# Suppression of Electromagnetic Crosstalk by Differential Excitation for SAW
Generation

Shunsuke Ota  Department of Electrical and Electronic Engineering, Tokyo
Institute of Technology, Tokyo 152-8550, Japan  National Institute of Advanced
Industrial Science and Technology (AIST), National Metrology Institute of
Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan  Yuma Okazaki
National Institute of Advanced Industrial Science and Technology (AIST),
National Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki
305-8563, Japan  Shuji Nakamura  National Institute of Advanced Industrial
Science and Technology (AIST), National Metrology Institute of Japan (NMIJ),
1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan  Takehiko Oe  National
Institute of Advanced Industrial Science and Technology (AIST), National
Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563,
Japan  Hermann Sellier  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38000 Grenoble, France  Christopher Bäuerle  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  Nobu-Hisa Kaneko
National Institute of Advanced Industrial Science and Technology (AIST),
National Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki
305-8563, Japan  Tetsuo Kodera  Department of Electrical and Electronic
Engineering, Tokyo Institute of Technology, Tokyo 152-8550, Japan  Shintaro
Takada  National Institute of Advanced Industrial Science and Technology
(AIST), National Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba,
Ibaraki 305-8563, Japan  Persent address: Department of Physics, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

Surface acoustic waves (SAWs) hold a vast potential in various fields such as
spintronics, quantum acoustics, and electron-quantum optics, but an
electromagnetic wave emanating from SAW generation circuits has often been a
major hurdle. Here, we investigate a differential excitation method of
interdigital transducers (IDTs) to generate SAWs while reducing the
electromagnetic wave. The results show that electromagnetic waves are
suppressed by more than $90\text{\,}\%$ in all directions. This suppression
overcomes the operating limits and improves the scalability of SAW systems.
Our results promise to facilitate the development of SAW-based applications in
a wide range of research fields.

Surface acoustic waves (SAWs) are a versatile phononic technology widely used
in both industrial applications and basic research Delsing et al. (2019). In
industry, they are crucial for frequency filtering in wireless devices IEE
and are used for the sensing and mixing of trace amounts of liquids Go et al.
(2017). In the realm of basic research, SAWs are leading to increasingly
impressive results in spintronics, such as spin currents Kobayashi et al.
(2017) and skyrmion generation Yokouchi et al. (2020), and in quantum
acoustics, such as coupling to superconducting qubits Gustafsson et al.
(2014); Manenti et al. (2017); Noguchi et al. (2017); Moores et al. (2018);
Bolgar et al. (2018) and developing a beam splitter Qiao et al. (2023).
Furthermore, in the field of electron-quantum optics, various studies have
been conducted Hermelin et al. (2011); McNeil et al. (2011); Stotz et al.
(2005); Bertrand et al. (2016); Takada et al. (2019); Jadot et al. (2021); Ito
et al. (2021); Edlbauer et al. (2021); Wang et al. (2022, 2023). In
combination with quantum dots on semiconductors, the transport of single
electrons across micrometer distances has been demonstrated Hermelin et al.
(2011); McNeil et al. (2011); Takada et al. (2019), as well as the coherent
transport of electron spins Bertrand et al. (2016); Jadot et al. (2021). The
use of SAWs in quantum information processing has been proposed for some time
Barnes et al. (2000); Foden et al. (2000); Rodriquez et al. (2005), and
recently the development of electron flying qubits Bäuerle et al. (2018);
Edlbauer et al. (2022) has been studied.

Meanwhile, such developments of SAWs often face the persistent problem of
electromagnetic wave crosstalk. High frequency components in experimental
circuits to generate SAWs emit electromagnetic waves while generating SAWs.
This electromagnetic wave is picked up by the metal gates of the target
structure, which are used to conduct the intended experiments (e.g., the metal
gates that define the quantum dots), and generates fluctuation of the electric
potential in the target. This undesired fluctuation not only hides the desired
effect by the SAW, but also interferes with the SAW to produce a negative
effect Kataoka et al. (2006); ota (2023). Indeed, this crosstalk is pronounced
and should be one of the reasons why quantum current sources using SAWs were
eventually largely abandoned, despite great efforts over the years Shilton et
al. (1996); Talyanskii et al. (1997); Cunningham et al. (1999, 2000); Ebbecke
et al. (2002); P. Utko and Lindelof (2003); Ford (2017).

In the field of electron-quantum optics, the crosstalk problem has been
avoided by shifting the arrival timing of the electromagnetic wave and the SAW
at the target structure where electrons are transported. This could be done by
making the length of the SAW generation signal sufficiently shorter than the
propagation time of the SAW from the generated position to the target
structure. Since the electromagnetic wave propagates with the speed of light
($3\text{\times}{10}^{8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$), which is 5
orders of magnitude faster than the speed of SAWs
($\sim$$3000\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$), it disappears
immediately after we stop applying the SAW generation signal and before the
SAW arrives at the target structure. Although this method allows for avoiding
critical influence of the crosstalk, it severely limits the timing of SAW
generation and continuous generation of SAW is prohibited. This inflexibility
is a clear disadvantage for various applications. For example, in electron-
quantum optics, it makes it difficult to scale up the system, where many
single electrons are transferred by SAWs at different timings. In this study,
we investigate the way not to avoid the crosstalk but to suppress it by
devising the SAW generation method. With the developed method we have
succeeded in radically reducing the electromagnetic crosstalk.

Figure 1: (a) Schematic diagram of the excitation method. Top: single
excitation, bottom: differential excitation. The SAW generation signal
$V(\phi)$ is input into the pads connecting to the comb-like electrodes of the
IDT, and the SAWs and electromagnetic waves (EM) are generated. In the case of
the single excitation, the electromagnetic wave originated from $V(\phi)$ is
emitted from the high-frequency circuit such as metal pads. In the case of the
differential excitation, the two types of electromagnetic waves originated
from $V(\phi)$ and $V(\phi+\pi)$ destructively interfere with each other and
cancel out. (b) Schematic diagram of the experimental setup and the device for
the differential excitation.

SAWs are generated by using comb-shaped electrodes called an interdigital
transducer (IDT) on a piezoelectric substrate. Top part of Fig. 1a shows a
typical scheme to generate SAWs, where one pad of the IDT is grounded while
the other is excited by an ac voltage. In this study, this scheme is called a
single excitation. Here, IDTs with a double-finger pattern are employed. This
type of IDT is often employed because it suppresses reflections of SAWs inside
the IDT and improves conduction efficiency. When we apply an ac voltage whose
frequency corresponds to the resonance frequency, $f_{\rm SAW}$, determined by
the period of the IDT fingers, $\lambda$, and the SAW velocity of the
substrate, $V_{\rm SAW}$, as $f_{\rm SAW}=v_{\rm SAW}/\lambda$ on the IDT,
SAWs generated from each finger constructively interfere and strong SAWs
propagate along the substrate towards the both directions. In this process,
electromagnetic waves with the same frequency, $f_{\rm SAW}$, are emitted from
the metal pad where the excitation voltage is applied. In this study, we
perform a differential excitation of the IDT as shown in the bottom part of
Fig. 1a to suppress the radiation of the electromagnetic waves. In the
differential excitation, the resonant ac voltage is applied to the both metal
pads with a same amplitude but a phase shift of $\pi$ between one to the
other. Since the shape of the applied ac field is same for the differential
excitation and the single excitation as depicted in Fig. 1a, SAWs are
generated as in the case of the single excitation. Different from the single
excitation, electromagnetic waves are emitted from the both pads. These
electromagnetic waves have a phase difference of $\pi$ and the amplitude is
expected to be equal if the shapes of the metal pads are symmetrical. Since
the wavelength of the electromagnetic waves, which is about
$100\text{\,}\mathrm{m}\mathrm{m}$ for $f_{\rm SAW}=$
$3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, are about 100 times longer than
the scale of the IDT, which is typically less than
$1\text{\,}\mathrm{m}\mathrm{m}$ for the IDT with a resonant frequency in the
GHz range, the two electromagnetic waves are expected to destructively
interfere. As a result, the electromagnetic wave radiated from the IDT should
be strongly suppressed.

Figure 2: Two-dimensional simulation results of the cancellation effect due to
the interference between two circular waves with opposite phases: Wave A and
Wave B. The wave centers correspond to metal pads
$110\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ apart on our device, where
a resonant ac voltage is applied to excite SAWs. Top left: Wave A - a
frequency $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, a phase 0, a velocity
$3\text{\times}{10}^{8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$, from point A
($0\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$,$55\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$).
Top right: Wave B - a frequency $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, a
phase $\pi$, a velocity
$3\text{\times}{10}^{8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$, from point B
($0\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$,$-55\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$).
Lower left: Combined Wave - interference pattern resulting from the
combination of Wave A and Wave B, showing the cancelation effect. Lower right:
Combined Wave (Detailed View) - Detailed interference pattern with a narrowed
amplitude range.

A simple simulation is performed to estimate the expected cancellation effect
of electromagnetic waves due to the differential excitation. In this
simulation, waves of the same frequency emitted from two different center
points are modeled as electromagnetic waves and their interference with each
other is calculated. Each center point is assumed to be the center of the
metal pads to which the resonant ac voltage is applied. In our sample design,
spacing of the two metal pads are
$110\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. The waves emitted from
both center points have a frequency of
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, an equal amplitude of 1, and a
velocity of $3\text{\times}{10}^{8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$.
The phase of the wave from center point A is set to 0 and the one from center
point B is set to $\pi$. This produces waves in perfectly opposite phases.
These waves are calculated on a $1\text{\,}\mathrm{m}$ square two-dimensional
plane so that the interference pattern could be easily checked. The circular
waves are calculated taking into account the phase difference that depends on
the distance from the center, and the amplitude is set to a constant value
independent of distance. Fig. 2 shows the respective electromagnetic waves
(Wave A and Wave B) in the top panels and the interference pattern resulting
from the synthesis of both waves (Combined Wave) in the lower panels. In the
lower left panel, the amplitude range of Combined Wave is set from -1 to 1,
and the cancellation effect of the combined electromagnetic wave can be
clearly observed. In the lower right panel, the amplitude range is set from
-0.01 to 0.01 in order to confirm the detailed interference pattern. From this
result, the cancellation effect of the electromagnetic waves is estimated to
be over $99\text{\,}\%$. The reason why a considerable suppression is
estimated after taking into account the phase shift due to the difference in
emission position of the two electromagnetic waves is that the difference in
emission position ($110\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) is
small enough for the wavelength of the
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ electromagnetic wave
($300\text{\,}\mathrm{m}\mathrm{m}$). Similar results are also obtained at
other frequencies commonly used for SAW generation signals
($3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ \-
$4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$).

Next, we perform measurement of the device with a semi-automatic probe station
at room temperature. The device is fabricated on a GaAs substrate and consists
of IDTs and metal pads connected to the IDTs. The electrodes of the IDTs are
fabricated using a standard electron-beam lithography with successive thin-
film evaporation (metalization Ti $3\text{\,}\mathrm{n}\mathrm{m}$, Al
$27\text{\,}\mathrm{n}\mathrm{m}$). The metal pads are fabricated using a
standard photolithography with successive thin-film evaporation (metalization
Ti $20\text{\,}\mathrm{n}\mathrm{m}$, Au $100\text{\,}\mathrm{n}\mathrm{m}$).
Two IDTs of the same design are placed facing each other in the direction of
SAW propagation, at a distance of
$1000\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, as shown in Fig. 1b. One
IDT acts as a SAW generator and the other as a SAW detector. The both IDTs are
designed to generate and detect SAWs at a resonant frequency of
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
($\lambda=$$2.86\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) and have 40
pairs of electrodes , an aperture of
$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. The SAW propagation
direction is set to [110] direction. Contact metal pads
($330\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ length and
$60\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ width) are connected to the
upper- and the lower-electrode sets of each IDT. High-frequency probes make
contact with these pads, thus establishing a connection to the IDTs. These
pads also act as a detector of electromagnetic waves. The resonant ac voltage
to excite SAWs is provided by an arbitrary waveform generator (AWG, Keysight
M8195A) and fed through different coaxial cables to the upper and lower
electrodes of the IDT, respectively, in the case of differential excitation. A
phase shifter (WAKA 02X0442-00) is connected to one of the input lines for
fine phase difference adjustment before the signal is input to the IDT. The
phase shifter is adjusted and fixed so that there is minimal phase
displacement at the probe when the same ac voltage is output from two channels
of AWG. On the detector side, the set of lower electrodes is grounded
(embedded in the surrounding ground pad). The set of upper electrodes is
connected to a high-speed sampling oscilloscope (Keysight N1094B DCA-M) to
observe the generated SAW and the electromagnetic wave. The detected signal is
amplified by a series of broadband amplifiers (SHF S126A, ZHL-4W-422+).

Figure 3: Time-resolved measurements on the IDT with the single excitation and
the differential excitation. Trace of the detector response for the resonant
ac voltage($1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, a time span
$60\text{\,}\mathrm{n}\mathrm{s}$, a peak-to-peak amplitude
$350\text{\,}\mathrm{m}\mathrm{V}$). The blue line represents the SAW and the
electromagnetic wave signals obtained from the single excitation, with the SAW
on the right and the previously detected electromagnetic wave on the left. The
red line represents the obtained signal from the differential excitation,
where the SAW amplitude is similar to the single excitation while the
electromagnetic component is significantly suppressed

.

SAWs and electromagnetic waves are generated and observed with the single and
differential excitation. To generate SAWs, we apply a sinusoidal signal at the
resonance frequency, $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, with a time
span of $60\text{\,}\mathrm{n}\mathrm{s}$ and a peak-to-peak amplitude of
$350\text{\,}\mathrm{m}\mathrm{V}$ in a repetition period of
$1520\text{\,}\mathrm{n}\mathrm{s}$. Here the time span of the signal,
$60\text{\,}\mathrm{n}\mathrm{s}$, is chosen to be longer than
$40\text{\,}\mathrm{n}\mathrm{s}$ which is the minimum length to fully excite
the IDT and to be shorter than the propagation time of SAWs between the IDTs,
$\sim$ $350\text{\,}\mathrm{n}\mathrm{s}$ which is calculated from the
distance between the IDTs
($1000\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) and the SAW speed in
GaAs ($\sim$ $2860\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$). Since the
electromagnetic waves emitted from the IDT propagate with the speed of light
($\sim$ $3\text{\times}{10}^{8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$), they
reach the other detector IDT almost instantaneously in less than
$10\text{\,}\mathrm{p}\mathrm{s}$. Therefore, when we perform a real-time
detection of the signal at the detector IDT, we first observe the
electromagnetic waves and later observe the SAW signal with a well-defined
separation between within the repetition period of
$1520\text{\,}\mathrm{n}\mathrm{s}$. Here we apply the signal with the same
amplitude for both the single and the differential excitation and hence the
detected SAW signal is expected to be double for the differential excitation
compared to the single. To compare the ratio between the amplitude of the SAW
signal and the electromagnetic waves for both excitation methods we double the
detected signal for the single excitation.

Fig. 3 shows the signal measured with the sampling oscilloscope. The blue line
is the data obtained when we perform the single excitation. The signal
observed in the earlier time comes from electromagnetic waves and the one
observed in approximately $350\text{\,}\mathrm{n}\mathrm{s}$ later comes from
SAWs. For the single excitation, the signal coming from electromagnetic waves
is as large as the one coming from SAWs. The red line shows the data obtained
when we perform the differential excitation. The signal coming from SAWs is as
large as the one from the single excitation. On the other hand, the signal
coming from electromagnetic waves is strongly suppressed as expected from our
simple simulation. Comparing the average values of the signal amplitude coming
from electromagnetic waves from the single and the differential excitation, it
is calculated that $92.4\text{\,}\%$ of the signal is suppressed for the case
of the differential excitation. In our simple simulation, the electromagnetic-
wave suppression of about $99\text{\,}\%$ is expected. The residual amount of
the electromagnetic waves can be attributed to minute discrepancies between
the two emitted electromagnetic waves due to the difference of response
functions of the metal pads. These discrepancies manifested as time-dependence
components that exhibited different phases and intensities and could not be
addressed by simple adjustments of the phase or the amplitude of the SAW
generation signal. Such fluctuations are considered to be intricately related
to high-frequency circuit components like a shape or a thickness of the metal
pads. It suggests that optimization of these components is essential for even
better suppression of the electromagnetic waves.

Figure 4: Spatial dependence of electromagnetic-wave suppression. A schematic
of the sample is shown in the middle. Surrounding plots are electromagnetic
waves detected on each metal pad with the single and the differential
excitation. The numbers of the metal pads in the schematic correspond to the
numbers of the plots. The suppression effect is almost same for all positions.
The change in the amplitude of the electromagnetic waves from the results in
Fig. 3 can be attributed to the different design of the metal pads for
detection.

Finally, we experimentally confirm the spatial dependence of the
electromagnetic-wave suppression. For that purpose, we prepared the other
device illustrated in the schematic in the middle of Fig. 4. On the left side,
an IDT and metal pads having the same design as the device measured in Fig. 3
are placed. On the right side, multiple metal pads
($220\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ in length and
$60\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ in width) are arranged to
pick up and detect electromagnetic waves. They are symmetrically arranged
about the horizontal line passing through the middle of the IDT. The signal
used to generate SAWs is the same as in Fig. 3. Each plot in Fig. 4 shows the
signal coming from electromagnetic waves detected by the individual metal pads
for the case of the single and the differential excitation. From these
results, it is confirmed that the electromagnetic-wave suppression occurs
similarly at all positions of the metal pads.

In conclusion, our study demonstrates the effectiveness of the differential
excitation in mitigating electromagnetic crosstalk in SAW-based devices. The
experimental results have shown that this approach is capable of nullifying
over $90\text{\,}\%$ of the undesirable electromagnetic components, achieving
a substantial reduction. This suppression of the electromagnetic crosstalk
will certainly contribute to the improvement of the accuracy to control single
flying electrons using SAW potentials Kataoka et al. (2006). Furthermore, the
spatial dependence of the electromagnetic wave cancellation effect has also
been investigated. Consistent reduction in electromagnetic waves has been
evident at all measured positions. This consistency is important for the
scalability of SAW-based devices. In addition, although our experiments have
been conducted at room temperature, we anticipate that the principles of the
electromagnetic-wave suppression with the differential excitation are equally
applicable in low-temperature environments. The electromagnetic-wave
suppression has been investigated with a double-finger IDT in this study.
However, it will be equally applicable to various types of IDTs Morgan (2007);
Lima et al. (2003); Schülein et al. (2015); Ekström et al. (2017); Dumur et
al. (2019). Our research focuses on fundamentally suppressing the
electromagnetic crosstalk that has long hindered advancements of SAW-based
single-electron control devices. The differential excitation of IDTs provides
a new perspective for addressing the problem of electromagnetic crosstalk in
SAW-based systems and opens up the possibility of application of SAWs in
various fields of research such as spintronics and quantum acoustics. For
example, in electron-quantum optics, the methodologies and insights obtained
in this study are expected to significantly improve the scalability of SAW-
based single-electron manipulation devices and make important contributions to
the evolution of sophisticated SAW quantum information processing devices.

Acknowledgments
S.O. acknowledge financial support from JST SPRING, Grant Number JPMJSP2106.
T.K. and S.T. acknowledge financial support from JSPS KAKENHI Grant Number
20H02559.
N.-H.K. acknowledges financial support from JSPS KAKENHI Grant Number
JP18H05258.
C.B. acknowledges financial support from the French Agence Nationale de la
Recherche (ANR), project QUABS ANR-21-CE47-0013-01 as well as funding from the
European Union’s H2020 research and innovation program under grant agreement
No 862683 ”UltraFastNano”.

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# On the extreme eigenvalues and asymptotic conditioning of a class of
Toeplitz matrix-sequences arising from fractional problems

M. Bogoya†111Dipartimento di Scienza ed Alta Tecnologia, Università
dell’Insubria, Via Valleggio 11, 22100 Como (ITALY).
×Departamento de Matemáticas, CINVESTAV, México D.F. (México).
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>S.M. Grudsky‡×
M. Mazza≀∗ S. Serra-Capizzano⋄∗

###### Abstract

The analysis of the spectral features of a Toeplitz matrix-sequence
$\left\\{T_{n}(f)\right\\}_{n\in\mathbb{N}}$, generated by a symbol $f\in
L^{1}([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided in
great detail in the last century, as well as the study of the conditioning,
when $f$ is nonnegative a.e. Here we consider a novel type of problem arising
in the numerical approximation of distributed-order fractional differential
equations (FDEs), where the matrices under consideration take the form

$\mathcal{T}_{n}=c_{0}T_{n}(f_{0})+c_{1}h^{h}T_{n}(f_{1})+c_{2}h^{2h}T_{n}(f_{2})+\cdots+c_{n-1}h^{(n-1)h}T_{n}(f_{n-1}),$

$c_{0},c_{1},\ldots,c_{n-1}\in[c_{*},c^{*}]$, $c^{*}\geqslant c_{*}>0$,
independent of $n$, $h=\frac{1}{n}$, $f_{j}\sim g_{j}$,
$g_{j}=|\theta|^{2-jh}$, $j=0,\ldots,n-1$. Since the resulting generating
function depends on $n$, the standard theory cannot be applied and the
analysis has to be performed using new ideas. Few selected numerical
experiments are presented, also in connection with matrices that come from
distributed-order FDE problems, and the adherence with the theoretical
analysis is discussed together with open questions and future investigations.

Keywords: Toeplitz sequences; algebra of matrix-sequences; generating
function; fractional operators.

AMS SC: 15B05; 15A18; 26A33

##  1 Introduction

In many practical applications it is required to solve numerically linear
systems of Toeplitz kind and of (very) large dimensions and hence several
specialized techniques of iterative type, such as preconditioned Krylov
methods and ad hoc multigrid procedures have been designed; we refer the
interested reader to the books [4, 14] and to the references therein. We
recall that such types of large Toeplitz linear systems emerge from specific
applications involving e.g. the numerical solution of (integro-) differential
equations and of problems with Markov chains. On the other hand, quite
recently, new examples of real world problems have emerged and among them we
can cite the modeling of anomalous diffusion processes, which naturally lead
to the use of fractional differential equations (FDEs). Also in this case we
encounter Toeplitz structures, which are dense since the fractional operators
are inherently nonlocal and hence new numerical challenges have come into the
play. In all the considered applications, the sizes of resulting matrices are
large and iterative solvers have to be considered, both for numerical
stability and computational issues. The convergence analysis of such iterative
procedures can be performed when we know the spectral features of the
considered coefficient matrices. This has been done in the recent literature
for certain constant-order FDE problems (see, e.g., [6, 7]), by exploiting the
well-established analysis of the spectral features of Toeplitz matrix-
sequences generated by Lebesgue integrable functions and the more recent
Generalized Locally Toeplitz theory [9].

Here we consider a novel type of problem arising in the numerical
approximation of distributed-order fractional operators. The latter can be
interpreted as a parallel distribution of derivatives of fractional orders,
whose most immediate application consists in the physical modeling of systems
characterized by a superposition of different processes operating in parallel.
As an example, we mention the application of fractional distributed-order
operators as a tool for accounting memory effects in composite materials [3]
or multi-scale effects [2]. For a detailed review on the topic we refer the
reader to [5]. The procedure to numerically approximate distributed-order
operators is made of two steps: 1) as the distributed-order operator consists
of a continuous distribution of fractional order, a numerical integration is
used to discretize the distributed-order operator into a multi-term constant-
order fractional derivative; 2) following the conversion of the distributed-
order operator into a multi-term fractional derivative, an approximation
method has to be used for discretizing each constant-order fractional
derivative composing the multi-term derivative.

In this paper, we focus on the case left open in [13] which arises when the
integral partition width used in step 1) is asymptotic to the discretization
step adopted in 2) and where the matrices under consideration take the form

$\mathcal{T}_{n}\mathrel{\mathop{\ordinarycolon}}=c_{0}T_{n}(f_{0})+c_{1}h^{h}T_{n}(f_{1})+c_{2}h^{2h}T_{n}(f_{2})+\cdots+c_{n-1}h^{(n-1)h}T_{n}(f_{n-1}),$
(1)

where $T_{n}(f_{j})$ is the Toeplitz matrix of size $n$ generated by $f_{j}$,
the constants $c_{0},c_{1},\ldots,c_{n-1}$ belong to the interval
$[c_{*},c^{*}]$, $c^{*}\geqslant c_{*}>0$, and are independent of $n$,
$h\mathrel{\mathop{\ordinarycolon}}=\frac{1}{n}$, $f_{j}\geqslant 0$ a.e. and
given $g_{j}(\theta)\mathrel{\mathop{\ordinarycolon}}=|\theta|^{2-jh}$,
$j=0,\ldots,n-1$ there exist positive constants $d_{*},d^{*}$ independent of
$j$ and $n$ satisfying

$d_{*}g_{j}\leqslant f_{j}\leqslant d^{*}g_{j}\quad\text{a.e.,}\quad
j=0,\ldots,n-1,$

(see e.g. [8, 10, 13, 12] and the references therein). Taking into
consideration the linearity of the operators $T_{n}(\cdot)$, $n\geqslant 1$,
since the resulting generating function

$F_{n}(\theta)\mathrel{\mathop{\ordinarycolon}}=\sum_{j=0}^{n-1}c_{j}h^{jh}f_{j}(\theta)$

depends on $n$, the standard theory cannot be applied in a straightforward
manner and the analysis has to be performed using new ideas. The description
and the exploitation of such new ideas is the main topic of this note. The
organization of the paper is quite simple. In Section 2 we briefly recall some
properties of Toeplitz matrices and the linear positive operator
$T_{n}(\cdot)$ that will be used in Section 3 to provide asymptotic estimates
for the extreme eigenvalues of $\mathcal{T}_{n}$ in (1). In Section 4 we show
and critically discuss few selected numerical examples for illustrating our
theoretical findings. Finally, Section 5 is devoted to conclusions and to
state few open problems.

##  2 A few preliminaries

Given a Lebesgue integrable function $f$ defined on $[-\pi,\pi]$, i.e. $f\in
L^{1}([-\pi,\pi])$, and periodically extended to the whole real line, let us
consider the Toeplitz matrix $T_{n}(f)$ of size $n$ generated by $f$. For any
$n$, the entries of $T_{n}(f)$ are defined via the Fourier coefficients
$\\{\mathfrak{a}_{k}(f)\\}_{k}$, $k\in\mathbb{Z}$, of $f$ in the sense that

$\left(T_{n}(f)\right)_{s,t}=\mathfrak{a}_{s-t}(f),\quad\quad
s,t\in\\{1,\ldots,n\\},$

and

$\mathfrak{a}_{k}(f)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)\textrm{e}^{-\textrm{i}k\theta}\operatorname{d\\!}\theta,\quad
k=0,\pm 1,\pm 2,\dots,\quad\textrm{i}^{2}=-1.$

The function $f$ is called the _generating function or symbol_ of $T_{n}(f)$.
In the general case $f$ is complex-valued and $T_{n}(f)$ is non-Hermitian for
all sufficiently large $n$. However if $f$ is real-valued a.e., then
$T_{n}(f)$ is Hermitian for all $n$. Furthermore, when $f$ is real-valued and
even a.e., the matrix $T_{n}(f)$ is (real) symmetric for all $n$ [9].

As it is well known in the literature the operator $T_{n}\colon
L^{1}([-\pi,\pi])\rightarrow\mathcal{M}_{n}({\mathbb{C}})$ is a linear
positive operator, briefly LPO, in the sense that:

Linearity.

$T_{n}(\alpha f+\beta g)=\alpha T_{n}(f)+\beta T_{n}(g)$, for any constants
$\alpha,\beta\in\mathbb{C}$, and any functions $f,g\in L^{1}([-\pi,\pi])$.

Positivity.

$T_{n}(f)$ is Hermitian nonnegative definite for any size $n\geqslant 1$ if
$f\geqslant 0$ a.e.

For more details on LPOs, we refer the reader to [16] and to the beautiful
book [11]. Here we just recall that the linearity is a trivial fact coming
from the linearity of the Fourier coefficients with respect to its argument
$f$, while the second property is less obvious. In reality, as proven in [16],
the positivity holds even in a stronger sense, because $f\geqslant 0$ a.e. and
$f$ not identically zero a.e. imply that $T_{n}(f)$ is positive definite for
any $n\geqslant 1$: conversely, $f\equiv 0$ a.e. means that $T_{n}(f)$ is
trivially the null matrix of size $n\geqslant 1$.

Furthermore, it is a simple check to show that linearity and positivity imply
monotonicity and hence from $f\geqslant g$ a.e. we deduce that
$T_{n}(f)-T_{n}(g)$ is nonnegative definite (respectively positive definite if
$f-g$ is not identically zero a.e.), that is $T_{n}(f)\geqslant T_{n}(g)$
($T_{n}(f)>T_{n}(g)$ if $f-g$ is not identically zero a.e.).

We conclude the current section by fixing the notation adopted throughout the
paper. Given a square complex matrix
$B\mathrel{\mathop{\ordinarycolon}}=[b_{s,t}]_{s,t=1}^{n}$, by $\|B\|_{p}$ we
mean the matrix norm induced by the vector $p$-norm, i.e.,

$\|B\|_{p}\coloneqq\sup\bigg{\\{}\frac{\|Bx\|_{p}}{\|x\|_{p}}\colon x\neq
0\bigg{\\}},$

and precisely $\|B\|_{1}=\max_{s}\sum_{t}|b_{s,t}|$ and
$\|B\|_{2}=\sigma_{\max}(B)$, also known as spectral norm. We recall that when
$B$ is Hermitian or symmetric we have $\|B\|_{2}=\rho(B)$, with $\rho(B)$
being the spectral radius of $B$ and, as a consequence, that
$\|B\|_{2}\leqslant\|B\|_{1}$. Finally, when considering an invertible matrix
$B$, with $\mu_{2}(B)$ we refer to the spectral condition number, i.e.,

$\mu_{2}(B)\mathrel{\mathop{\ordinarycolon}}=\|B^{-1}\|_{2}\|B\|_{2}=\frac{\sigma_{\max}(B)}{\sigma_{\min}(B)}.$

##  3 Analysis of the problem

The set of properties introduced in Section 2 allows one to deduce the
following general result regarding the considered class of quite involved
matrices (1), arising in the numerical approximation of distributed-order FDEs
(see [10]).

######  Theorem 3.1.

For $j=0,\ldots,n-1$, let $f_{j}$ be an essentially bounded function defined
on $[-\pi,\pi]$ such that there exist positive constants $d_{*},d^{*}>0$,
independent of $j$ and of $n$, for which

$d_{*}g_{j}(\theta)\leqslant f_{j}(\theta)\leqslant d^{*}g_{j}(\theta)\
\mbox{a.e.},\ \ \
g_{j}(\theta)\mathrel{\mathop{\ordinarycolon}}=|\theta|^{2-jh}.$

Set

$\tilde{\mathcal{T}}_{n}\mathrel{\mathop{\ordinarycolon}}=c_{0}T_{n}(g_{0})+c_{1}h^{h}T_{n}(g_{1})+c_{2}h^{2h}T_{n}(g_{2})+\cdots+c_{n-1}h^{(n-1)h}T_{n}(g_{n-1}),$
(2)

where the constants $c_{0},c_{1},\ldots,c_{n-1}$ belong to the interval
$[c_{*},c^{*}]$, $c^{*}\geqslant c_{*}>0$, and are independent of $n$ and
$h\mathrel{\mathop{\ordinarycolon}}=\frac{1}{n}$. Then $\mathcal{T}_{n}$ as in
(1) and $\tilde{\mathcal{T}}_{n}$ are both Hermitian positive definite and in
addition

$d_{*}\tilde{\mathcal{T}}_{n}\leqslant\mathcal{T}_{n}\leqslant
d^{*}\tilde{\mathcal{T}}_{n}.$

Consequently we have

$\displaystyle d_{*}\lambda_{\min}(\tilde{\mathcal{T}}_{n})$
$\displaystyle\leqslant\lambda_{\min}(\mathcal{T}_{n})\leqslant$
$\displaystyle d^{*}\lambda_{\min}(\tilde{\mathcal{T}}_{n}),$ $\displaystyle
d_{*}\lambda_{\max}(\tilde{\mathcal{T}}_{n})$
$\displaystyle\leqslant\lambda_{\max}(\mathcal{T}_{n})\leqslant$
$\displaystyle d^{*}\lambda_{\max}(\tilde{\mathcal{T}}_{n}),$
$\displaystyle\frac{d_{*}}{d^{*}}\cdot\mu_{2}(\tilde{\mathcal{T}}_{n})$
$\displaystyle\leqslant\mu_{2}(\mathcal{T}_{n})\leqslant$
$\displaystyle\frac{d^{*}}{d_{*}}\cdot\mu_{2}(\tilde{\mathcal{T}}_{n}).$

###### Proof..

Taking into account that $f_{j},g_{j}$ are nonnegative a.e. and not
identically zero a.e, from the fact $T_{n}(\cdot)$ is LPO for any size $n$ and
in the strong sense, we infer that $T_{n}(f_{j})$ is Hermitian positive
definite for $j=0,\ldots,n-1$, and that $T_{n}(g_{j})$ is real symmetric
positive definite, owing to the additional fact that $g_{j}$ is also even for
any $j=0,\ldots,n-1$. Consequently, both $\mathcal{T}_{n}$ and
$\tilde{\mathcal{T}}_{n}$ are positive definite, the first generically
Hermitian and the second being also real symmetric, since they are linear
combinations with positive coefficients of Hermitian positive definite
matrices (real symmetric positive definite matrices in the second case).

Now, any linear combination with positive coefficients of LPOs is a new LPO so
that the relationships

$F_{n}(\theta)\mathrel{\mathop{\ordinarycolon}}=\sum_{j=0}^{n-1}c_{j}h^{jh}f_{j}(\theta),\qquad\tilde{F}_{n}(\theta)\mathrel{\mathop{\ordinarycolon}}=\sum_{j=0}^{n-1}c_{j}h^{jh}g_{j}(\theta),$

$\mathcal{T}_{n}=T_{n}(F_{n})$,
$\tilde{\mathcal{T}}_{n}=T_{n}(\tilde{F}_{n})$,
$d_{*}\tilde{F}_{n}(\theta)\leqslant F_{n}(\theta)\leqslant
d^{*}\tilde{F}_{n}(\theta)$ a.e. are implied. As a consequence, taking into
account the monotonicity, we deduce

$d_{*}\tilde{\mathcal{T}}_{n}\leqslant\mathcal{T}_{n}\leqslant
d^{*}\tilde{\mathcal{T}}_{n},$ (3)

while the other four relations are directly implied by the latter one in (3),
with derivation steps as done in [15], in a slightly different setting. With
this the proof of the theorem is concluded. ∎

The previous reduction theorem allows one to shift the focus of the considered
class of problems $\mathcal{T}_{n}$, to the simpler class of matrices
$\tilde{\mathcal{T}}_{n}$, with reference to the specific generating function
$\tilde{F}_{n}(\theta)$.

For the very same reason, since the constants $c_{0},c_{1},\ldots,c_{n-1}$
belong to the interval $[c_{*},c^{*}]$, $c^{*}\geqslant c_{*}>0$, and are
independent of $n$, with $h=\frac{1}{n}$, the asymptotic study can be
simplified further and reduced to the analysis of $\hat{\mathcal{T}}_{n}$ with
$\hat{\mathcal{T}}_{n}\mathrel{\mathop{\ordinarycolon}}=T_{n}(\hat{F}_{n})$,
and

$\hat{F}_{n}(\theta)\mathrel{\mathop{\ordinarycolon}}=\sum_{j=0}^{n-1}h^{jh}g_{j}(\theta),$
(4)

in accordance to the following result, whose proof mimics the same steps of
the previous one.

######  Theorem 3.2.

For $j=0,\ldots,n-1$, let $c_{j}$ be positive numbers for which there exist
positive constants $c^{*},c_{*}$ independent of $n$ with

$0<c_{*}\leqslant c_{j}\leqslant c^{*}.$

Set

$\hat{\mathcal{T}}_{n}\mathrel{\mathop{\ordinarycolon}}=T_{n}(g_{0})+h^{h}T_{n}(g_{1})+h^{2h}T_{n}(g_{2})+\cdots+h^{(n-1)h}T_{n}(g_{n-1}),$

Then $\hat{\mathcal{T}}_{n}$ and $\tilde{\mathcal{T}}_{n}$ as in (2) are both
Hermitian positive definite and in addition

$c_{*}\hat{\mathcal{T}}_{n}\leqslant\tilde{\mathcal{T}}_{n}\leqslant
c^{*}\hat{\mathcal{T}}_{n}.$

Consequently we have

$\displaystyle c_{*}\lambda_{\min}(\hat{\mathcal{T}}_{n})$
$\displaystyle\leqslant\lambda_{\min}(\tilde{\mathcal{T}}_{n})\leqslant$
$\displaystyle c^{*}\lambda_{\min}(\hat{\mathcal{T}}_{n}),$ $\displaystyle
c_{*}\lambda_{\max}(\hat{\mathcal{T}}_{n})$
$\displaystyle\leqslant\lambda_{\max}(\tilde{\mathcal{T}}_{n})\leqslant$
$\displaystyle c^{*}\lambda_{\max}(\hat{\mathcal{T}}_{n}),$
$\displaystyle\frac{c_{*}}{c^{*}}\cdot\mu_{2}(\hat{\mathcal{T}}_{n})$
$\displaystyle\leqslant\mu_{2}(\tilde{\mathcal{T}}_{n})\leqslant$
$\displaystyle\frac{c^{*}}{c_{*}}\cdot\mu_{2}(\hat{\mathcal{T}}_{n}).$

The rest of the section deals with the more specific case of the asymptotic
spectral analysis of $\hat{\mathcal{T}}_{n}$. The next result shows asymptotic
estimates for the extreme eigenvalues of $\hat{\mathcal{T}}_{n}$. We aim at
studying the asymptotic behavior of the extreme eigenvalues of
$T_{n}(\hat{F}_{n})$ and in particular at proving that
$\lambda_{\min}(T_{n}(\hat{F}_{n}))\sim h$. For reaching this goal we consider
two preparatory lemmas.

######  Lemma 3.1.

For all $\alpha_{j}\in(0,1]$ and all natural $n$, we have

$T_{n}(|\theta|^{2-\alpha_{j}})h^{\alpha_{j}}\leqslant
T_{n}(|\theta|^{2})+R_{n,j},$

with $R_{n,j}$ a real matrix such that

$\rho(R_{n,j})=\|R_{n,j}\|_{2}\leqslant\frac{\alpha_{j}}{3\pi
n^{2}(3-\alpha_{j})}.$

###### Proof..

We first observe that
$|\theta|^{2-\alpha_{j}}h^{\alpha_{j}}\leqslant|\theta|^{2}$ if and only if
$h\leqslant|\theta|$, therefore

$|\theta|^{2-\alpha_{j}}h^{\alpha_{j}}\leqslant|\theta|^{2}+\psi_{n,j}(\theta)$

where
$\psi_{n,j}(\theta)\coloneqq\chi_{[-h,h]}(\theta)\big{\\{}|\theta|^{2-\alpha_{j}}h^{\alpha_{j}}-|\theta|^{2}\big{\\}}$.
Since $T_{n}\colon L^{1}(-\pi,\pi)\to\mathcal{M}_{n}(\mathbb{C})$ is a LPO we
deduces that

$T_{n}(|\theta|^{2-\alpha_{j}})h^{\alpha_{j}}\leqslant
T_{n}(|\theta|^{2})+R_{n,j},$

where $R_{n,j}\coloneqq T_{n}(\psi_{n,j})$ which is a real and symmetric
matrix since $\psi_{n,j}$ is a real-valued and even function.

Now we evaluate $\rho(R_{n,j})=\|R_{n,j}\|_{2}$ and for doing that we
calculate the $k$-th Fourier coefficient of $\psi_{n,j}$, which we called
$\mathfrak{a}_{k}(\psi_{n,j})$,

$\displaystyle|\mathfrak{a}_{k}(\psi_{n,j})|=$
$\displaystyle\bigg{|}\frac{1}{2\pi}\int_{-h}^{h}(|\theta|^{2-\alpha_{j}}h^{\alpha_{j}}-|\theta|^{2})\textrm{e}^{-\textrm{i}k\theta}\operatorname{d\\!}\theta\bigg{|}$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{2\pi}\int_{-h}^{h}|\theta|^{2-\alpha_{j}}(h^{\alpha_{j}}-|\theta|^{\alpha_{j}})\operatorname{d\\!}\theta$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{h}(h^{\alpha_{j}}\theta^{2-\alpha_{j}}-\theta^{2})\operatorname{d\\!}\theta$
$\displaystyle=$ $\displaystyle\frac{\alpha_{j}}{3\pi n^{3}(3-\alpha_{j})}.$

Therefore,

$\rho(R_{n,j})=\|R_{n,j}\|_{2}\leqslant\|R_{n,j}\|_{1}=\max_{s}\sum_{k=0}^{n-1}|\mathfrak{a}_{s-k}(\psi_{n,j})|\leqslant\frac{\alpha_{j}}{3\pi
n^{2}(3-\alpha_{j})},$

finishing the proof. ∎

######  Lemma 3.2.

For an entire number $q\geqslant 1$ let
$M_{n,q}\coloneqq\frac{1}{q}\sum_{j=0}^{q-1}T_{n}(|\theta|^{2-\alpha_{j}})h^{\alpha_{j}}$
with $\alpha_{0},\ldots,\alpha_{q-1}\in[0,1]$. Then

$M_{n,q}\leqslant T_{n}(|\theta|^{2})+R_{n},$

where $R_{n}\coloneqq\sum_{j=0}^{q-1}R_{n,j}$ with $R_{n,j}$ as in Lemma 3.1,
and

$\displaystyle\lambda_{\min}(M_{n,q})\leqslant$
$\displaystyle\lambda_{\min}(T_{n}(|\theta|^{2}))+\frac{h^{2}}{\pi
q}\sum_{j=0}^{q-1}\frac{\alpha_{j}}{\alpha_{j}+1}$ $\displaystyle\sim$
$\displaystyle h^{2}.$

###### Proof..

Let $x_{0}\in\mathbb{R}^{n}$ be an eigenvector of $T_{n}(|\theta|^{2})$
related to $\lambda_{0}\coloneqq\lambda_{\min}(T_{n}(|\theta|^{2}))$, that is
$T_{n}(|\theta|^{2})x_{0}=\lambda_{0}x_{0}$. Assume also that
$\|x_{0}\|_{2}=1$. From Lemma 3.1 we know that

$\displaystyle M_{n,q}=$
$\displaystyle\frac{1}{q}\sum_{j=0}^{q-1}T_{n}(|\theta|^{2-\alpha_{j}})h^{\alpha_{j}}$
$\displaystyle\leqslant$ $\displaystyle
T_{n}(|\theta|^{2})+\frac{1}{q}\sum_{j=0}^{q-1}R_{n,j}$ $\displaystyle=$
$\displaystyle T_{n}(|\theta|^{2})+\frac{1}{q}R_{n}.$

Hence we obtain

$\displaystyle\lambda_{\min}(M_{n,q})=$
$\displaystyle\min_{\|x\|_{2}=1}x^{*}M_{n,q}x\leqslant x_{0}^{*}M_{n,q}x_{0}$
$\displaystyle\leqslant$ $\displaystyle
x_{0}^{*}T_{n}(|\theta|^{2})x_{0}+\frac{1}{q}x_{0}^{*}R_{n}x_{0}$
$\displaystyle=$ $\displaystyle\lambda_{0}+\frac{1}{q}\rho(R_{n})$
$\displaystyle=$
$\displaystyle\lambda_{0}+\frac{1}{q}\sum_{j=0}^{q-1}\|R_{n,j}\|_{2}$
$\displaystyle\leqslant$
$\displaystyle\lambda_{0}+\frac{\alpha_{j}}{3\pi(3-\alpha_{j})}h^{2}.$

Finally, using that $\lambda_{0}\sim h^{2}$ we finish the proof. ∎

######  Theorem 3.3.

Let $M_{n}\coloneqq hT_{n}(\hat{F}_{n})$. Then

$\lambda_{\min}(M_{n})\leqslant\lambda_{\min}(T_{n}(|\theta|^{2}))+\frac{h^{3}}{\pi}\sum_{j=0}^{n-1}\frac{jh}{jh+1}\sim
h^{2}.$

###### Proof..

It is sufficient to invoke the Lemma 3.2 with $q=n$ and $\alpha_{j}=jh$. ∎

##  4 Numerical experiments

In this section we numerically check the theoretical findings provided in
Section 3 and provide an heuristic analysis. With this aim we first calculate
the entries of the matrices $T_{n}(\hat{F}_{n})$ and $T_{n}(\eta)$ for
$\eta(\theta)\mathrel{\mathop{\ordinarycolon}}=\theta^{2}$. From (4), the
$k$-th Fourier coefficient of the function $\hat{F}_{n}(\theta)$ named
$\mathfrak{a}_{k}(\hat{F}_{n})$ can be obtained as

$\mathfrak{a}_{k}(\hat{F}_{n})=\sum_{j=0}^{n-1}h^{jh}\mathfrak{a}_{k}(g_{j}),$

where $\mathfrak{a}_{k}(g_{j})$ is the $k$-th Fourier coefficient of the
function $g_{j}(\theta)=|\theta|^{2-jh}$. Additionally,
$\mathfrak{a}_{k}(g_{j})$ can be exactly calculated with

$\displaystyle\mathfrak{a}_{k}(g_{j})=$
$\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\theta^{2-jh}\cos(k\theta)\operatorname{d\\!}\theta$
$\displaystyle=$ $\displaystyle\frac{\pi^{2-jh}}{3-jh}\
{}_{1}H_{2}\Big{(}\frac{1}{2}(3-jh);\frac{1}{2},\frac{1}{2}(5-jh);-\frac{k^{2}\pi^{2}}{4}\Big{)}$

where ${}_{p}H_{q}$ is the well-known Generalized Hypergeometric function
which can be found as an internal function, e.g., in the Mathematica software.
Thus we can exactly calculate the entries of the Toeplitz matrix
$T_{n}(\hat{F}_{n})$.

On the other hand, the Fourier coefficients of $\eta$ can be exactly
calculated by

$\mathfrak{a}_{k}(\eta)=\begin{cases}\mfrac{2}{k^{2}}\cos(\pi
k)+\mfrac{k^{2}\pi^{2}-2}{k^{3}\pi}\sin(\pi k),&k\neq 0;\\\\[5.69054pt]
\mfrac{\pi^{2}}{3},&k=0.\end{cases}$

We now present an heuristic finding which is related to the Avram–Parter
formula. Let $q\in\mathbb{N}$ be a sufficiently large number, and consider the
related symbol $\hat{F}_{q}$, that is

$\hat{F}_{q}(\theta)\coloneqq\sum_{j=0}^{q-1}\frac{|\theta|^{2-\frac{j}{q}}}{q^{\frac{1}{q}}}=\theta^{2}\frac{1-\frac{1}{q|\theta|}}{1-\big{(}\frac{1}{q|\theta|}\big{)}^{\frac{1}{q}}}.$

According to [1, Th.3.2], for every $x\in(0,1)$, the Avram–Parter formula
shows that

$\lim_{n\to\infty}\lambda_{\lceil xn\rceil}(T_{n}(\hat{F}_{q}))=Q_{q}(x),$ (5)

where $Q_{q}\colon[0,\pi]\to\mathbb{R}$ is the quantile function related to
$\hat{F}_{q}$ which in this case is given by $Q_{q}(x)\coloneqq\hat{F}_{q}(\pi
x)$. Using the estimation

$\Big{\\{}1-\Big{(}\frac{1}{\pi
jqh}\Big{)}^{\frac{1}{q}}\Big{\\}}^{-1}=\frac{q}{\log(\pi
jqh)}\Big{\\{}1+O\Big{(}\frac{1}{q}\log(\pi
jqh)\Big{)}\Big{\\}}\quad\mbox{as}\quad q\to\infty,$

a simple asymptotic calculation produces

$\displaystyle Q_{q}(jh)$ $\displaystyle=$ $\displaystyle(\pi
jh)^{2}\frac{1-\frac{1}{\pi jqh}}{1-\big{(}\frac{1}{\pi
jqh}\big{)}^{\frac{1}{q}}}$ $\displaystyle=$ $\displaystyle\frac{\pi jh(\pi
jqh-1)}{\log(\pi jqh)}+O\Big{(}(\pi jh)^{2}+\frac{\pi jh}{q}\Big{)}.$

Abusing of the limit (5), let’s take $q=n$ and $x=jh$ to obtain

$\lambda_{j}(T_{n}(\hat{F}_{n}))\approx Q_{n}(jh)=\frac{\pi j(\pi
j-1)}{\log(\pi j)}h+O\big{(}(jh)^{2}\big{)},$

which gives us the heuristic calculations

$\lambda_{\min}(T_{n}(\hat{F}_{n}))\approx\frac{\pi(\pi-1)}{\log(\pi)}h+O(h^{2})\quad\mbox{and}\quad\lambda_{\max}(T_{n}(\hat{F}_{n}))\approx\frac{\pi^{2}n}{\log(\pi
n)}+O(1).$ (6)

Table 1 shows the extreme eigenvalues of $T_{n}(\hat{F}_{n})$ for different
values of $n$. It agrees with the heuristic results in (6) and, in particular,
shows that $\mu_{2}(T_{n}(\hat{F}_{n}))=O\big{(}\frac{(\pi n)^{2}}{\log(\pi
n)}\big{)}$. Interestingly, the constant $\frac{\pi(\pi-1)}{\log(\pi)}\approx
5.8774$ agrees with the row $\lambda^{*}_{\min}$ since we expect a very slow
convergence in this cases.

$n$ | 64 | 128 | 256 | 512 | 1024 | 2048
---|---|---|---|---|---|---
$\lambda_{\min}$ | $7.8737\\!\cdot\\!10^{-2}$ | $3.9480\\!\cdot\\!10^{-2}$ | $1.9768\\!\cdot\\!10^{-2}$ | $9.8914\\!\cdot\\!10^{-3}$ | $4.9476\\!\cdot\\!10^{-3}$ | $2.4743\\!\cdot\\!10^{-3}$
$\lambda^{*}_{\min}$ | $5.0392$ | $5.0534$ | $5.0607$ | $5.0644$ | $5.0663$ | $5.0673$
$\lambda_{\max}$ | $120.9373$ | $212.8457$ | $380.1275$ | $687.1094$ | $1\,254.2460$ | $2\,307.9670$
$\lambda^{*}_{\max}$ | $1.015$ | $1.010$ | $1.006$ | $1.004$ | $1.002$ | $1.001$
$\mu_{2}$ | $1\,535.9667$ | $5\,391.2724$ | $19\,229.1665$ | $69\,465.1987$ | $253\,507.4186$ | $932\,790.7960$
$\mu^{*}_{2}$ | $0.2015$ | $0.1999$ | $0.1989$ | $0.1982$ | $0.1978$ | $0.1976$
Table 1: The extreme eigenvalues and the conditioning of $T_{n}(\hat{F}_{n})$
for different values of $n$. Here we used the notation
$\lambda^{*}_{\min}=n\lambda_{\min}$, $\lambda^{*}_{\max}=\frac{\log(\pi
n)}{\pi^{2}n}\lambda_{\max}$, and $\mu^{*}_{2}=\frac{\log(\pi n)}{(\pi
n)^{2}}\mu_{2}$.

Having in mind that matrices in the form of $T_{n}(\hat{F}_{n})$ arise within
distributed-order fractional problems which are typically ill-conditioned, we
conclude this section providing a short discussion on the conditioning of
$T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n})$, i.e., we check how the spectrum changes
when preconditioning with a Laplacian-like matrix. Table 2 shows the extreme
eigenvalues of $T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n})$ for different values of
$n$ and it suggests that

$\lambda_{\min}(T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n}))=O\Big{(}\frac{n}{\log
n}\Big{)}\quad\mbox{and}\quad\lambda_{\max}(T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n}))=O\Big{(}\frac{n^{2}}{\log(\pi
n)}\Big{)}.$

From this results we expect that
$\mu_{2}(T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n}))=O(n)$ which means that
$\eta(\theta)$ is not acting as an effective preconditioning function. Figure
1 shows the eigenvalues of $T_{n}(\hat{F}_{n})$ and of
$T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n})$ for $n=1024$ and from them, it is
immediately clear that the largest eigenvalue of $T_{n}(\hat{F}_{n})$ is
magnified after preconditioning and that the preconditioned spectrum is far
from being well-clustered.

$n$ | 64 | 128 | 256 | 512 | 1024 | 2048
---|---|---|---|---|---|---
$\lambda_{\min}$ | 15.4546 | 26.5447 | 46.4058 | 82.3378 | 147.9148 | 268.6040
$\lambda^{\dagger}_{\min}$ | 1.004 | 1.006 | 1.005 | 1.003 | 1.001 | 1.000
$\lambda_{\max}$ | 1 793.0355 | 6 479.2722 | 23 557.6771 | 86 165.4914 | 316 954.9557 | 1 175 952.6713
$\lambda^{\dagger}_{\max}$ | 2.3217 | 2.3715 | 2.4048 | 2.4268 | 2.4412 | 2.4587
$\mu_{2}$ | 116.0198 | 244.0896 | 507.6456 | 1 046.4872 | 2 142.8207 | 4 378.0162
$\mu^{\dagger}_{2}$ | 1.8128 | 1.9069 | 1.9830 | 2.0439 | 2.0926 | 2.1377
Table 2: The extreme eigenvalues and the conditioning of
$T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n})$ where $\eta(\theta)=\theta^{2}$, for
different values of $n$. Here we used the notation
$\lambda^{\dagger}_{\min}=\frac{\log n}{n}\lambda_{\min}$,
$\lambda^{\dagger}_{\max}=\frac{\log(\pi n)}{n^{2}}\lambda_{\max}$, and
$\mu^{\dagger}_{2}=\frac{1}{n}\mu_{2}$. Figure 1: The eigenvalues of
$T_{n}(\hat{F}_{n})$ and of $T_{n}^{-1}(\eta)T_{n}(\hat{F}_{n})$ where
$\eta(\theta)=\theta^{2}$, for $n=1024$. Regular scale (top) and logarithmic
scale (bottom).

##  5 Conclusions

In this note we have considered a novel type of spectral problem involving
Toeplitz structures and arising in the numerical approximation of distributed-
order fractional differential equations. The spectral study of the matrices
under consideration has been reduced to the study of the matrix

$T_{n}(g_{0})+h^{h}T_{n}(g_{1})+h^{2h}T_{n}(g_{2})+\cdots+h^{(n-1)h}T_{n}(g_{n-1}),$

where $h=\frac{1}{n}$, $g_{j}(\theta)=|\theta|^{2-jh}$, and $j=0,\ldots,n-1$.
Because the resulting generating function depends on $n$, the standard theory
cannot be applied and the analysis has been performed using new ideas. Few
selected numerical experiments have been presented and critically discussed.
Many open questions remain: we can consider for instance the use of the given
spectral information in a context of Krylov preconditioning or multigrid
design for large linear systems coming from FDEs with distributed order and
the extension of the proposed analysis in a multidimensional context, when the
spatial derivatives belong to a $d$-dimensional domain with $d\geqslant 2$.
These lines of research will be the subject of future investigations.

## Acknowledgement

The third and fourth authors are members of the INdAM research group GNCS. The
work of the third author was partly supported by the GNCS-INdAM Young
Researcher Project 2020 titled “Numerical methods for image restoration and
cultural heritage deterioration”.

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