}[0,1]$ and is more general concept that relation of
inclusion.

It is clear that if $\mathfrak{F}_{1}\subset\mathfrak{F}_{2}$ and
$f\in\text{Int}(\mathfrak{F}_{1})$ then $f\in\text{Int}(\mathfrak{F}_{2})$ –
just use the definition of function limit over filter. So we can formulate
next easy proposition.

######  Proposition 3.

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a function, $\mathfrak{F}_{1},\
\mathfrak{F}_{2}$ be filters on $\text{TP}[0,1]$ such that $\mathfrak{F}_{1}\
\subset\mathfrak{F}_{2}$ and $f\in\text{Int}(\mathfrak{F}_{1})$. Then
$f\in\text{Int}(\mathfrak{F}_{2})$.

######  Theorem 13.

Let $\mathfrak{F}_{1},\mathfrak{F}_{2}$ be filters on $[0,1]$. Let
$f:[0,1]\rightarrow{\mathbb{R}}$ be a bounded function. Let
$I=\lim\limits_{\mathfrak{F}_{1}}S(f,\Pi,T)$ and
$\mathfrak{F}_{2}\succ_{\rho}\mathfrak{F}_{1}$. Then
$I=\lim\limits_{\mathfrak{F}_{2}}S(f,\Pi,T)$.

###### Proof.

Denote $C:=\sup\limits_{t\in[0,1]}|f(t)|$.

We have to proof that for every $\varepsilon>0$ there exists
$B\in\mathfrak{F}_{2}$ such that for each $(\Pi_{B},T_{B})\in B$ we have
$|S(f,\Pi_{B},T_{B})-I|<\varepsilon$.

We know that for every $\varepsilon>0$ there exists $A\in\mathfrak{F}_{1}$
such that for each $(\Pi_{1},T_{1})\in A$ we have
$|S(f,\Pi_{1},T_{1})-I|<\varepsilon$.

Now for an arbitrary $\varepsilon>0$ and $A\in\mathfrak{F}_{1}$ found above
one can find $A_{2}\in\mathfrak{F}_{2}$ such that for all $(\Pi_{2},T_{2})\in
A_{2}$ there is $(\Pi_{1},T_{1})\in A_{1}$ such that
$\rho((\Pi_{1},T_{1}),(\Pi_{2},T_{2}))<\varepsilon$. Then put $B:=A_{2}$. Then
for all $(\Pi_{B},T_{B})\in B$ we have $(\Pi_{1},T_{1})\in A_{1}$ such that

$\displaystyle|S(f,\Pi_{B},T_{B})-I|=$
$\displaystyle|S(f,\Pi_{B},T_{B})-S(f,\Pi_{1},T_{1})+S(f,\Pi_{1},T_{1})-I|\leqslant$
$\displaystyle|S(f,\Pi_{B},T_{B})-S(f,\Pi_{1},T_{1})|+|S(f,\Pi_{1},T_{1})-I|=$
$\displaystyle\sum_{t\in T_{B}\cap
T_{1}}|f(t)|\cdot|\Delta_{B}(t)-\Delta_{1}(t)|+\sum_{t\in T_{B}\setminus
T_{1}}|f(t)|\cdot\Delta_{B}(t)+$ $\displaystyle\sum_{t\in T_{1}\setminus
T_{B}}|f(t)|\cdot\Delta_{1}(t)+\varepsilon\leqslant
C\cdot\rho((\Pi_{B},T_{B}),(\Pi_{1},T))+\varepsilon\leqslant$ $\displaystyle
C\varepsilon+\varepsilon\leqslant\varepsilon(1+C).$

∎

##  4\. Exactly tagged filters

In this part of our paper we consider problems deals filter integration of
unbounded functions.

######  Definition 14.

Let $\mathfrak{B}$ be a filter base on $TP[0,1]$. We say that $\mathfrak{B}$
is exactly tagged if there exist $A\subset[0,1]$ – a strictly decreasing
sequence of numbers such that for each $B\in\mathfrak{B}$ and for every
$(\Pi,T)\in B$ we have that $T\cap A=\emptyset$.

######  Definition 15.

We say that filter $\mathfrak{F}$ on $TP[0,1]$ is exactly tagged if there
exists exactly tagged base $\mathfrak{B}$ of $\mathfrak{F}$.

######  Theorem 16.

If filter $\mathfrak{F}$ on $TP[0,1]$ is exactly tagged then there exists
unbounded function $f:[0,1]\rightarrow{\mathbb{R}}$ such that $f\in
Int(\mathfrak{F})$.

###### Proof.

Denote
$\displaystyle{\mathbb{N}}^{-1}=\left\\{\frac{1}{n}\right\\}_{n\in{\mathbb{N}}}$
and consider next filter base $\mathfrak{B}=(B_{n})_{n\in{\mathbb{N}}}$ on
$TP[0,1]$:
$\displaystyle B_{1}=\left\\{(\Pi,T):T\cap{\mathbb{N}}^{-1}=\emptyset\text{
and }d(\Pi)<1\right\\}$;
$\displaystyle B_{2}=\left\\{(\Pi,T):T\cap{\mathbb{N}}^{-1}=\emptyset\text{
and }d(\Pi)<\frac{1}{2}\right\\}$;
$\displaystyle B_{3}=\left\\{(\Pi,T):T\cap{\mathbb{N}}^{-1}=\emptyset\text{
and }d(\Pi)<\frac{1}{3}\right\\}$;
…
$\displaystyle B_{m}=\left\\{(\Pi,T):T\cap{\mathbb{N}}^{-1}=\emptyset\text{
and }d(\Pi)<\frac{1}{m}\right\\}$.

Consider now

$f(t)=\begin{cases}\displaystyle n,\text{ if }t=\frac{1}{n},\
n\in{\mathbb{N}}\\\ 0,\text{ otherwise}\end{cases}.$

Then for each $n\in{\mathbb{N}}$ and for every $(\Pi,T)\in B_{n}$ we have that
$S(f,\Pi,T)=0$, so $\lim\limits_{\mathfrak{B}}S(f,\Pi,T)=0$. ∎

For a tagged partition $(\Pi,T)$ of $[0,1]$ and $\tau\in[0,1]$ denote
$\ell(\Pi,T,\tau)$ the number which is equal to the length of the segment
$\Delta\in\Pi$, for which $\tau\in\Delta$, if $\tau\in T$. If $\tau\notin T$,
we put $\ell(\Pi,T,\tau)=0$. In this notation

$S(f,\Pi,T)=\sum_{t\in[0,1]}f(t)\ell(\Pi,T,t).$

######  Theorem 17.

For a filter $\mathfrak{F}$ on $TP[0,1]$ the following assertions are
equivalent:

  1. (1)

There exists an unbounded function $f:[0,1]\rightarrow[0,+\infty)$ such that
$S(f,\Pi,T)$ is $\mathfrak{F}$-bounded;

  2. (2)

There exists a countable subset $\\{t_{n}\\}_{n\in{\mathbb{N}}}\subset[0,1]$
such that there is $A\in\mathfrak{F}$ such that for every $(\Pi,T)\in A$

$\sum_{n\in{\mathbb{N}}}n\cdot\ell(\Pi,T,t_{n})<1.$

###### Proof.

(1)$\Rightarrow$(2): Let $f$ be a non-negative, unbounded function on $[0,1]$
such that there is $C>0$ and $B\in\mathfrak{F}$ such that for each $(\Pi,T)\in
B$ we have $\sum\limits_{t\in[0,1]}f(t)\cdot\ell(\Pi,T,t)<C$. As $f$ is
unbounded, there exists $(\alpha_{n})\subset[0,1]$ such that for every
$n\in{\mathbb{N}}$ $f(\alpha_{n})\geqslant Cn$. Then there exists
$(\alpha_{n})\subset[0,1]$, $C>0$, there is $A\in\mathfrak{F}$, $A:=B$ such
that for all $(\Pi,T)\in A$ we obtain:

$\displaystyle\sum_{t\in[0,1]}n\cdot\ell(\Pi,T,\alpha_{n})\leqslant\sum_{n\in{\mathbb{N}}}\frac{f(\alpha_{n})}{C}\cdot\ell(\Pi,T,\alpha_{n})\leqslant$
$\displaystyle\frac{1}{C}\sum_{t\in[0,1]}f(t)\cdot\ell(\Pi,T,t)<\frac{1}{C}\cdot
C=1.$

(2)$\Rightarrow$(1): Let there exists a countable subset
$\\{t_{n}\\}_{n\in{\mathbb{N}}}\subset[0,1]$ and $C>0$ such that there is
$A\in\mathfrak{F}$ such that for every $(\Pi,T)\in A$
$\sum\limits_{n\in{\mathbb{N}}}n\cdot\ell(\Pi,T,t_{n})<C$. Consider function

$f(t)=\begin{cases}n,\text{ if }t=\alpha_{n},\ n\in{\mathbb{N}}\\\ 0,\text{ if
}t\neq\alpha_{n}\end{cases}.$

Obviously, $f(t)$ is unbounded. Then there is $C>0$ and there is
$B\in\mathfrak{F}$, $B:=A$ such that for every $(\Pi,T)\in A$

$\displaystyle\sum_{t\in[0,1]}f(t)\cdot\ell(\Pi,T,t)\leqslant\sum_{n\in{\mathbb{N}}}f(\alpha_{n})\cdot\ell(\Pi,T,\alpha_{n})\leqslant$
$\displaystyle\sum_{n\in{\mathbb{N}}}n\cdot\ell(\Pi,T,\alpha_{n})<C$

. ∎

##  5\. Integration over filter on a subsegment

Our next goal is as follows: if function $f$ is integrable on $[0,1]$ over
filter $\mathfrak{F}$ on $\text{TP}[0,1]$ then for an arbitrary
$[\alpha,\beta]\subset[0,1]$ function $f$ is is integrable on $[\alpha,\beta]$
over filter $\mathfrak{F}$.

To achieve this purpose we need to construct some restriction of filter
$\mathfrak{F}$ on subsegment $[\alpha,\beta]\subset[0,1]$. Now we present how
we can construct such restriction.

Consider an arbitrary $[\alpha,\beta]\subset[0,1]$. We consider only $T$ such
that $T\cap(\alpha,\beta)\neq\emptyset$. Consider an arbitrary $(\Pi,T)\in
TP[0,1]$.

We have four cases:

  1. (1)

$\min\\{T\cap(\alpha,\beta)\\}>\min\\{\Pi\cap(\alpha,\beta)\\}$
$\max\\{T\cap(\alpha,\beta)\\}<\max\\{\Pi\cap(\alpha,\beta)\\}$;

  2. (2)

$\min\\{T\cap(\alpha,\beta)\\}>\max\\{\Pi\cap(0,\alpha)\\}$
$\max\\{T\cap(\alpha,\beta)\\}<\max\\{\Pi\cap(\alpha,\beta)\\}$;

  3. (3)

$\min\\{T\cap(\alpha,\beta)\\}>\min\\{\Pi\cap(\alpha,\beta)\\}$
$\max\\{T\cap(\alpha,\beta)\\}<\min\\{\Pi\cap(\beta,1)\\}$;

  4. (4)

$\min\\{T\cap(\alpha,\beta)\\}>\max\\{\Pi\cap(0,\alpha)$
$\max\\{T\cap(\alpha,\beta)\\}<\min\\{\Pi\cap(\beta,1)\\}$.

We have to construct a restriction of $(\Pi,T)$ on $[\alpha,\beta]$. In each
of four described cases we have such $(\Pi_{k},T_{k})\in TP[\alpha,\beta]$,
$k=1,2,3,4$:

  1. (1)

$\Pi_{1}=\bigg{(}\Pi\setminus\big{(}(\Pi\cap[0,\alpha))\cup(\Pi\cap(\beta,1))\cup\min\\{\Pi\cap(\alpha,\beta)\\}\cup\max\\{\Pi\cap(\alpha,\beta)\\}\big{)}\bigg{)}\cup\\{\alpha,\beta\\}$
$T_{1}=T\setminus\big{(}(T\cap[0,\alpha))\cup(T\cap(\beta,1])\big{)}$;

  2. (2)

$\Pi_{2}=\bigg{(}\Pi\setminus\big{(}(\Pi\cap[0,\alpha))\cup\max\\{\Pi\cap(\alpha,\beta)\\}\cup(\Pi\cap(\beta,1))\big{)}\bigg{)}\cup\\{\alpha,\beta\\}$
$T_{2}=T_{1}$;

  3. (3)

$\Pi_{3}=\bigg{(}\Pi\setminus\big{(}(\Pi\cap[0,\alpha))\cup\min\\{\Pi\cap(\alpha)\\}\cup(\Pi\cap[\beta,1))\big{)}\bigg{)}\cup\\{\alpha,\beta\\}$
$T_{3}=T_{1}$;

  4. (4)

$\Pi_{4}=\bigg{(}\Pi\setminus\big{(}(\Pi\cap[0,\alpha))\cup(\Pi\cap[\beta,1))\big{)}\bigg{)}\cup\\{\alpha,\beta\\}$
$T_{4}=T_{1}$.

Now if we have an arbitrary filter $\mathfrak{F}$ on $TP[0,1]$ we can
construct filter $\mathfrak{F}_{[\alpha,\beta]}$ on $TP[\alpha,\beta]$,
induced with $\mathfrak{F}$ in such way: consider an arbitrary
$A\in\mathfrak{F}$ and for each $(\Pi,T)\in A$ we have to execute an
algorithm, described above. For each $A\in\mathfrak{F}$ denote
$A_{\alpha}^{\beta}$ the restriction of $A$ on $[\alpha,\beta]$, described
above.

######  Definition 18.

Let $\mathfrak{F}$ be a filter on $TP[0,1]$, $[\alpha,\beta]\subset[0,1]$. We
call the filter $\mathfrak{F}$ $[\alpha,\beta]$-complemented if for each
$A\in\mathfrak{F}$, for every $(\Pi_{1},T_{1}),\ (\Pi_{2},T_{2})\in
A_{\alpha}^{\beta}$ there exists $(\Pi^{*},T^{*})\in TP[0,\alpha]$ and
$(\Pi^{**},T^{**})\in TP[\beta,1]$ such that

$(\Pi^{*},T^{*})\cup(\Pi_{1},T_{1})\cup(\Pi^{**},T^{**})\in A,$
$(\Pi^{*},T^{*})\cup(\Pi_{2},T_{2})\cup(\Pi^{**},T^{**})\in A.$

Here we present promised result about filter integration on subsegment.

######  Theorem 19.

Let $f:[0,1]\rightarrow{\mathbb{R}}$, $\mathfrak{F}$ be a filter on $TP[0,1]$
such that for each $[\alpha,\beta]\subset[0,1]$ $\mathfrak{F}$ is
$[\alpha,\beta]$-complemented. Let $f$ is integrated of $[0,1]$ with respect
to $\mathfrak{F}$. Then for every $[\alpha,\beta]\subset[0,1]$ $f$ is
integrated on $[\alpha,\beta]$ with respect to $\mathfrak{F}$

###### Proof.

We know that for an arbitrary $\varepsilon>0$ there exists $A\in\mathfrak{F}$
such that for all $(\Pi_{1},T_{1}),(\Pi_{2},T_{2})\in A$ we have:
$|S(f,\Pi_{1},T_{1})-S(f,\Pi_{2},T_{2})|<\varepsilon$.

Let fix $\varepsilon>0$ and consider an arbitrary
$[\alpha,\beta]\subset[0,1]$. For $A\in\mathfrak{F}$ consider an arbitrary
$(\Pi^{1},T^{1}),(\Pi^{2},T^{2})\in A_{\alpha}^{\beta}$. As $\mathfrak{F}$ is
$[\alpha,\beta]$-complemented we can find $(\Pi^{*},T^{*})\in A_{0}^{\alpha}$
and $(\Pi^{**},T^{**})\in A_{\beta}^{1}$ such that
$(\Pi_{11},T_{11}):=(\Pi^{*},T^{*})\cup(\Pi^{1},T^{1})\cup(\Pi^{**},T^{**})\in
A$ and
$(\Pi_{22},T_{22}):=(\Pi^{*},T^{*})\cup(\Pi^{2},T^{2})\cup(\Pi^{**},T^{**})\in
A$. Then
$\varepsilon>|S(f,\Pi_{11},T_{11})-S(f,\Pi_{22},T_{22})|=|S(f,\Pi^{1},T^{1})-S(f,\Pi^{2},T^{2})|$.

∎

Acknowledgment. This paper is partially supported by a grant from Akhiezer
Foundation (Kharkiv). The author is thankful to his parents for their support
and his scientific adviser, professor Vladimir Kadets for his constant help
with this project. Also author thanks the Defense Forces of Ukraine for the
defence and fight against Russian aggressors.

## References

  * [1] V. Kadets, A course in Functional Analysis and Measure Theory. Translated from the Russian by Andrei Iacob. Universitext. Cham: Springer. xxii, 539 p. (2018).
  * [2] V. Kadets, D. Seliutin, Completeness in topological vector spaces and filters on $\mathbb{N}$, Bulletin of the Belgian Mathematical Society, Simon Stevin 28(4).
  * [3] Listán-García, M.C. $f$-statistical convergence, completeness and $f$-cluster points, Bull. Belg. Math. Soc. Simon Stevin, 23, 235–245 (2016).


# Pseudodifferential operators with completely periodic symbols

Gianluca Garello  Mathematics Department, University of Turin, via Carlo
Alberto 10, I-10123 Torino, Italy. (ORCID: 0000-0002-8636-2998 )
<EMAIL_ADDRESS>and  Alessandro Morando  DICATAM, University of
Brescia, via Valotti 9, I-25133 Brescia, Italy. (ORCID: 0000-0002-6800-7490)
<EMAIL_ADDRESS>
###### Abstract.

Motivated by the recent paper of Boggiatto-Garello in J. Pseudo-Differ. Oper.
Appl. 11 (2020), 93-117, where a Gabor operator is regarded as
pseudodifferential operator with symbol $p(x,\omega)$ periodic on both the
variables, we study the continuity and invertibility, on general time
frequency invariant spaces, of pseudodifferential operators with completely
periodic symbol and general $\tau$ quantization.

Keywords. Gabor Frames, Modulation Spaces, Periodic Distributions,
Pseudodifferential Operators .
MSC2020. 35S05, 42C15, 47B90.

###### Contents

  1. 1 Introduction.
  2. 2 Preliminaries
    1. 2.1 Notations and basic tools
    2. 2.2 Time frequency shifts (tfs)
    3. 2.3 Modulation spaces
    4. 2.4 Periodic distributions
  3. 3 Pseudodifferential Operators with periodic symbol
  4. 4 Continuity
    1. 4.1 The case of Fourier Multipliers
  5. 5 Invertibility
  6. A On periodic distributions and distributions on the torus

##  1\. Introduction.

Consider the formal expression of the Gabor operator:
$Sf=\sum\limits_{h,k\in\mathbb{Z}^{d}}(f,g_{h,k})_{{}_{L^{2}}}g_{h,k}$,
$g_{h,h}(t)=e^{2\pi i\beta k\cdot t}g(t-\alpha h)$,
$\alpha,\beta\in\mathbb{R}_{+}$, and that of the pseudodifferential operator
with Kohn-Nirenberg quantization: $\displaystyle{a(x,D)\varphi(x)=\iint
e^{2\pi i\omega\cdot(x-t)}a(x,\omega)f(t)\,dt\,d\omega}$, where
$f\in\mathcal{S}(\mathbb{R}^{d})$,
$a(x,\omega)\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ and the integration is
intended in distribution sense. In the recent work [2] it is proven that
$S=a(x,D)$, where

(1) $a(x,\omega)=\sum_{h,k\in\mathbb{Z}^{d}}e^{-2\pi i(x-\alpha
h)\cdot(\omega-\beta k)}g(x-\alpha h)\bar{\hat{g}}(\omega-\beta k),$

with suitable decay conditions at infinitive of $g$, $\hat{g}$, and
convergence in $L^{\infty}(\mathbb{R}^{2d})$. Then using the Calderón -
Vaillancourt Theorem about $L^{2}$ continuity of pseudodifferential operators,
see [18], one can prove that for $\alpha+\beta$ less than a suitable positive
constant $C_{g}$, depending only on the decay at infinitive of $g,\hat{g}$ and
some of their derivatives, the Gabor operator is invertible as a bounded
linear operator on $L^{2}(\mathbb{R}^{d})$. Then as well known in frame
theory, the Gabor system $\mathcal{\\{}g_{h,k}\\}_{h,k\in\mathbb{Z}^{d}}$
realizes to be a frame.
The literature about Gabor frame theory is wide, we quote here only the
monographies [6], [13], [15], [5]. Among others, the problem of finding
conditions on the parameters $\alpha,\beta$ in order to obtain Gabor frames is
a challenging one, see for example [10], [14] and the references therein.
Notice now that the symbol in (1) is completely periodic, with period $\alpha$
with respect to the spatial variable $x$ and period $\beta$ with respect to
$\omega$. For all the reasons listed above, we think it should be of some
relevance to develop the study of pseudodifferential operators with completely
periodic symbols, their continuity and possible invertibility in $L^{2}$ or
more general function spaces.
Concerning symbols independent of $x$, that is Fourier multipliers, we quote
the papers [12], [20], where the periodic case in considered.
Wider is the literature concerning the pseudodifferential operators on compact
Lie groups, see the fundamental book of Ruzhansky-Turunen [25], which have as
particular case symbols periodic in $x$ and discrete (non periodic) in
$\omega$. Also interesting is the reversed case where the symbols are discrete
in $x$ and periodic in $\omega$; the related operators are called in this case
”pseudo-difference” operators, see [3], [19]. About pseudodifferential
operators on generalized spaces, e.g. modulation spaces, we refer to the
following papers [27], [28], [8], [1], [22], [23], [4], [11], [7], [9].
The plan of the paper is the following: in §2 we give the notations and
definitions; then we review some basic facts about periodic distributions with
respect to a general invertible matrix and their Fourier transform. In §3 we
introduce the pseudodifferential operators with general $\tau$ quantization
and for the case of periodic symbols we provide a representation formula,
obtained by linear combination of time frequency shift operators. At the end,
respectively in §4 and §5 we set the results of continuity and invertibility
on general families of time frequency shift invariant spaces. The Appendix A
is devoted to give some technicalities in order to compare periodic
distributions on $\mathbb{R}^{n}$ and distribuitions on the $n$ dimensional
torus $\mathbb{T}^{n}$.

##  2\. Preliminaries

###  2.1. Notations and basic tools

In whole the paper we will use the following notations and tools:

  * •

$\mathbb{R}^{n}_{0}=\mathbb{R}^{n}\setminus\\{0\\}$,
$\mathbb{Z}^{n}_{0}=\mathbb{Z}^{n}\setminus\\{0\\}$;

  * •

$\langle x\rangle=\sqrt{1+|x|^{2}}$, where $|x|$ is the Euclidean norm of
$x\in\mathbb{R}^{n}$;

  * •

$x\cdot\omega=\langle x,\omega\rangle=\sum_{j=1}^{n}x_{j}\omega_{j}$;
$x,\omega\in\mathbb{R}^{n}$;

  * •

$(f,g)=\int f(x)\bar{g}(x)\,dx$ is the inner product in
$L^{2}(\mathbb{R}^{n})$;

  * •

$\mathcal{F}f(\omega)=\hat{f}(\omega)=\int f(x)e^{-2\pi ix\cdot\omega}\,dx$
the Fourier transform of $f\in\mathcal{S}(\mathbb{R}^{n})$, with the well
known extension to $u\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$.

The polynomial weight function $v$ is defined for some $s\geq 0$ by

(2) $v(z)=(1+|z|^{2})^{s/2}\,,\quad\forall\,z\in\mathbb{R}^{n}\,.$

A non negative measurable function $m=m(z)$ on $\mathbb{R}^{n}$ is said to be
a polynomially moderate (or temperate) weight function if there exists a
positive constant $C$ such that

(3) $m(z_{1}+z_{2})\leq Cv(z_{1})m(z_{2})\quad\text{for
all}\,z_{1},z_{2}\in\mathbb{R}^{n}.$

For other details about weight functions see [13, §11.1].

In the following we will use in many cases the matrix in $GL(2d)$

(4) $\mathcal{J}=\left(\begin{array}[]{cc}0&-I\\\ I&0\end{array}\right),$

which defines the symplectic form, see [13, §9.4],

(5) $[z_{1},z_{2}]:=\langle
z_{1},\mathcal{J}z_{2}\rangle=x_{2}\cdot\omega_{1}-x_{1}\cdot\omega_{2}\quad,\quad
z_{1}=(x_{1},\omega_{1}),\,z_{2}=(x_{2},\omega_{2})\in\mathbb{R}^{2d}.$

###  2.2. Time frequency shifts (tfs)

For $z=(x,\omega)\in\mathbb{R}^{2d}$ we define the operators:

(6) $\displaystyle T_{x}f(t)=f(t-x)$ $\displaystyle\text{(translation)};$ (7)
$\displaystyle M_{\omega}f(t)=e^{2\pi i\omega\cdot t}f(t)$
$\displaystyle\text{(modulation)},$ (8)
$\displaystyle\pi_{z}f=M_{\omega}T_{x}f$ $\displaystyle\text{(time frequency
shift)},$

with suitable extension to distributions in
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$.
For $u\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$,
$z=(x,\omega)\in\mathbb{R}^{2d}$, the next properties easily follow:

(9) $\displaystyle T_{x}M_{\omega}u=e^{-2\pi ix\cdot\omega}M_{\omega}T_{x}u,$
(10) $\displaystyle\mathcal{F}(T_{x}u)=M_{-x}\mathcal{F}u,$
$\displaystyle\mathcal{F}^{-1}(T_{x}u)=M_{x}\mathcal{F}^{-1}u,$ (11)
$\displaystyle\mathcal{F}(M_{\omega}u)=T_{\omega}\mathcal{F}u,$
$\displaystyle\mathcal{F}^{-1}(M_{\omega}u)=T_{-\omega}\mathcal{F}^{-1}u,$
(12) $\displaystyle\mathcal{F}(\pi_{z}u)=e^{2\pi
ix\cdot\omega}\pi_{\mathcal{J}^{T}z}\mathcal{F}u;$
$\displaystyle\mathcal{F}^{-1}(\pi_{z}u)=e^{2\pi
ix\cdot\omega}\pi_{\mathcal{J}z}\mathcal{F}^{-1}u.$

###  2.3. Modulation spaces

######  Definition 2.1.

For a fixed nontrivial function $g$ the short-time Fourier transform (or Gabor
transform) of a function $f$ with respect to $g$ is defined as

(13) $V_{g}f(z):=(f,\pi_{z}g)=\int_{\mathbb{R}^{d}}f(t)e^{-2\pi i\omega\cdot
t}\overline{g(t-x)}dt\,,\quad\mbox{for}\,\,z=(x,\omega)\in\mathbb{R}^{2d}\,,$

whenever the integral can be considered, also in weak distribution sense.

When $f,g\in L^{2}(\mathbb{R}^{d})$ , $V_{g}f$ is a uniformly continuous
function on $\mathbb{R}^{2d}$, $V_{g}f\in L^{2}(\mathbb{R}^{2d})$ and
$\|V_{g}f\|_{L^{2}}=\|f\|_{L^{2}}\|g\|_{L^{2}}$. See e.g. [13, §3].

######  Definition 2.2.

For a fixed $g\in\mathcal{S}(\mathbb{R}^{d})\setminus\\{0\\}$ and
$p,q\in[1,+\infty]$, the $m-$weighted modulation space
$M^{p,q}_{m}(\mathbb{R}^{d})$ consists of all tempered distributions
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

(14)
$\|f\|_{M^{p,q}_{m}}:=\left(\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}|V_{g}f(x,\omega)|^{p}m(x,\omega)^{p}dx\right)^{q/p}d\omega\right)^{1/q}<+\infty\,,$

(with expected modification in the case when at least one among $p$ or $q$
equals $+\infty$).

The definition of the space $M^{p,q}_{m}$ is independent of the choice of the
window $g$, different windows $g$ provide equivalent norms and $M^{p,q}_{m}$
turns out to be a Banach space. In the case of $p=q$ we denote
$M^{p}_{m}:=M^{p,p}_{m}$, when $m(x,\omega)\equiv 1$ we write $M^{p,q}$.

For more details about modulation spaces see [13, §6.1, §11].

###  2.4. Periodic distributions

We say that a distribution $u\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$ is
periodic (of period 1) if

(15) $T_{\kappa}u=u\quad\text{for any}\,\,\kappa\in\mathbb{Z}^{n}.$

Notice that $u$ is in this case a tempered distribution in
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$, then its Fourier transform $\hat{u}$
can be considered. Moreover it can be shown that

(16) $\hat{u}=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(u)\delta_{\kappa},$

where the series converges in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$,

(17) $c_{\kappa}(u):=\langle u,\phi e^{-2\pi
i\langle\cdot,\kappa\rangle}\rangle=\widehat{u\phi}(\kappa),\\\ $

and $\phi\in C^{\infty}_{0}(\mathbb{R}^{n})$ satisfies

(18) $\sum_{\kappa\in\mathbb{Z}^{n}}\phi(x-\kappa)=1.$

For the details see Hörmander [16, §7.2]. Now by a straightforward application
of Fourier inverse transform we obtain

(19) $u=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(u)e^{2\pi
i\langle\cdot,\kappa\rangle},$

with convergence in $\mathcal{S}^{\prime}$ and $c_{\kappa}(u)$ defined in
(17).
Notice that a general periodic distribution $u$ can be regarded as a
distribution on the torus $\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$, that
is a linear continuous form on $C^{\infty}(\mathbb{T}^{n})$. In the following
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$ will be the topological dual space of
$C^{\infty}(\mathbb{T}^{n})$. Thus the coefficients in the expansion (19) can
be regarded as the Fourier coefficients of $u$, namely

(20) $c_{\kappa}(u)=\langle u,e^{-2\pi
i\langle\cdot,\kappa\rangle}\rangle_{{}_{\mathbb{T}^{n}}},$

where $\langle\cdot,\cdot\rangle_{\mathbb{T}^{n}}$ denotes the duality pair
between $\mathcal{D}^{\prime}(\mathbb{T}^{n})$ and
$C^{\infty}(\mathbb{T}^{n})$. Agreeing with (20), for $f\in
L^{1}(\mathbb{T}^{n})$, using (18), we get

(21) $c_{\kappa}(f)=\int_{\mathbb{R}^{n}}f(x)\phi(x)e^{-2\pi
ix\cdot\kappa}\,dx=\int_{[0,1]^{n}}f(x)e^{-2\pi ix\cdot\kappa}\,dx.$

In Appendix A we clarify how to make rigorous the above calculations in
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$, see in particular (75).
For a detailed discussion about distributions on the torus one can see the
book of M. Ruzhansky and V. Turunen [25].

Consider now and in the whole paper $L=(a_{ij})\in GL(n)$, the space of
invertible matrices of size $n\times n$.
For $\mathbb{T}^{n}_{L}:=\mathbb{R}^{n}/L\mathbb{Z}^{n}$, we still identify
the set of $L$-periodic distributions, that is
$u\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$ such that, for any
$\kappa\in\mathbb{Z}^{n},T_{L\kappa}u=u$, with the space
$\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ of linear continuous forms on
$C^{\infty}(\mathbb{T}^{n}_{L})$. Notice that also in this case
$\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})\subset\mathcal{S}^{\prime}(\mathbb{R}^{n})$.
For any $u\in\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ observe that
$v=u(L\cdot)$ is a $1$-periodic distribution. Applying then the Fourier
expansion $v=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(v)e^{2\pi
i\langle\cdot,\kappa\rangle}$, $c_{\kappa}(v)=\langle v,e^{-2\pi
i\langle\cdot,\kappa\rangle}\rangle_{\mathbb{T}^{n}}$, we obtain

(22) $u=v(L^{-1}\cdot)=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(v)e^{2\pi
i\langle
k,L^{-1}\cdot\rangle}=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(v)e^{2\pi
i\langle L^{-T}k,\cdot\rangle},$

where $L^{-T}:=(L^{-1})^{T}$ denotes the transposed of the inverse matrix of
$L$ and

(23) $c_{\kappa}(v)=\langle u(L\cdot),e^{-2\pi
i\langle\kappa,\cdot\rangle}\rangle_{\mathbb{T}^{n}}=\frac{1}{|\textup{det}L|}\langle
u,e^{-2\pi i\langle L^{-T}\kappa,\cdot\rangle}\rangle_{\mathbb{T}^{n}_{L}}.$

Thus we obtain for any $u\in\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ the
Fourier expansion

(24) $u=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa,L}(u)e^{2\pi i\langle
L^{-T}k,\cdot\rangle},$

with the Fourier coefficients

(25) $c_{\kappa,L}(u):=c_{\kappa}(u(L\cdot))=\frac{1}{|\textup{det}L|}\langle
u,e^{-2\pi i\langle L^{-T}\kappa,\cdot\rangle}\rangle_{\mathbb{T}^{n}_{L}}.$

For short in the following we set $c_{\kappa}(u)=c_{\kappa,L}(u)$.
Consider $L^{p}(\mathbb{T}^{n}_{L})$, $1\leq p<\infty$, the set of measurable
L-periodic functions on $\mathbb{R}^{n}$ such that
$\|f\|_{L^{p}(\mathbb{T}^{n}_{L})}=\int_{L[0,1]^{n}}|f(x)|^{p}\,dx<\infty$,
with obvious modification for the definition of
$L^{\infty}(\mathbb{T}^{n}_{L})$.
Then for $f\in L^{1}(\mathbb{T}^{1}_{L})$ the following:

(26) $f(x)=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}(f)e^{2\pi
iL^{-T}\kappa\cdot x}$

holds with convergence in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$, and

(27) $c_{\kappa}(f)=\frac{1}{|\textup{det}L|}\int_{L[0,1]^{n}}e^{2\pi
iL^{-T}\kappa\cdot x}f(x)\,dx\\\ $

######  Remark 1.

It can be useful to write the Fourier expansion of
$u\in\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ in terms of the lattice
$\Lambda=L\mathbb{Z}^{n}$:

(28) $u=\sum_{\mu\in\Lambda^{\bot}}\hat{u}(\mu)e^{2\pi
i\langle\mu,\cdot\rangle},$

with

(29) $\hat{u}(\mu):=\frac{1}{\textup{vol}(\Lambda)}\langle u,e^{-2\pi
i\langle\mu,\cdot\rangle}\rangle_{\mathbb{T}^{n}_{L}}.$

Here $\Lambda^{\perp}:=L^{-T}\mathbb{Z}^{n}$ and
$\textup{vol}(\Lambda):=|\textup{det}L|=\text{meas}\,(L[0,1]^{n})$ are
respectively called dual lattice and volume of $\Lambda$.

######  Example 1.

For $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{R}^{n}$, $\alpha_{j}>0$,
let us consider the diagonal matrix $A\in GL(n)$, together with its inverse

(30) $A=\left(\begin{array}[]{cccc}\alpha_{1}&\dots&0&\\\
\vdots&\ddots&\vdots&\\\ 0&\dots&\alpha_{n}&\end{array}\right)\quad;\quad
A^{-1}=\left(\begin{array}[]{cccc}\frac{1}{\alpha_{1}}&\dots&0&\\\
\vdots&\ddots&\vdots&\\\ 0&\dots&\frac{1}{\alpha_{n}}&\end{array}\right)$

and introduce for $\kappa\in\mathbb{Z}^{n}$ the following notations, $\alpha
k:=A\kappa=(\alpha_{1}k_{1},\dots,\alpha_{n}k_{n})$;
$\frac{\kappa}{\alpha}:=A^{-1}k=\left(\frac{\kappa_{1}}{\alpha_{1}},\dots,\frac{\kappa_{n}}{\alpha_{n}}\right)$,
$\prod\alpha:=\prod_{j=1}^{n}\alpha_{j}$ . Consider now an $A$-periodic
function $f$ which satisfies
$\int_{0}^{\alpha_{1}}\dots\int_{0}^{\alpha_{n}}|f(x)|\,dx_{1}\dots
dx_{n}<+\infty$, then directly from (26), (27) we obtain

(31) $\displaystyle f(x)=\sum_{\kappa\in\mathbb{Z}^{n}}c_{k}(f)e^{2\pi
i\frac{k}{\alpha}\cdot x},$ (32) where (34) $\displaystyle
c_{k}(f)=\frac{1}{\prod\alpha}\int_{0}^{\alpha_{1}}\dots\int_{0}^{\alpha_{n}}f(x)e^{-2\pi
i\frac{\kappa}{\alpha}\cdot x}\,dx_{1}\dots dx_{n}.$

##  3\. Pseudodifferential Operators with periodic symbol

We say $\tau$ pseudodifferential operator, $0\leq\tau\leq 1$, with symbol
$p(z)=p(x,\omega)\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$, the operator
acting from $\mathcal{S}(\mathbb{R}^{d})$ to
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ defined by

(35)
$\textup{Op}_{\tau}(p)u(x):=\int_{\mathbb{R}^{d}_{\omega}}\int_{\mathbb{R}^{d}_{y}}e^{2\pi
i(x-y)\cdot\omega}p\left((1-\tau)x+\tau y,\omega\right)u(y)\,dy\,d\omega,\quad
u\in\mathcal{S}(\mathbb{R}^{d}).$

The formal integration must be understood in distribution sense. For the
definition and development of pseudodifferential operators see the basic texts
[26], [17].

For $I$ and $0$ respectively the identity and null matrices of dimension
$d\times d$, let us introduce the $d\times 2d$ matrices

(36) $I_{1}=(I,0),\quad I_{2}=(0,I)$

######  Proposition 3.1.

Consider $p\in\mathcal{D}^{\prime}(\mathbb{T}^{2d}_{L})$, $L\in GL(2d)$. Then
for any $0\leq\tau\leq 1$ and $u\in\mathcal{S}(\mathbb{R}^{d})$ we can write

(37)
$\textup{Op}_{\tau}(p)u=\sum_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{{2\pi
i\tau}\langle
I_{2}L^{-T}\kappa\,,\,I_{1}L^{-T}\kappa\rangle}\pi_{{}_{\mathcal{J}L^{-T}\kappa}}u,$

where $c_{\kappa}(p)$ are the Fourier coefficients defined in (25) and
$\mathcal{J}$ the matrix introduced in (4).

###### Proof.

Using (24), (25) we perform the Fourier expansion of the symbol $p$

(38) $p=\sum_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi i\langle
L^{-T}\kappa,\cdot\rangle}$

with convergence in $\mathcal{S}^{\prime}(\mathbb{R}^{2d})$. Then for any
$u\in\mathcal{S}(\mathbb{R}^{d})$ we get

(39) $\textup{Op}_{\tau}(p)u(x)=\iint e^{2\pi
i(x-y)\cdot\omega}\sum_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
iL^{-T}\kappa\cdot\left((1-\tau)x+\tau y,\omega\right)}u(y)\,dy\,d\omega.$

Considering the decomposition

(40) $L^{-T}=\left(\begin{array}[]{c}I_{1}L^{-T}\\\
I_{2}L^{-T}\end{array}\right),$

we obtain

(41) $L^{-T}\kappa\cdot\left((1-\tau)x+\tau
y,\omega\right)=I_{1}L^{-T}\kappa\cdot\left((1-\tau)x+\tau
y\right)+I_{2}L^{-T}\kappa\cdot\omega.$

Let us set for simplicity of notation $L_{j}^{-T}=I_{j}L^{-T}$, $j=1,2$. Then
in view of convergence in $\mathcal{S}^{\prime}$ and formal integration in
distribution sense it follows

(42)
$\begin{array}[]{l}\begin{array}[]{ll}\textup{Op}_{\tau}(p)u(x)=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)\int\\!\\!\\!\int
e^{2\pi i(x-y)\cdot\omega}&e^{2\pi iL^{-T}_{1}\kappa\cdot\left((1-\tau)x+\tau
y\right)}\times\\\ &\times e^{2\pi
iL^{-T}_{2}\kappa\cdot\omega}u(y)\,dy\,d\omega=\end{array}\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)\int e^{2\pi
ix\cdot\omega}e^{2\pi iL^{-T}_{2}\kappa\cdot\omega}\int e^{-2\pi
iy\cdot\omega}e^{2\pi iL^{-T}_{1}\kappa\cdot\left((1-\tau)x+\tau
y\right)}u(y)\,dy\,d\omega=\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
iL^{-T}_{1}\kappa\cdot\left((1-\tau)x\right)}\int e^{2\pi
i\left(x+L^{-T}_{2}\kappa\right)\cdot\omega}\int e^{-2\pi i\left(\omega-\tau
L^{-T}_{1}\kappa\right)\cdot y}u(y)\,dy\,d\omega=\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
i(1-\tau)L^{-T}_{1}\kappa\cdot x}\int e^{2\pi
i\left(x+L^{-T}_{2}\kappa\right)\cdot\omega}\hat{u}\left(\omega-\tau
L^{-T}_{1}\kappa\right)\,d\omega=\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
i(1-\tau)L^{-T}_{1}\kappa\cdot x}\int e^{2\pi
i\left(x+L^{-T}_{2}\kappa\right)\cdot\omega}T_{\tau
L^{-T}_{1}\kappa}\hat{u}(\omega)\,d\omega=\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
i(1-\tau)L^{-T}_{1}\kappa\cdot x}\int e^{2\pi
i\left(x+L^{-T}_{2}\kappa\right)\cdot\omega}\widehat{M_{\tau
L^{-T}_{1}\kappa}u}(\omega)\,d\omega=\\\ \\\
=\displaystyle\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
i(1-\tau)L^{-T}_{1}\kappa\cdot x}\left(M_{\tau
L^{-T}_{1}\kappa}u\right)\left(x+L^{-T}_{2}\kappa\right)=\\\ \\\
\displaystyle=\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)e^{2\pi
i(1-\tau)L^{-T}_{1}\kappa\cdot x}\,T_{-L^{-T}_{2}\kappa}M_{\tau
L^{-T}_{1}\kappa}u(x)=\\\ \\\
\displaystyle=\sum\limits_{\kappa\in\mathbb{Z}^{2d}}c_{\kappa}(p)M_{(1-\tau)L^{-T}_{1}\kappa}T_{-L^{-T}_{2}\kappa}M_{\tau
L^{-T}_{1}\kappa}u(x).\end{array}$

The proof ends by observing that, thanks to (9),

(43)
$\begin{array}[]{l}M_{(1-\tau)L^{-T}_{1}\kappa}T_{-L^{-T}_{2}\kappa}M_{\tau
L^{-T}_{1}\kappa}=\\\ \\\ =e^{-2\pi i\langle-L^{-T}_{2}\kappa,\tau
L^{-T}_{1}\kappa\rangle}M_{(1-\tau)L^{-T}_{1}\kappa}M_{\tau
L^{-T}_{1}\kappa}T_{-L^{-T}_{2}\kappa}=\\\ \\\ =e^{2\pi i\tau\langle
L^{-T}_{2}\kappa,L^{-T}_{1}\kappa\rangle}M_{L^{-T}_{1}\kappa}T_{-L^{-T}_{2}\kappa}=e^{2\pi
i\tau\langle
L^{-T}_{2}\kappa,L^{-T}_{1}\kappa\rangle}\pi_{\mathcal{J}L^{-T}\kappa}.\end{array}$

∎

######  Remark 2.

Consider $\mu=L^{-T}\kappa\in\Lambda^{\perp}$. In view of (37) the
pseudodifferential operator $\textup{Op}_{\tau}$ may be written in lattice
notation

(44) $\textup{Op}_{\tau}(p)=\sum_{\mu\in\Lambda^{\perp}}\hat{p}(\mu)e^{2\pi
i\tau\langle I_{2}\mu,I_{1}\mu\rangle}\pi_{\mathcal{J}\mu}.$

######  Example 2.

Let $a=(a_{1},\dots a_{d})$, $b=(b_{1},\dots,b_{d})$ be two vectors in
$(\mathbb{R}\setminus\\{0\\})^{d}$. Using the notation in Example 1, we say
that a symbol $p\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ is $ab$-periodic if,
for any $\kappa=(h,k)\in\mathbb{Z}^{2d}$,
$p(\cdot+ah,\cdot+bk)=p(\cdot,\cdot)$. Considering the matrix

(45) $L=\left(\begin{array}[]{cc}A&0\\\ 0&B\end{array}\right),$

with $A,B$ diagonal matrices defined as in (30), it is trivial to show that
$I_{1}L^{-T}\kappa=\frac{h}{a}$ and $I_{2}L^{-T}\kappa=\frac{k}{b}$. Then for
any $u\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$

(46) $\pi_{\mathcal{J}L^{-T}\kappa}u=M_{\frac{h}{a}}T_{-\frac{k}{b}}u=e^{2\pi
i\langle\frac{h}{a},\cdot\rangle}u(\cdot+\frac{k}{b}).$

Thus for any $0\leq\tau\leq 1$ we have

(47) $\textup{Op}_{\tau}p(u)=\sum_{(h,k)\in\mathbb{Z}^{2d}}c_{h,k}(p)e^{2\pi
i\tau\langle\frac{h}{a},\frac{k}{b}\rangle}e^{2\pi
i\langle\frac{h}{a},\cdot\rangle}u(\cdot+\frac{k}{b}),$

with convergence in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and $c_{h,k}(p)$
defined by formal integration

(48) $c_{h,k}(p)=\frac{1}{|\prod
ab|}\int_{0}^{a_{1}}\\!\\!\\!\\!\\!\\!\\!\dots\\!\\!\int_{0}^{a_{d}}\\!\\!\\!\\!\int_{0}^{b_{1}}\\!\\!\\!\\!\\!\\!\\!\dots\\!\\!\int_{0}^{b_{d}}p(x,\omega)e^{-2\pi
i\left(\frac{h}{a}\cdot x+\frac{k}{b}\cdot\omega\right)}\,dx\,d\omega\,\,.$

##  4\. Continuity

We say that a Banach space $\mathcal{S}(\mathbb{R}^{d})\hookrightarrow
X\hookrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$, with
$\mathcal{S}(\mathbb{R}^{d})$ dense in $X$, is time frequency shifts invariant
(tfs invariant from now on) if for some polynomial weight function $v$ and
$C>0$

(49) $\|\pi_{z}u\|_{X}\leq Cv(z)\|u\|_{X},\quad u\in X,\quad
z\in\mathbb{R}^{2d}.$

######  Example 3.

  * •

The $m-$weighted modulation spaces $M^{p,q}_{m}(\mathbb{R}^{d})$,
$p,q\in[1,+\infty]$ are time frequency shifts invariant, see [13, Theorem
11.3.5].

  * •

The $m-$weighted Lebesgue space $L^{p}_{m}(\mathbb{R}^{d})$ and $m-$weighted
Fourier-Lebesgue space $\mathcal{F}L^{p}_{m}(\mathbb{R}^{d})$ are respectively
defined as the sets of measurable functions and tempered distributions in
$\mathbb{R}^{d}$, making finite the norms
$\|f\|_{L^{p}_{m}}:=\|m(\cdot,\omega_{0})f\|_{L^{p}}$ and
$\|f\|_{\mathcal{F}L^{p}_{m}}=\|m(x_{0},\cdot)\hat{f}\|_{L^{p}}$, whatever are
$(x_{0},\omega_{0})\in\mathbb{R}^{2d}$. (Equivalent norms in
$L^{p}_{m}(\mathbb{R}^{d})$ and $\mathcal{F}L^{p}_{m}(\mathbb{R}^{d})$ should
correspond to different choices of $(x_{0},\omega_{0})$. See [23, Remark 1.1])

For $z=(x,\omega)\in\mathbb{R}^{2d}$, assuming $(x_{0},\omega_{0})=(0,0)$ and
using (3), we compute:

(50)
$\begin{array}[]{ll}\|\pi_{z}f\|^{p}_{L^{p}}=\|M_{\omega}T_{x}f\|^{p}_{L^{p}_{m}}&=\int
m(t,0)^{p}|f(t-x)|^{p}\,dt=\int m(t+x,0)^{p}|f(t)|^{p}\,dt\\\ &\leq
C^{p}v(x,0)^{p}\int
m(t,0)^{p}|f(t)|^{p}\,dt=C^{p}v(x,0)^{p}\|f\|^{p}_{L^{p}_{m}}\end{array}$

and

(51)
$\begin{array}[]{l}\|\pi_{z}f\|^{p}_{\mathcal{F}L^{p}_{m}}=\|M_{\omega}T_{x}f\|^{p}_{\mathcal{F}L^{p}_{m}}=\int
m(0,t)^{p}|\widehat{M_{\omega}T_{x}f(t)}|^{p}\,dt\\\ =\int
m(0,t)^{p}|T_{\omega}M_{-x}\hat{f}(t)|^{p}\,dt=\int
m(0,t)^{p}|\hat{f}(t-\omega)|^{p}\,dt\\\ \leq C^{p}v(0,\omega)^{p}\int
m(0,t)^{p}|\hat{f}(t)|^{p}\,dt=C^{p}v(0,\omega)^{p}\|f\|^{p}_{\mathcal{F}L^{p}_{m}}.\end{array}$

Then $L^{p}_{m}(\mathbb{R}^{d})$ and $\mathcal{F}L^{P}_{m}(\mathbb{R}^{d})$
are time frequency shifts invariant for any $p\in[1,+\infty]$.

In both the examples the positive constants $C$ are directly obtained by (3)
and depend only on the weights $m$.

######  Theorem 4.1.

Let $X$ be a time frequency shifts invariant space, $L\in GL(2d)$,
$p\in\mathcal{D}^{\prime}(\mathbb{T}^{2d}_{L})$. Assume that the Fourier
coefficients $c_{\kappa}(p)$ defined in (25) satisfy,

(52)
$\|c_{\kappa}(p)\|_{\ell^{1}_{L,v}}:=\sum_{\kappa\in\mathbb{Z}^{2d}}v\left(\mathcal{J}L^{-T}\kappa\right)|c_{\kappa}(p)|<+\infty.$

Then for any $\tau\in[0,1]$ the operator $\textup{Op}_{\tau}(p)$ is bounded on
$X$ and

(53) $\|\textup{Op}_{\tau}(p)\|_{\mathcal{L}(X)}\leq
C\|c_{\kappa}(p)\|_{\ell^{1}_{L,v}},$

Where $C$ is the constant in (49).

In lattice terms, see (44), we can write

(54)
$\|c_{\kappa}(p)\|_{\ell^{1}_{L,v}}=\sum_{\mu=\in\Lambda^{\perp}}|\hat{p}(\mu)|v(\mathcal{J}\mu):=\|\hat{p}(\mu)\|_{\ell^{1}_{v}},$

where $\mu=L^{-T}\kappa$, $\kappa\in\mathbb{Z}^{2d}$.

###### Proof.

Using Proposition 3.1 and in view of the tfs invariance (49) we obtain for
$u\in\mathcal{S}(\mathbb{R}^{d})$

(55)
$\|\textup{Op}_{\tau}(p)u\|_{X}\leq\sum_{\kappa\in\mathbb{Z}^{2d}}|c_{\kappa}(p)|\|\pi_{\mathcal{J}L^{-T}\kappa}u\|_{X}\leq
C\sum_{\kappa\in\mathbb{Z}^{2d}}|c_{\kappa}(p)|v(\mathcal{J}L^{-T}\kappa)\|u\|_{X},$

where $C$ is the constant in (49). The proof follows from the density of
$\mathcal{S}(\mathbb{R}^{d})$ in $X$. ∎

###  4.1. The case of Fourier Multipliers

Assume now that the symbol is independent of $x$, namely consider a Fourier
multiplier $\sigma=\sigma(\omega)\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$,
$P-$periodic, with $P\in GL(d)$, that is

(56) $T_{Pk}\sigma=\sigma\,,\quad\forall\,k\in\mathbb{Z}^{d}\,.$

In such a case the related pseudodifferential operator, as a linear bounded
operator from $\mathcal{S}(\mathbb{R}^{d})$ to
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$, reads as

(57)
$\sigma(D)u=\mathcal{F}^{-1}(\sigma\hat{u})\,,\quad\forall\,u\in\mathcal{S}(\mathbb{R}^{d})\,.$

Inserting within (57) the Fourier expansion of $\sigma$

(58) $\sigma(\omega)=\sum\limits_{k\in\mathbb{Z}^{d}}c_{k}(\sigma)e^{2\pi
iP^{-T}k\cdot\omega}\,,$

where the series is convergent in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, we
compute

(59)
$\sigma(D)u=\mathcal{F}^{-1}(\sigma\hat{u})=\sum\limits_{k\in\mathbb{Z}^{d}}c_{k}(\sigma)T_{-P^{-T}k}u\,,\quad\mbox{for
any}\,\,u\in\mathcal{S}(\mathbb{R}^{d})\,,$

with convergence of the series in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$; in
(57), (58), $c_{k}(\sigma)$ stand as usual for the Fourier coefficients of
$\sigma$.
The following result shows that in the case of Fourier multiplier operators
the sufficient boundedness condition given in Theorem 4.1 is also necessary
for $\sigma(D)$ to be extended as a linear bounded operator in the weighted
Lebesgue space $L^{1}_{v}(\mathbb{R}^{d})$, where $v=v(x)$ is a polynomial
weight function in $\mathbb{R}^{d}$.

######  Proposition 4.1.

Let $\sigma=\sigma(\omega)\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ be
$P-$periodic for $P\in GL(d)$. If we assume that $\sigma(D)$ extends to a
bounded operator in $L^{1}_{v}(\mathbb{R}^{d})$, that is

(60) $\|\sigma(D)u\|_{L^{1}_{v}}\leq
C\|u\|_{L^{1}_{v}}\,,\quad\forall\,u\in\mathcal{S}(\mathbb{R}^{d})\,,$

for a constant $C>0$, then

(61) $\sum_{k\in\mathbb{Z}^{d}}v\left(P^{-T}k\right)|c_{k}(\sigma)|<+\infty.$

###### Proof.

It is enough to evaluate $\sigma(D)$ on a non negative continuous function
$\tilde{u}\in L^{1}_{v}(\mathbb{R}^{d})$ supported on the compact set
$\mathcal{P}_{0}:=P^{-T}([0,1]^{d})$, such that $\|u\|_{L^{1}_{v}}=1$ 111Such
a function $u$ can be defined by
$\tilde{u}(x):=|\mbox{det}P|\frac{\psi(P^{T}x)}{v(x)}$, where $\psi$ is any
non negative smooth function supported on the unit cube $Q=[0,1]^{d}$ such
that $\int\psi(z)dz=1$.. The function $T_{-P^{T}k}\tilde{u}$ will be then
supported on $\mathcal{P}_{k}:=\mathcal{P}_{0}-P^{-T}k$ for all
$k\in\mathbb{Z}^{d}$. Since the set collection
$\\{\mathcal{P}_{k}\\}_{k\in\mathbb{Z}^{d}}$ defines a covering on
$\mathbb{R}^{d}$ such that $\mathcal{P}_{k}\cap\mathcal{P}_{h}$ has zero
Lebesgue measure, whenever $k\neq h$, we get

(62)
$\|\sigma(D)\tilde{u}\|_{L^{1}_{v}}=\sum\limits_{k\in\mathbb{Z}^{d}}\int_{\mathcal{P}_{k}}v(x)|\sigma(D)\tilde{u}(x)|dx\,.$

It is even clear that $\sigma(D)\tilde{u}$ reduces to
$c_{k}(\sigma)T_{-P^{-T}k}\tilde{u}$ in the interior of the set
$\mathcal{P}_{k}$, for each $k$, so that by change of integration variable and
sub-multiplicativity of $v$, we get

(63)
$\begin{split}\|\sigma(D)\tilde{u}\|_{L^{1}_{v}}=\sum\limits_{k\in\mathbb{Z}^{d}}|c_{k}(\sigma)|\int_{\mathcal{P}_{k}}v(x)|\tilde{u}(x+P^{-T}k)|dx\\\
=\sum\limits_{k\in\mathbb{Z}^{d}}|c_{k}(\sigma)|\int_{\mathcal{P}_{0}}v(y-P^{-T}k))|\tilde{u}(y)|dy\geq\frac{1}{K}\sum\limits_{k\in\mathbb{Z}^{d}}|c_{k}(\sigma)|v(P^{-T}k)\,,\end{split}$

with $K>0$ depending only on $P$ and $v$. ∎

######  Remark 3.

Combining Proposition 4.1 and Theorem 4.1 we obtain that, in the case of
Fourier multipliers, condition (61) is actually equivalent to the continuity
in any tfs invariant Banach space, as defined in (49); here translation
invariance of the space is enough, due to the Fourier multiplier structure
(59).
Instead, condition (52) is no longer necessary for continuity of periodic
pseudodifferential operators, with $x$-dependent symbol, in tfs invariant
spaces. It can be easily shown by taking a symbol of the following form
$p(x,\omega)=\nu(x)\sigma(\omega)$ where $\nu$ is a function in
$L^{\infty}({\mathbb{T}})$, such that the sequence of its Fourier coefficients
$\\{c_{k}(\nu)\\}\notin\ell^{1}(\mathbb{Z})$ and $\sigma(\omega)$ satisfies
(61). For instance we could take $\nu(x)=1$ for $0\leq x<1/2$, $\nu(x)=0$ for
$1/2\leq x<1$, repeated by periodicity. It is straightforward to check that
the pseudodifferential operator $p(x,D)$ maps continuously
$L^{p}_{v}(\mathbb{R})$ into itself, whenever $1\leq p<+\infty$. On the other
hand we compute at once that

(64)
$\sum_{(h,k)\in\mathbb{Z}^{2}}v(k)|c_{(h,k)}(p)|=\sum_{h\in\mathbb{Z}}|c_{h}(\nu)|\sum_{k\in\mathbb{Z}}v(k)|c_{k}(\sigma)|=+\infty.$

##  5\. Invertibility

For the study of invertibility condition of pseudodifferential operators we
will make use of the well known properties of the von Neumann series in Banach
algebras, see e.g. [24], in the following version.

######  Proposition 5.1.

Consider $x\in\mathcal{A}$, where $(\mathcal{A},\|\cdot\|)$ is a Banach
algebra on the field of complex numbers, with multiplicative identity $e$. If
there exists $c\in\mathbb{C}\setminus\\{0\\}$ such that $\|e-cx\|<1$, then $x$
is invertible in $\mathcal{A}$ and

(65) $x^{-1}=c\sum_{n=0}^{\infty}(e-cx)^{n}.$

######  Theorem 5.1.

Let $X$ be a tfs invariant space, $L\in GL(2d)$,
$p\in\mathcal{D}^{\prime}(\mathbb{T}^{2d}_{L})$. Assume that the Fourier
coefficients $c_{\kappa}(p)$, $\kappa\in\mathbb{Z}^{2d}$, satisfy

(66) $c_{0}(p)\neq
0\quad\text{and}\quad\sum_{\kappa\in\mathbb{Z}^{2d}_{0}}|c_{\kappa}(p)|v\left(\pi_{\mathcal{J}L^{-T}\kappa}\right)<\frac{|c_{0}(p)|}{C},$

where $C$ is the constant in (49). Then for any $0\leq\tau\leq 1$

  * i)

the operator $\textup{Op}_{\tau}(p)$ is invertible in $\mathcal{L}(X)$;

  * ii)

the norm in $\mathcal{L}(X)$ of the inverse operator satisfies the following
estimate

(67)
$\|(\textup{Op}_{\tau}(p))^{-1}\|_{\mathcal{L}(X)}\leq\dfrac{1}{\left(1+Cv(0)\right)|c_{0}(p)|-C\|c_{k}(p)\|_{\ell^{1}_{L,m}}}.$

Notice that, according to the previous estimate, the invertibility of
$\textup{Op}_{\tau}(p)$ is independent of the quantization $\tau$.

###### Proof.

Our goal is to estimate the operator norm
$\|I-c\,\textup{Op}_{\tau}(p)\|_{\mathcal{L}(X)}$, for any $0\leq\tau\leq 1$,
and $c$ suitable non vanishing constant. Let us consider the definition of
Fourier coefficient (25). Since the monochromatic signals $e^{-2\pi i\langle
L^{-T}\kappa,x\rangle}$ are $L$-periodic, it easily follows that $c_{0}(1)=1$
and $c_{\kappa}(1)=0$ when $\kappa\in\mathbb{Z}^{2d}_{0}$ . Thus
$c_{\kappa}(1-cp)=-cc_{\kappa}(p)$, when $\kappa\neq 0$ and
$c_{0}(1-cp)=1-c\,c_{0}(p)$. Assuming that $\langle
p,1\rangle_{\mathbb{T}_{L}^{2d}}\neq 0$, thus $c_{0}(p)=\frac{\langle
p,1\rangle_{\mathbb{T}^{2d}_{L}}}{\textup{det}L}\neq 0$, and setting
$c=\frac{1}{c_{0}(p)}$, the symbol of the operator
$I-\frac{1}{c_{0}(p)}\textup{Op}_{\tau}(p)$ admits the following Fourier
coefficients

(68)
$c_{\kappa}\left(1-\frac{p}{c_{0}(p)}\right)=\left\\{\begin{array}[]{ll}0&\text{when}\,\,\kappa=0\\\
-\frac{c_{\kappa}(p)}{c_{0}(p)}&\text{when}\,\,\kappa\neq
0\,.\end{array}\right.$

The following estimate then follows directly from Theorem 4.1,

(69)
$\begin{array}[]{ll}\|I-\frac{1}{c_{0}(p)}\textup{Op}_{\tau}(p)\|_{\mathcal{L}(X)}&\leq
C\|c_{k}(1-\frac{p}{c_{0}(p)})\|_{\ell^{1}_{L,m}}=\\\ \\\
&=\frac{C}{|c_{0}(p)|}\sum\limits_{\kappa\in\mathbb{Z}^{2d}_{0}}|c_{\kappa}(p)|v\left(\mathcal{J}L^{-T}\kappa\right),\end{array}$

where $C$ is the constant in (49). Thus i) directly follows from Proposition
5.1 .
Thanks to the assumption (66), the estimate (69) and Proposition 5.1, the
inverse operator $(\textup{Op}_{\tau}(p))^{-1}$ can be expanded in Neumann
series

(70)
$(\textup{Op}_{\tau}(p))^{-1}=\frac{1}{c_{0}(p)}\sum_{n=0}^{+\infty}\left(I-\frac{1}{c_{0}(p)}\textup{Op}_{\tau}(p)\right)^{n},$

then using again (69) we have

(71)
$\begin{array}[]{ll}\|(\textup{Op}_{\tau}(p))^{-1}\|_{\mathcal{L}(X)}&\leq\frac{1}{|c_{0}(p)|}\sum\limits_{n=0}^{+\infty}\|I-\frac{1}{c_{0}(p)}\textup{Op}_{\tau}(p)\|_{\mathcal{L}(X)}^{n}\\\
\\\
&\leq\frac{1}{|c_{0}(p)|}\sum\limits_{n=0}^{+\infty}\frac{C^{n}}{|c_{0}(p)|^{n}}\left(\sum\limits_{\kappa\in\mathbb{Z}^{2d}_{0}}|c_{k}(p)|v\left(\mathcal{J}L^{-T}\kappa\right)\right)^{n}\\\
&\leq\frac{1}{|c_{0}(p)|}\sum\limits_{n=0}^{+\infty}\left(\frac{C}{|c_{0}(p)|}(\|c_{\kappa}(p)\|_{\ell^{1}_{L,m}}-|c_{0}(p)|v(0))\right)^{n}\\\
\\\
&\leq\frac{1}{|c_{0}(p)|}\dfrac{1}{1-\frac{C}{|c_{0}(p)|}\left(\|c_{\kappa}(p)\|_{\ell^{1}_{L,m}}-|c_{0}(p)|v(0)\right)}\\\
\\\
&=\dfrac{1}{\left(1+Cv(0)\right)|c_{0}(p)|-C\|c_{k}(p)\|_{\ell^{1}_{L,m}}},\end{array}$

which proves ii). ∎

##  Appendix A On periodic distributions and distributions on the torus

This section is devoted to shortly review some known facts about the
comparison between the space of periodic distributions in $\mathbb{R}^{n}$,
see Section 2.4, and the space $\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ of
distributions on the torus
$\mathbb{T}^{n}_{L}:=\mathbb{R}^{n}/L\mathbb{Z}^{n}$, in order to justify the
identification of the aforementioned spaces, that we have implicitly assumed
in the whole paper. Recall that $\mathcal{D}^{\prime}(\mathbb{T}^{n}_{L})$ is
the space linear continuous forms on the function space
$C^{\infty}(\mathbb{T}^{n}_{L})$ (the latter being endowed with its natural
Fréchet space topology). As already mentioned in Section 2.4, the reader is
referred to Ruzhansky - Turunen [25] for a thorough study of distributions on
the torus.
For the rest, the results collected herebelow come essentially from making
explicit some of the results established in Hörmander [16, Section 7.2].
It is well understood that functions on the torus $\mathbb{T}^{n}_{L}$ can be
naturally identified with $L-$periodic functions in $\mathbb{R}^{n}$. Below,
we will illustrate a way to extend the same identifications to all
$L-$periodic distributions in $\mathbb{R}^{n}$. This extension to
distributions can be made by a duality argument, as it is customary. Thought
the following arguments work in the case of $L-$periodicity, with arbitrary
invertible matrix $L\in GL(n)$, just for simplicity we will restrict to the
case of $L=I_{n}$ the $n\times n$ identity matrix, leading to $1-$periodic
functions and distributions.
So let us first consider a $1-$periodic measurable function $f=f(x)$ in
$\mathbb{R}^{n}$ such that $f\in L^{1}([0,1]^{n})$; of course, such a function
$f$ is a $1-$periodic tempered distribution in $\mathbb{R}^{n}$. On the other
hand, when identified (as usual) with an integrable function on the torus
$\mathbb{T}^{n}$, $f$ defines an element of
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$, whose action on test functions
$\psi\in C^{\infty}(\mathbb{T}^{n})$ is given by

$\langle f,\psi\rangle_{\mathbb{T}^{n}}:=\int_{[0,1]^{n}}f(x)\psi(x)dx\,;$

in order to avoid confusion, here and below
$\langle\cdot,\cdot\rangle_{\mathbb{T}^{n}}$ stands for the duality pair
between $\mathcal{D}^{\prime}(\mathbb{T}^{n})$ and
$\mathcal{C}^{\infty}(\mathbb{T}^{n})$, whereas $\langle\cdot,\cdot\rangle$ is
denoting the dual pair between $\mathcal{S}(\mathbb{R}^{n})$ and
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$.
Testing $f$ against an arbitrary function
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$ we compute

(72) $\begin{split}\langle
f,\varphi\rangle&=\int_{\mathbb{R}^{n}}f(x)\varphi(x)dx=\sum\limits_{\kappa\in\mathbb{Z}^{n}}\int_{[0,1]^{n}+\kappa}f(x)\varphi(x)dx\\\
&=\sum\limits_{\kappa\in\mathbb{Z}^{n}}\int_{[0,1]^{n}}f(y+\kappa)\varphi(y+\kappa)dy=\int_{[0,1]^{n}}\sum\limits_{\kappa\in\mathbb{Z}^{n}}f(y+\kappa)\varphi(y+\kappa)dy\\\
&=\int_{[0,1]^{n}}f(y)\sum\limits_{\kappa\in\mathbb{Z}^{n}}\varphi(y+\kappa)dy\,,\end{split}$

where the countable-additivity of the Lebesgue integral, together with
Fubini’s theorem to interchange the sum and the integral, and the periodicity
of $f$ are used.
The function

$\varphi_{\rm
per}(x):=\sum\limits_{\kappa\in\mathbb{Z}^{n}}\varphi(x+\kappa)\,,\quad
x\in\mathbb{R}^{n}\,,$

which appears in the last line in (72), is a $1-$periodic $C^{\infty}$function
that can be canonically identified with a (unique) element in
$C^{\infty}(\mathbb{T}^{n})$ ); we call it $1-$periodization of $\varphi$. It
is fairly easy to check that the mapping $\varphi\mapsto\varphi_{\rm per}$ is
continuous as a (linear) operator from $\mathcal{S}(\mathbb{R}^{n})$ to
$C^{\infty}(\mathbb{T}^{n})$ (thus from $C^{\infty}_{0}(\mathbb{R}^{n})$ to
$C^{\infty}(\mathbb{T}^{n})$, as well). Therefore the calculations above show
naturally the way to define a linear mapping $\mathcal{T}$ from the space
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$ to the subspace of
$\mathcal{D}^{\prime}(\mathbb{R}^{n})$ consisting of $1-$periodic
distributions (which are automatically tempered distributions), just by
setting for $U\in\mathcal{D}^{\prime}(\mathbb{T}^{n})$

(73) $\langle\mathcal{T}(U),\varphi\rangle:=\langle U,\varphi_{\rm
per}\rangle_{\mathbb{T}^{n}}\,,\quad\forall\,\varphi\in\mathcal{S}(\mathbb{R}^{n})\,.$

It is easy to verify that $\mathcal{T}$ acts continuously from
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$ to $1-$periodic distributions in
$\mathbb{R}^{n}$. Moreover (72) shows that $\mathcal{T}(U)$ properly reduces
to the $1-$periodic tempered distribution in $\mathbb{R}^{n}$ corresponding to
a function $U\in L^{1}(\mathbb{T}^{n})$.
It is a little less obvious that $\mathcal{T}$ is invertible, so that it
actually defines an isomorphism. This can be proved by noticing that every
function $\psi\in C^{\infty}(\mathbb{T}^{n})$ can be regarded as the
$1-$periodization of (at least) one function
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$, namely $\psi=\varphi_{\rm per}$. To
see this, consider a function $\phi\in C^{\infty}_{0}(\mathbb{R}^{n})$
satisfying (18) and set

(74) $\varphi:=\phi\psi\,,$

for any function $\psi\in C^{\infty}(\mathbb{T}^{n})$ (identified with its
$1-$periodic $C^{\infty}-$counterpart in $\mathbb{R}^{n}$). Of course,
$\varphi$ defined above belongs to $C^{\infty}_{0}(\mathbb{R}^{n})$ so it is
rapidly decreasing in $\mathbb{R}^{n}$; moreover, in view of (18) and the
periodicity of $\psi$, we get for any $x\in\mathbb{R}^{n}$

$\varphi_{\rm
per}(x)=\sum\limits_{\kappa\in\mathbb{Z}^{n}}\varphi(x+\kappa)=\sum\limits_{\kappa\in\mathbb{Z}^{n}}\phi(x+\kappa)\psi(x+\kappa)=\psi(x)\sum\limits_{\kappa\in\mathbb{Z}^{n}}\phi(x+\kappa)=\psi(x)\,,$

showing that $\psi$ is actually the $1-$periodization of $\varphi$. This leads
to associate to any periodic distribution
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ a linear form $U$ on
$C^{\infty}(\mathbb{T}^{n})$ by setting for every $\psi\in
C^{\infty}(\mathbb{T}^{n})$

(75) $\langle U,\psi\rangle_{\mathbb{T}^{n}}:=\langle u,\varphi\rangle\,,$

being $\varphi=\varphi(x)$ the rapidly decreasing (actually compactly
supported smooth) function in $\mathbb{R}^{n}$ associated to $\psi$ as in
(74). In order to give consistency to the definition of $U$, we must prove
that it is independent of $\varphi$. Let us notice that, thanks to (17), any
function $\varphi\in\mathcal{S}(\mathbb{R}^{n})$, whose $1-$periodization is
given by $\psi\in C^{\infty}(\mathbb{T}^{n})$, satisfies the following:

$\hat{\varphi}(\kappa)=c_{\kappa}(\psi)\,,\quad\forall\,\kappa\in\mathbb{Z}^{n}\,.$

Thus using the Fourier expansion (19) we obtain

$\langle u,\varphi\rangle=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}\langle
e^{2\pi
i\langle\cdot,\kappa\rangle},\varphi\rangle=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}\hat{\varphi}(\kappa)=\sum_{\kappa\in\mathbb{Z}^{n}}c_{\kappa}c_{\kappa}(\psi)\,.$

This shows the consistency of the definition of $U$ above; the continuity of
the linear form $U$ on $C^{\infty}(\mathbb{T}^{n})$ also easily follows, so
that $U\in\mathcal{D}^{\prime}(\mathbb{T}^{n})$.
The mapping $u\mapsto U$ defined on $1-$periodic distributions in
$\mathbb{R}^{n}$ by (75) provides a linear continuous operator from the space
of $1-$periodic (tempered) distributions in $\mathbb{R}^{n}$ to
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$, which is actually the inverse of the
operator $\mathcal{T}$ introduced before, see (73). This equivalently shows
that $\mathcal{T}$ is an isomorphism, up to which $1-$periodic distributions
in $\mathbb{R}^{n}$ can be thought to be elements of
$\mathcal{D}^{\prime}(\mathbb{T}^{n})$ and viceversa.
Thus a $1-$periodic distribution $u\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$
can be regarded as a linear continuous form on $C^{\infty}(\mathbb{T}^{n})$;
in particular this makes rigorous the testing of $u$ against $1-$periodic
smooth functions in $C^{\infty}(\mathbb{T}^{n})$, such as $e^{-2\pi
i\langle\kappa,\cdot\rangle}$, with $\kappa\in\mathbb{Z}^{n}$, providing
meanwhile an explicit explanation of formula (20) for the Fourier coefficients
of $u$.

## Acknowledgments

The research of A. Morando is partially supported by the Italian MUR Project
PRIN prot. 20204NT8W4.
G. Garello is supported by the Local Research Grant of University of Torino.

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