# Design, Modelling and Control of an Amphibious Quad-Rotor for Pipeline
Inspection

Petar Durdevic
Department of Energy Technology
Aalborg University
Esbjerg, Denmark 6700
<EMAIL_ADDRESS>
& Shaobao Li
Department of Energy Technology
Aalborg University
Esbjerg, Denmark 6700
<EMAIL_ADDRESS>
& Daniel O. Arroyo
Department of Energy Technology
Aalborg University
Esbjerg, Denmark 6700
<EMAIL_ADDRESS>

###### Abstract

Regular inspections are crucial to maintaining waste-water pipelines in good
condition. The challenge is that inside a pipeline the space is narrow and may
have a complex structure. The conventional methods that use pipe robots with
heavy cables are expensive, time-consuming, and difficult to operate. In this
work, we develop an amphibious system that combines a quad-copter with a
surface vehicle, creating a hybrid unmanned aerial floating vehicle (HUAFV).
Nonlinear dynamics of the HUAFV are modeled based on the dynamic models of
both operating modes. The model is validated through experiments and
simulations. A PI controller designed and tuned on the developed model is
implemented onto a prototype platform. Our experiments demonstrate the
effectiveness of the new HUAFV’s modeling and design.

_Keywords_ Robotics $\cdot$ UAV $\cdot$ HUAFV $\cdot$ Model

##  1 Introduction

Denmark has around 80,400 km of sewerage pipes connecting 6$\%$ of its land
area Miljøstyrelsen (Online: accessed 06 January 2023); DANVA (Online:
accessed 12 December 2022) and renovating this system was estimated to cost
(150M USD) DANVA (Online: accessed 12 December 2022) in 2018. Moreover, poorly
maintained pipes can lead to high groundwater infiltration, which is a
significant problem in Denmark. An official report estimates the infiltration
to be anywhere between 150-400$\%$ of the sewage, which is around 150-200M m3
of infiltrated water. This number is expected to rise due to climate change
DANVA (Online: accessed 12 December 2022). The consequence is a high hydraulic
load on the wastewater treatment plants, leading to reduced treatment
efficiency.

Sewer pipelines require periodic inspections and early fault detection to
prevent groundwater infiltration. In the last decades, an extensive variety of
robotic systems have been developed to perform inspections due to their
increasingly lower cost and flexibility. In general, robotic systems for in-
pipe inspections can be classified as pig robot type, wheeled, caterpillar,
wall-press, walking, inchworm, or screw type robots H. R. Choi and S. G. Roh
(2007). Among all robotic solutions, cable-tethered or remotely operated
vehicles (ROV) wheeled robots are most commonly used, Nassiraei et al. (2007);
Ahrary et al. (2007). Wheeled robotic systems have limitations, in spite of
their simplicity. For instance, they can operate only in dry pipes and may not
be able to crawl up in pipeline structures with high inclinations, or in ill-
constrained pipelines that are broken. One of the problems that ROVs encounter
in the pipelines of city sewage systems is sand sediments that tend to
accumulate over time making ROV’s movement difficult. Due to this, sewage
inspections in certain municipalities in Denmark are performed by closing off
the pipeline under consideration. Additionally, the sand and debris must be
washed out with water and the water has to be drained. This operation requires
abundant amounts of water, time, and energy. To address the limitations that
crawling and submersible robots have, Vertical Takeoff and Landing (VTOL) and
Unmanned Air Vehicles (UAV) capable of flying inside the pipelines have
emerged. These commercial semi-autonomous drones such as Elios 2 have been
used to inspect London city sewage DroneFlight (Online: accessed 20 July 2021)
and Barcelona Flyability (Online: accessed 14 October 2019). However, flying
inside a pipeline poses great challenges due to its internally constrained
space. But since the drone needs to keep hovering over the water’s surface,
the flight time is reduced.

Some commercial designs such as the Splash-drone Swellpro (Online: accessed 27
September 2019); Lloyd et al. (2017), are capable of flying and landing in
water. The Aqua-copter, the QuadH20 and the Mariner Quad-copter are drones
that are designed to fall into the water, Quadh2o (Online: accessed 14 October
2019); Swellpro (Online: accessed 14 October 2019). The aforementioned
systems, however, are designed for manual piloting, and thus not suitable for
flying inside pipelines where wireless signals could be blocked.

Other autonomous solutions such Aquatic-Drones (Online: accessed 20 July 2021)
have been designed to monitor waterways, ports, and sea. This paper presents
the design and modeling of a fully autonomous HUAFV, which we call the Quad-
float. This platform is efficient for sailing in the pipelines, flying to
access difficult areas, or being deployed down a man-hole when required.

To our knowledge, no research work has addressed the problem of designing
HUAFVs capable of navigating inside confined areas, such as pipelines, nor on
the control and navigation of HUAFVs on water surfaces.

A survey of early hybrid systems is presented in Yang et al. (2015). Other
research articles have discussed other more recent types of hybrid systems
called: Hybrid Unmanned Aerial Underwater Vehicles (HUAUV), Hybrid Aerial
Underwater Vehicles (HAUV), and unmanned aerial underwater vehicles (UAUV) as
was reported in Drews et al. (2014); Neto et al. (2015); da Rosa et al.
(2018); Maia et al. (2017); Lu et al. (2019); Mercado et al. (2019); Zha et
al. (2019); Ma et al. (2018a); Zubi et al. (2018); Ma et al. (2018b).

The Loon Copter described in Zubi et al. (2018) is an autonomous quadcopter
with active buoyancy control that is capable of performing aerial, water-
surface, and subaquatic diving. A closed-loop control system is used to
perform aerial and water-surface missions and an open-loop control system is
used for diving. A similar Hybrid Unmanned Aerial Underwater Vehicle is
presented in Ma et al. (2018b) where a dynamic model of the transition from
air to underwater media process and its control system is developed and tested
by simulation.

In Lu et al. (2019) a hybrid aerial underwater vehicle (HAUV) is considered,
which uses four rotors both for underwater and air navigation. The model is
extended with time-varying added mass and damping, additionally, it is assumed
that the restoring torques are zero, i.e. $\tau_{R}=0$, since the center of
buoyancy and gravity coincide. Similar ideas have been explored in the field
of HUAUV, Drews et al. (2014); Neto et al. (2015); da Rosa et al. (2018); Maia
et al. (2017); Mercado et al. (2019); Zha et al. (2019), where the restoring
torques are not considered, and only the restoring forces due to buoyancy are
considered. The reason for this is that these works focus on submersible
vehicles. In our work, the restoring torques are crucial due to surface-
dwelling, as the movement in $x$ and $y$ is generated by changing the pitch
and roll angle respectively, and thus the restoring torque dynamic is crucial.
The restoring torques are modeled following the meta-critic restoring forces
Fossen (1994).

The Quad-float presented in this paper is a novel amphibious drone aimed at
navigating autonomously inside pipelines, capable of flying and navigating by
floating on the water’s surface. However, this work addresses only the
modeling issues for the flotation paradigm. A nonlinear dynamic model of the
platform was developed based on quad-copter rigid body kinematics, which
includes the drag forces and torques and the restoring forces and torques
associated with floating vehicles. A novelty of this Quad-float is the
addition of the effect of restoring torques in its model. A PID controller was
designed to control the position of the vehicle while it is floating. A
prototype of the Quad-float was built and used for model validation and
control implementation, using inbuilt LiDAR measurements for vehicle
localization.

##  2 The Quad-float Concept

Figure 1: Sketch of the Quad-float in a pipeline environment floating in a
layer of water, with the relevant kinematic and kinetic parameters and forces.

The Quad-float is an amphibious vehicle, that combines a quad-copter and a
flotation device. During the flight, the Quad-float has the same six degrees
of freedom (DOFs) as the quad-copter. When in water, it moves like a boat,
which is normally only modeled with 3 DOFs Fantoni et al. (1999). However, in
the relatively small Quad-float, the rotation in roll and pitch cannot be
neglected due to the nonlinear couplings between the four rotors. Therefore, a
6-DOF model will also be used to describe the motion in water.

In this paper, an important addition to the standard quad-copter model is the
inclusion of buoyancy and thus meta-critic stability and hydrodynamic forces.
During the flight, a quad-copter movement is affected by aerodynamic forces,
torques, and the gravitational force $f_{g}$. However, the aerodynamic forces
can be neglected due to the relatively low velocity during flight.

######  Remark.

A crucial design feature of the Quad-float is that the center of buoyancy
(CB), must be placed below the center of gravity (CG) to achieve meta-critic
stability, as shown in figure 2.

Figure 2: Meta-critic Stability

It is expected that when the Quad-float lands in the water, the hydrodynamic
and static forces and torques will be dominant, similarly to those affecting a
flotation vehicle. Figure 1 illustrates the Quad-float when floating on water
inside a pipeline.

##  3 Designed System

The proposed amphibious platform consists of three major parts: a frame
attached to a pontoon and the electronics equipment housed in two waterproof
compartments.

###  3.1 Frame

The Quad-float follows the X type design Zhang et al. (2014). This setup was
chosen as it gives us the largest actuation force for relatively low
individual motor rotation velocity.

###  3.2 Pontoon

The pontoon is made from a 50 mm Styrofoam sheet, which has been cut into the
shape of the quad-rotor’s frame.

###  3.3 Control and Propulsion System

The Quad-float is equipped with four Emax RS2205-2600KV brushless DC motors
(BLDC), controlled by an Airbot typhoon 32 v2 $4\times 35$ A electronic speed
controller (ESC). A 10 DOFs inertial measurement unit (IMU) is used (Bosch
BNO055). An ST VL53L0X LiDAR is used for relative $x$ and $y$ distance
measurements An NXP MK66FX1M0VMD18 Kinetis K66: 180 MHz Cortex-M4F micro-
controller unit (MCU) is used as the processing unit

##  4 Dynamic Model of the Quad-Float

Several general assumptions are made to simplify the modeling problem.

######  Assumption 1.

The vehicle is symmetrical with a symmetrical mass distribution.

######  Assumption 2.

The motor dynamics are fast in comparison to system dynamics and can be
ignored.

######  Assumption 3.

As the vehicle travels at a relatively low speed in the water, the added mass
can be neglected.

######  Assumption 4.

The water surface is considered to be flat and without waves.

######  Remark.

Assumptions 1-3 are valid in general. Regarding Assumption 4, experiments were
done in a flat water surface such that the wave disturbance induced by the
motion of the Quad-float was relatively small and thus can be neglected. This
simplification was made to reduce the complexity of the model. It is worth
mentioning that in a real environment wave disturbances may affect the Quad-
float due to water flow. To address this problem a disturbance estimator
should be designed for disturbance rejection. This work will be done in a
future paper.

###  4.1 Rigid Body Kinematics

A quad-copter can be described as a rigid body, with 6 DOFs. The generalized
coordinates of a quad-copter are
$\zeta=[x,y,z,\phi,\theta,\psi]^{T}\in\mathbb{R}^{6}$ defined in the inertial
frame, where $p=[x,y,z]^{T}$ denotes the position referred to surge, sway, and
heave, respectively and $\eta=[\phi,\theta,\psi]^{T}$ denotes the Euler angles
referred to roll, pitch, and yaw, respectively. The kinetic and potential
energies, $(K,P)$, represented in the generalized coordinates can be computed
through the Lagrangian:

$L(\zeta,\dot{\zeta})=K-P=K_{trans}+K_{rot}-P,$ (1)

where the potential energy is defined as $P=-mg_{z}p_{z}+G\zeta$, where $m$ is
the mass of the vehicle, $g_{z}$ is the gravitational acceleration and
$G\zeta$ are the restoring forces. The translational kinetic energy is defined
as $K_{trans}=\frac{1}{2}m\dot{p}^{T}\dot{p}$, and the rotational kinetic
energy is defined as $K_{rot}=\frac{1}{2}\omega_{b}^{T}I\omega_{b}$, in which
$\omega_{b}$ is the angular velocity resolved in the body fixed frame
$(*_{b})$, and $I$ is the inertia tensor which due to the symmetry of the
vehicle can be a diagonal matrix defined as:

$I=diag[I_{xx},I_{yy},I_{zz}].$ (2)

The angular velocity $\omega_{b}$ is related to the generalized coordinates
angular velocity vector $\eta$ through a kinematic relationship:

$\dot{\eta}=W^{-1}(\eta)\omega_{b}$ (3)

where $W$ is the Euler angle transformation matrix from the body frame to the
inertial frame, refer to equation 4, using the $[z,y,x]$ convention Fossen
(1994); Castillo et al. (2004):

$W=\begin{bmatrix}1&0&-s_{\theta}\\\ 0&c_{\phi}&c_{\theta}s_{\phi}\\\
0&-s_{\phi}&c_{\theta}c_{\phi}\end{bmatrix},$ (4)

where $c_{*}=cos(*)$ and $s_{*}=sin(*)$. The inertia tensor can then be
represented in the generalized coordinates, i.e. $\eta$, as:

$\mathbb{J}=\mathbb{J}(\eta)=W^{T}IW$ (5)

leading to:

$K_{rot}=\frac{1}{2}\dot{\eta}^{T}\mathbb{J}\dot{\eta},$ (6)

which gives the final Lagrangian

$\displaystyle
L=K-P=\frac{1}{2}m\dot{p}^{T}\dot{p}+\frac{1}{2}\dot{\eta}^{T}\mathbb{J}\dot{\eta}-mg_{z}p_{z}+G\zeta$
(7)

###  4.2 Euler-Lagrange

After applying the Euler-Lagrange equation we get the full dynamics with
external generalized forces:

$\frac{d}{dt}\frac{\partial L}{\partial\dot{\zeta}}-\frac{\partial
L}{\partial\zeta}=\begin{bmatrix}F_{p}\\\ \tau\end{bmatrix}$ (8)

where the translational force $F_{p}=FR\in R^{3}$ with $R$ being rotational
matrix, defined by equation (9), Choi and Ahn (2014); Goldstein et al. (2002).
Here $F=[0,0,T_{tot}]$ is referred to as the translational force with
$T_{tot}$ as the total thrust generated by the four propellers, and
$\tau=[\tau_{\phi},\tau_{\theta},\tau_{\psi}]$ the generalized moments for the
attitude control. F and $\tau$ will be discussed in section 4.6.

$R=\begin{bmatrix}c_{\psi}c_{\theta}&c_{\psi}s_{\phi}s_{\theta}-c_{\phi}s_{\psi}&s_{\phi}s_{\psi}+c_{\phi}c_{\psi}s_{\theta}\\\
c_{\theta}s_{\psi}&c_{\phi}c_{\psi}+s_{\phi}s_{\psi}s_{\theta}&c_{\phi}s_{\psi}s_{\theta}-c_{\psi}s_{\phi}\\\
-s_{\theta}&c_{\theta}s_{\phi}&c_{\phi}c_{\theta}\end{bmatrix}$ (9)

As we have no cross-terms in the kinetic energy combining $\dot{P}$ and
$\dot{\eta}$ from equation (7), the Euler-Lagrange equations can be divided
into translational and rotational dynamics, Garcia et al. (2006).

$m\ddot{P}+mge^{3}=F_{p},\hskip
14.22636pt\mathbb{J}\ddot{\eta}+\dot{\mathbb{J}}\dot{\eta}-\frac{1}{2}\frac{\partial}{\partial\eta}(\dot{\eta}^{T}\mathbb{J})=\tau.$
(10)

We can represent the Coriolis/Centripetal vector, $C(\eta,\dot{\eta})$, as is
equation (11), which includes the gyroscopic and the centrifugal terms Choi
and Ahn (2014); Castillo et al. (2005); Garcia et al. (2006); Raffo et al.
(2010).

$C(\eta,\dot{\eta})=\dot{\mathbb{J}}\dot{\eta}-\frac{1}{2}\frac{\partial}{\partial\eta}(\dot{\eta}^{T}\mathbb{J}).$
(11)

Thus, we can simplify the kinetics term from (10) to:

$\mathbb{J}\ddot{\eta}+C(\eta,\dot{\eta})=\tau.$ (12)

###  4.3 Hydrodynamic Forces and Moments - Damping

Damping is a non-conservative force and is added as an external force. For a
surface vehicle, damping is induced by a multitude of external effects, and
the total hydrodynamic damping can be formulated as follows Fossen (2011):

$D(\dot{p})\triangleq
D_{P}(\dot{p})+D_{S}(\dot{p})+D_{W}(\dot{p})+D_{M}(\dot{p})$ (13)

where $D_{P}(\dot{p})$, $D_{S}(\dot{p})$, $D_{W}(\dot{p})$ and
$D_{M}(\dot{p})$ are the radiation-induced potential damping, linear and
quadratic skin friction, wave drift damping, and vortex shedding damping,
respectively Fossen (1994). The hydrodynamic damping $D(\dot{p})$ affects the
motion of the vehicle in a highly nonlinear and coupled fashion. Practically,
it is not trivial to determine the higher-order terms and the off-diagonal
terms in $D(\dot{p})$ Fossen (1994). We, therefore, choose to simplify the
damping $D(\dot{p})$ by assuming that it is non-coupled and thus we can
simplify $D(\dot{p})$ to:

$D=diag[D_{X},D_{Y},D_{Z},D_{K},D_{M},D_{N}]$ (14)

where $[D_{X},D_{Y},D_{Z},D_{K},D_{M},D_{N}]$ are the damping forces in the 6
DOFs. Identification of the drag coefficients is, again, not trivial, and is
in this work done by a trial and error method based on experience with the
experimental setup, alternatively, the parameters can be identified from data.
$D$ can be represented as the translational and rotational parts as follows:

$\displaystyle D_{p}$ $\displaystyle=diag[D_{X},D_{Y},D_{Z}],$ (15)
$\displaystyle D_{\eta}$ $\displaystyle=diag[D_{K},D_{M},D_{N}].$

###  4.4 Hydrodynamic Forces and Moments - Restoring Forces and Moments

Quad-float is a surface vehicle and will thus be affected by the same
restoring forces as a ship in $[\phi,\theta,z]$. The vehicle is designed such
that the weight and the buoyancy are in balance:

$mg=\rho g\nabla,$ (16)

where $\nabla$ is the volume of the displaced fluid. Due to restoring forces,
the system will be open-loop stable in $[\phi,\theta,z]$, referred to as meta-
critic stability. The restoring forces are governed by the meta-centric height
$M_{i}$ the center of buoyancy (CB) and the center of gravity (CG) Fossen
(1994, 2011).

The restoring forces in $z$ and the moments in $[\phi,\theta]$ can be written
as follows Fossen (2011):

$\displaystyle Z_{restoring}$ $\displaystyle=-\rho gA_{wp}z,$ (17)
$\displaystyle K_{restoring}$ $\displaystyle=-\rho
g\nabla\overline{GM_{T}}sin\phi,$ $\displaystyle M_{restoring}$
$\displaystyle=-\rho g\nabla\overline{GM_{L}}sin\theta.$

where $A_{wp}=LH$ is the water plane area, $\rho$ is the density of the
displaced fluid, $g$ is gravity and $\overline{GM_{T}}$ and
$\overline{GM_{L}}$ are the transverse and longitudinal meta-centric height
(m), respectively (i.e. the distance between the meta-center $M_{i}$ and CG),
refer to figure 2. For surface vessels, we can use a linear approximation,
following the assumptions in Fossen (1994):

  * •

$\phi$, $\theta$, $z$ $\approx 0$.

  * •

$\int{}_{0}^{z}A_{wp}(z_{\zeta})dz_{\zeta}\approx A_{wp}(0)z$.

  * •

$sin(\theta)\approx\theta,cos(\theta)\approx 1$,
$sin(\phi)\approx\phi,cos(\phi)\approx 1$.

Then we have $g(\zeta)\approx G\zeta$ and thus $g(\zeta)$ becomes:

$G=diag[0,0,\rho gA_{wp}(0),\rho g\nabla\overline{GM_{T}},\rho
g\nabla\overline{GM_{L}},0],$ (18)

where $G$ can be represented as the translational and rotational parts as
follows:

$\displaystyle G_{p}$ $\displaystyle=diag[0,0,\rho gA_{wp}(0)],$ (19)
$\displaystyle G_{\eta}$ $\displaystyle=diag[\rho
g\nabla\overline{GM_{T}},\rho g\nabla\overline{GM_{L}},0].$

###  4.5 Vectorial Representation of the Dynamics

The model in terms of translation and rotation forces can be structured as
follows:

$\displaystyle m\ddot{p}$ $\displaystyle=F_{p}-mg_{z}-D_{p}\dot{p}-G_{p}p,$
(20) $\displaystyle\mathbb{J}\ddot{\eta}$
$\displaystyle=\tau-C(\eta,\dot{\eta})\dot{\eta}-D_{\eta}\dot{\eta}-G_{\eta}\eta.$

The addition of the restoring forces and moments $(G_{p}p,G_{\eta}\eta)$ and
the damping forces and moments $(D_{p}\dot{p},D_{\eta}\dot{\eta})$ distinguish
this model from the standard quad-rotor model. From a general control
perspective, the model has been expanded to include the un-damped natural
frequency $\omega_{n}$ and the damping ratio. In essence, this enables the
presented system to be stable, unlike the standard quad-rotor definition,
changing the control objective whilst in water.

###  4.6 Motor External Forces

The propulsion on a quad-rotor consists of four rotor-blades, with a constant
pitch, that is attached to a Brush-Less Direct Current (BLDC) motor’s axis.
Thus by alternating the angular speed of the motor $\omega_{i,i=1,\cdots,4}$,
the individual motor’s thrust $T_{i,i=1,\cdots,4}$ and torque
$\tau_{i,i=1,\cdots,4}$ are alternated.

This leads to the following generalized forces and moments with respect to
$z_{b}$ and $\eta$.

$\begin{bmatrix}T_{tot}\\\ \tau\end{bmatrix}=\begin{bmatrix}T_{tot}\\\
\tau_{\phi}\\\ \tau_{\theta}\\\
\tau_{\psi}\end{bmatrix}=\begin{bmatrix}\sum{}_{i=1}^{4}T_{i}\\\
(-T_{1}-T_{3}+T_{2}+T_{4})\cdot l_{x}/2\\\ (T_{1}+T_{2}-T_{3}-T_{4})\cdot
l_{y}/2\\\ (\tau_{m_{1}}-\tau_{m_{2}}+\tau_{m_{3}}-\tau_{m_{4}})\end{bmatrix}$
(21)

##  5 Experimental Model Validation

To validate the model, experiments were performed in a pool with a diameter of
1.5 $m$, a wall height of 0.3 $m$, and a water depth of 0.08 $m$.

Three 1.5 second impulse response experiments were performed for roll, pitch,
and yaw with the following amplitudes: 0.1 $N/m$, 0.1 $N/m$, 0.005 $N/m$
respectively. These values were heuristically determined by performing some
experiments, as they had a significant effect on the system.

Figure 3: Model validation regarding rolling, where a step input on
$\tau_{\phi}$ was applied.

###  5.1 Validation Result Discussion

The model performed well regarding the roll and pitch angles, as seen in
figures 3, 4, with a relatively good fit to the experimental data obtained. An
input in roll torque influences the pitch and yaw, and vice-versa. This can be
attributed to multiple causes: i) the complex nonlinear coupling in the system
which is not represented by the model, ii) due to the motor’s velocity and
iii) the unsymmetrical nature of the physical structure of the platform. The
translational performance of the model was evaluated using Lidar measurements,
and as can be seen in the results, a steady state is not reached due to the
size of the pool used in the experiments, which has only 25 cm of travel
distance available. Because the measurements are performed with a LiDAR, the
distance is slightly offset due to the rotation in $\psi$. The model’s fit to
the experimental data regarding an input in $\tau_{\psi}$, is relatively good,
with a small offset where the experimental data is changing faster than the
simulation result.

Figure 4: Model validation regarding pitching, where a step input on
$\tau_{\theta}$ was applied Figure 5: Model validation regarding yawing, where
a step input on $\tau_{\psi}$ was applied

##  6 Control System Design

Since the aerial flight control is no different from existing algorithms
Michael et al. (2010), in this work we focus only on the sailing control,
where the quad-float moves like a boat. When in water the Quad-float is open-
loop stable in $[z,\phi,\theta]$, thus our objective is to control
$\bar{p}=[x_{d},y_{d}]$ and $\bar{\eta}=[\psi_{d}]$ and track the references
$\bar{p}_{d}=[x_{d},y_{d}]$ and $\bar{\eta}_{d}=[\psi_{d}]$. The Quad-float is
under-actuated wrt. $[\bar{p}]$, in the same manner as the quad-copter and to
achieve the translational movement in $\bar{p}$ the force $F_{x}$ and $F_{y}$
is required which is governed by non-linear coupling

$\displaystyle F_{x}$
$\displaystyle=T_{tot}\cdot(s(\phi)s(\psi)+c(\phi)c(\psi)s(\theta)),$ (22)
$\displaystyle F_{y}$
$\displaystyle=-T_{tot}\cdot(c(\psi)s(\phi)-c(\phi)s(\psi)s(\theta)).$

i.e. to obtain $F_{x}$ and $F_{y}$ requires the system to be actuated in
$[\theta,\phi]$, which is achieved through an input in
$[\tau_{\phi},\tau_{\theta}]$ following the system model, (20). In the system
model from equation (20) a rotational movement in $\psi$ is achieved through
$[\tau_{\phi}]$. The two feedback control loops are considered separately,
i.e. the desired positions for $\bar{p}_{d}=[x_{d},y_{d}]$ and
$\bar{\eta}_{d}=[\psi_{d}]$. The control structure can be structured as
follows:

$\displaystyle\tau_{\bar{p}}$
$\displaystyle=[\tau_{\phi},\tau_{\theta}]=e_{\bar{p}}\cdot K_{{\bar{p}}},$
(23) $\displaystyle\tau_{\bar{\eta}}$
$\displaystyle=[\tau_{\psi}]=e_{\bar{\eta}}\cdot K_{{\bar{\eta}}}.$

where $e_{\bar{p}}=\bar{p}_{d}-\bar{p}$ and
$e_{\bar{\eta}}=\bar{\eta}_{d}-\bar{\eta}$ are the tracking errors regarding
$\bar{p}$ and $\bar{\eta}$ respectively, and $K_{*}$ is the feedback control
parameter. Here we refer to $K_{p}$ as the translational controller i.e. with
respect to $x$ and $y$, i.e. $K_{x}$ and $K_{y}$, and to $K_{\eta}$ as the
rotational controller with respect to $\psi$, i.e. $K_{\psi}$.

In the current work, the feedback control parameters $K_{*}$ are substituted
by a proportional-integral controller.

###  6.1 Controller Tuning

During the early experiments with the platform, we experienced that the system
would become unstable if it reached angles above 0.3 rad due to the physical
position of the CG relative to the CB and the design of the Pontoon. In order
to keep the system stable the translational controller was tuned, such that
the $\phi<0.1rad$ and $\theta<0.1rad$ for steps of 0.1 $m$. Initially, we
chose a rise-time of 2.5 $s$, but it was reduced to 5 $s$ due to issues with
the physical setup. Although the $\psi$ actuation is decoupled from the $\phi$
and the $\theta$ actuation, the physical construction of the pontoons does not
allow for aggressive $\psi$ actuation. Some experiential tests were performed
to test the platform, where steps of 0.2 rad with a rise-time of 1 $s$ were
found to be safe and used as a controller requirement. The controller
parameters were designed using Matlab’s PID tuning toolbox.

###  6.2 Control Simulation and Experimental Results

In the current work, we consider two operating conditions: i) navigate within
the pipeline in the translational plane $[x,y]$ to ‘locate the damage’, ii)
adjust the $\psi$ angle $[\psi]$ to ‘inspect the damage’. Two scenarios were
chosen to evaluate the tracking performance of the controller regarding the
position $y$ and angle $\psi$. Additionally, the same reference signals are
used both in the simulations and in the experiments and are shown in equations
(24) and (25).

$y_{d}=\begin{cases}0m&\quad\text{if }t<t_{step}\\\ 0.1m&\quad\text{if
}t>t_{step}\end{cases}$ (24) $\psi_{d}=\begin{cases}0rad&\quad\text{if
}0s<t<5s\\\ 0.1745rad&\quad\text{if }5s<t<10s\\\ 0.1745\cdot
2rad&\quad\text{if }10s<t<15s\\\ \vdots\end{cases}$ (25)

####  6.2.1 Simulation Results

The simulation results are shown in figures 6 and 7, regarding the step
response in $y$ and angle $\psi$, respectively. The controller performs
according to the design specifications in both cases.

Figure 6: Simulation Results: Reference tracking performance of the controller
regarding $y_{d}$. Figure 7: Simulation Results: Reference tracking
performance of the controller regarding $\psi_{d}$.

####  6.2.2 Experimental Results

The same controller parameters are implemented onto the hardware platform, and
the controller was evaluated following the same reference signals, specified
in equations (24) and (25). The experiments were performed in the same pool as
was used for the model validations, the experimental results are respectively
displayed in figures 8 and 9.

Table 1: Control Comparison: simulation and experimental results $y$ | $y_{sim}$ | $y_{exp}$ |  $\%$ deviation
---|---|---|---
Rise-time [0-100 $\%$] | 5.38 $s$ | 5.1313 $s$ | 4.62 $\%$
Peak-time | 8.9 $s$ | 10.1373 $s$ | 13.9 $\%$
Peak | 0.16 $m$ | 0.15 $m$ | 6.25 $\%$
Settling-time [2$\%$] | 148 $s$ |  (71.35) $s$ |  (51.79) $\%$
$\psi$ | $\psi_{sim}$ | $\psi_{exp}$ |  $\%$ deviation
Rise-time [0-100 $\%$] | 0.38 $s$ | 0.4452 $s$ | 18.42 $\%$
Peak-time | 0.68 $s$ | 0.6988 $s$ | 2.94 $\%$
Peak | 0.26 $rad$ | 0.2204 $rad$ | 15.38 $\%$
Settling-time [2$\%$] | 3.75 $s$ |  (5.06) $s$ |  (34.93) $\%$

Controller performance measures are shown in table 1, showing the rise-time,
peak-time, peak value, and settling time for the simulation and experimental
results. It is clear that the transient performance of experimental results is
similar to the simulated results, indicating that the model is a good
representation of the true system. In the simulations, the translational
controller tracks the reference, and the $\phi$ angle is kept below 0.3 $rad$.
In the experiment, the $\phi$ angle never reaches more than 0.1 $rad$, which
is good for system stability and shows that the system has the potential to be
more aggressive.

In the case of the rotational controller, the simulations show a rise time of
0.38 $s=t_{100}$ and a settling time of 3.75 $s$, which are good performance
measures considering that the system actuates 0.1745 $rad$ at each step.
Similar behavior is seen in the experimental data only with a slightly slower
response. During the experiments, it was observed that the actuation system
was very powerful and could perform much more than is seen here.

Note that the settling time of the translational controller was never reached
due to the accuracy of the measurement, which is 0.01m. Therefore the settling
time, shown in the parenthesis, is the final value of $y$ in the experiment.
In addition, in the experimental results the rotational controller is affected
by oscillatory behavior which results in it not settling at a steady state
value, and thus the settling time, in parenthesis, is the final value of
$\psi$ in the step, i.e. after 5 $s$. The rise-time, peak-time, and peak
values regarding the rotational and the translational controller are similar
in the experimental and simulation results, none deviating more than 20$\%$.

Figure 8: Experimental result: step test in $y$ direction, (the angle offset
is not adjusted for tilt) Figure 9: Experimental result: step test in $\psi$
direction

##  7 Discussion

###  7.1 Model Discussion

The comparison between the simulated model response to the experimental data
collected from the designed platform, seen in figures 3,4 and 5, indicates
that the model fits well with the experimental results. A few exceptions are
listed below, together with potential causes.

####  7.1.1 $\phi,\theta$

The experimental data has an $\approx 0.18$ $s$ and $\approx 0.19$ $s$ delay
for the two angles respectively, this is caused by the initial build-up of
thrust from the propeller, this dynamic was not included in the model. The
second observation is the oscillations in $\phi$ $\&$ $\theta$, this is caused
by the waves in the pool and increases when the vehicle moves and creates
additional waves in the pool which bounce off the pool wall. The reason is
that as the system is under-actuated, a movement in the translational
directions $x$ and $y$ is created by altering the $\theta$ and $\phi$ angles
respectively. This motion displaces the fluid below the pontoons and thus
creates a trough after which follows a crest and the frequency of the
platform’s angular rotation around $x$ and $y$ will be translated into the
wave frequency.

####  7.1.2 $\psi$

The experimental data and the simulation have the same response at
$\tau_{\psi}\neq 0$, but at $\tau_{\psi}=0$ after 5 $s$ the experimental data
is damped faster, both reach approximately the same steady-state value. This
could be caused by coupling in the damping.

####  7.1.3 $x,y$

The simulated translational position follows the experimental data well,
deviating as the distance grows. 111The distance measurement is affected by
the measurement method; where the distance to the wall is measured with the
LiDAR, and as the platform changes its $\psi$ position the distance changes.

###  7.2 Controller Discussion

The rotational controller’s performance demonstrates the capability of the
Quad-float for fast reference tracking. This is desirable for pipeline
inspections as the orientation of the Quad-float for inspections is crucial.
The experimental analysis of the controllers indicates that the disturbance
has an impact on the controller’s reference tracking as both the rotational
and the translational controller are affected by oscillations.

The oscillations produced are body-induced waves, i.e. waves created by the
movement of the body in the water pool and the reflections of the waves in the
pool’s wall. The oscillations can for example be seen in figure 8 from 80 $s$
to 90 $s$ in the $\phi$ angle. As a consequence, the system does not reach the
1 $\%$ steady state value during the experiments but instead oscillates with
$\approx$ $10\%$ from the steady state.

##  8 Conclusion

This work presents the design and testing of a hybrid unmanned aerial floating
vehicle (HUAFV), named the Quad-float. The purpose of this vehicle is to
provide a platform that can navigate pipelines flooded with water by floating
on the water’s surface A dynamic model of the Quad-float has been developed
and experimentally validated, on the constructed platform. The model was used
as a simulation platform for testing and tuning a PI control structure, which
was later implemented on the platform. Experiments were performed to test the
controller’s translational and rotational tracking performance. The
translational tracking performance was achieved according to design
specifications and is limited by the hardware design. Improvements in the
hardware design, to achieve a larger pitch and roll angle, will allow for a
larger translational force and thus a faster system response. As the concept
presented in the paper is new, there are several challenges that should still
be solved and thus there is potential for improvement. In the current work
only the control system for moving on the water surface was developed, future
work will explore designing a controller for taking off and landing on a
moving surface within a confined environment.

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# PBR, nonreality and entangled measurement

Gábor Hofer-Szabó HUN-REN Research Center for the Humanities, Budapest, email:
<EMAIL_ADDRESS>
###### Abstract

In a recent paper, Cabbolet argues that the PBR theorem is nonreal since in
the ensemble interpretation of quantum mechanics the entangled measurement
used in the derivation of the PBR theorem is nonexisting. However, Cabbolet
(1) doesn’t provide any argument for the nonexistence of entangled
measurements beyond the incompatibility of the existence of entangled
measurements and the existence of $\psi$-epistemic models which we already
know from the PBR theorem; and (2) he doesn’t show why it is more reasonable
to abandon entangled measurements instead of $\psi$-epistemic models. Hence,
the PBR theorem remains intact.

Keywords: PBR theorem, ensemble interpretation, $\psi$-epistemic/$\psi$-ontic,
entangled measurement

The question of what exactly Einstein’s ensemble interpretation of quantum
mechanics (QM) consisted of is a delicate question [2]. Nevertheless, in the
modern framework of operational theories and ontological models [6], the
Einsteinian view has become to be identified as a $\psi$-epistemic ontological
model for QM [3]. The $\psi$-epistemic models, however, are ruled out by the
PBR theorem [5]. But PBR’s argument has an implicit assumption: the existence
of entangled measurements. Thus, the PBR theorem can be read as an argument
establishing a contradiction between the existence of $\psi$-epistemic models
and the existence of entangled measurements. PBR resolves the contradiction by
abandoning $\psi$-epistemic models. This is a plausible choice since we have
no direct empirically access to ontological models while routinely apply
entangled measurements in QM.

In a recent paper [1], Cabbolet chooses the opposite way out of the
contradiction. He explicitly constructs a classical model mimicking PBR’s
bipartite ensembles. The model is "$\psi$-epistemic" in the sense that there
are individual systems in each ensemble that share two properties mimicking
PBR’s two preparations, $\ket{0}$ and $\ket{+}$. Cabbolet then argues that
there is no entangled measurement for this model. Consequently, the PBR
theorem is nonreal since in $\psi$-epistemic models entangled measurements are
nonexisting.

Cabbolet’s argument consist in the construction of a specific "entangled
measurement" on his classical ensembles and showing that this measurement will
not yield an outcome for a $1/4$ of the cases. Consequently, this measurement
cannot exist. This is a strange way of proving nonexistence. One usually
doesn’t prove $\lnot\exists x.Px$ by showing that $\lnot Pa$ for a given $a$.
But the important point is this: Cabbolet’s example is unnecessary since he
need not prove in any way that the existence of $\psi$-epistemic models rule
out the existence of entangled measurements. We know that from the PBR
theorem. The PBR theorem is just a contradiction between the existence of
$\psi$-epistemic models and entangled measurements. What he needs to show is
why it is more rational to abandon entangled measurements instead of
$\psi$-epistemic models. To this question, however, Cabbolet does not give an
answer.

Cabbolet’s strategy against PBR is a kind of "your modus ponens is my modus
tollens". PBR takes it for granted that entangled measurements do exist and
use the theorem to argue against $\psi$-epistemic models for QM. In contrast,
Cabbolet explicitly constructs a classical $\psi$-epistemic toy model and
argues that there can exist no entangled measurements. But the question of the
existence of entangled measurements is a factual question. Experimenters
routinely perform measurements in different entangled bases. On the other
hand, Cabbolet’s toy model is not a full-fledged $\psi$-epistemic model for
QM. So the decision between abandoning epistemic models or entangled
measurements is not as symmetric as it seems from the pure logic of the PBR
argument.

The main problems with Cabbolet’s paper is that (1) it doesn’t provide any
argument for the nonexistence of entangled measurements beyond the
incompatibility of the existence of entangled measurements and
$\psi$-epistemic models which we already know from the PBR theorem; and (2) it
doesn’t show why it is more reasonable to abandon entangled measurements
instead of $\psi$-epistemic models. Therefore, I think the PBR theorem and its
usual interpretation remains intact.

I also have some other problems with Cabbolet’s paper. Namely, his "entangled
measurement" (i) is not the entangled measurement of PBR and (ii) from its
nonexistence it does not follow that there are no entangled measurements in
general. To tackle these problems, I need to go deeper into Cabbolet’s paper.
But (i)-(ii) are only side problems since they all concern Cabbolet’s specific
strategy to prove that entangled measurements are nonexisting. But, as I said
above, to show this Cabbolet should provide a $\psi$-epistemic model for QM.
So my main criticism is problem (1) and (2) above.

Let us start with some terminology. An ontological (hidden variable) model
identifies the quantum state with a preparation procedure generating an
ensemble of systems each in a given ontic (hidden) states $\lambda$. The model
fixes the ontology as a set of ontic states $\Lambda=\\{\lambda\\}$ and
associates with every (pure) quantum state $\ket{\psi}$ a probability
distribution $\mu_{\psi}(\lambda)$ of ontic states. If these ontic states
uniquely determine the outcome of every measurement, then the ontological
model is called outcome-deterministic [6]. If the support of the probability
distribution $\mu_{\psi}(\lambda)$ and $\mu_{\psi^{\prime}}(\lambda)$ of two
different quantum states $\ket{\psi}$ and $\ket{\psi^{\prime}}$ can overlap,
then we call the model $\psi$-epistemic, otherwise $\psi$-ontic [3]. The PBR
theorem shows that QM does not allow for a $\psi$-epistemic ontological model.

The argument runs as follows. In Step 1, consider two quantum states
(preparation procedures) $\ket{0}$ and
$\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ and suppose ad absurdum that the
support of the associated probability distributions $\mu_{0}(\lambda)$ and
$\mu_{+}(\lambda)$ overlap. Next, consider four ensembles each consisting of
independently prepared pairs of systems corresponding to the following four
quantum states:

$\displaystyle\ket{0}\otimes\ket{0}\qquad\ket{0}\otimes\ket{+}\qquad\ket{+}\otimes\ket{0}\qquad\ket{+}\otimes\ket{+}$

Since the probability distributions $\mu_{0}(\lambda)$ and $\mu_{+}(\lambda)$
overlap, there will be a pair in each of the four ensembles with the same
ontic state $\lambda$ lying in the overlap region.

Now, suppose we perform an entangled measurement on this very pair of systems.
The outcomes of this measurement are represented in QM by the following four
entangled states:

$\displaystyle\ket{\xi_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}\ket{0}\otimes\ket{1}+\ket{1}\otimes\ket{0}\big{)}$
$\displaystyle\ket{\xi_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}\ket{0}\otimes\ket{-}+\ket{1}\otimes\ket{+}\big{)}$
$\displaystyle\ket{\xi_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}\ket{+}\otimes\ket{1}+\ket{-}\otimes\ket{0}\big{)}$
$\displaystyle\ket{\xi_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}\ket{+}\otimes\ket{-}+\ket{-}\otimes\ket{+}\big{)}$

where $\\{\ket{0},\ket{1}\\}$ is a basis in the qubit space and
$\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$. Since the entangled states
"antidistinguish" the preparations (Leifer, 2014), that is each entangled
state $\ket{\xi_{i}}$ is orthogonal to one of the above quantum states

$\displaystyle\ket{\xi_{1}}$ $\displaystyle\perp$
$\displaystyle\ket{0}\otimes\ket{0}$ $\displaystyle\ket{\xi_{2}}$
$\displaystyle\perp$ $\displaystyle\ket{0}\otimes\ket{+}$
$\displaystyle\ket{\xi_{3}}$ $\displaystyle\perp$
$\displaystyle\ket{+}\otimes\ket{0}$ $\displaystyle\ket{\xi_{4}}$
$\displaystyle\perp$ $\displaystyle\ket{+}\otimes\ket{+}$

therefore our pair of systems cannot yield result 1 when prepared in the state
$\ket{0}\otimes\ket{0}$, cannot yield result 2 when prepared in the state
$\ket{0}\otimes\ket{+}$, and so on. But the outcome of the measurement cannot
depend on which procedure the pair was previously prepared by; it can depend
only on the ontic state of the pair. (This assumption is called
"$\lambda$-sufficiency" by Spekkens.) Thus, the measurement cannot yield an
outcome. Contradiction. Hence, $\mu_{0}(\lambda)$ and $\mu_{+}(\lambda)$
cannot overlap. In Step 2, PBR derives a similar contradiction using multiple
tensor products for any two distinct quantum states, hence ruling out a
$\psi$-epistemic interpretation of QM in general.

Now, I turn to Cabbolet’s paper.

Cabbolet mimics Step 1 of PBR’s bipartite ensembles by two large batches of
bolts. Each bolt can be of type M12 or type M10 and can be either sellable or
unsellable. We perform measurements on the bolts. Let $A$ denote the
measurement of the type of the bolts with outcomes $0$ (type M12) and $1$
(type M10) and let $B$ (in Cabbolet: measurement $P$) denote the measurement
of the sellability of the bolts with outcomes $+$ (sellable) and $-$
(unsellable). I assume that Cabbolet’s example with the bolts should not be
read literally. In this case, namely, one would not be surprised that there is
no entangled measurement which can be performed on such classical systems as
bolts. This reading is reinforced also by the fact that in the description of
the measurement statistics Cabbolet applies the formalism of QM. Thus, I will
take Cabbolet’s model with the bolts simply metaphorically as an illustration
of a part of a $\psi$-epistemic ontological model of QM and I take the
measurements $A$ and $B$ simply as measurements on a quantum system,
disregarding the specific details of his example.

Consider then two ensembles of quantum systems, one prepared in the state
$\ket{0}$, the other prepared in the state $\ket{+}$. Suppose we perform two
measurement $A$ and $B$ on the systems which are represented in QM by the
Pauli matrices $\sigma_{z}$ and $\sigma_{x}$. If we perform measurement $B$ on
the first ensemble, we obtain the outcome $+$ in half of the cases and $-$ in
the other half. Similarly, if we perform a measurement $A$ on the second
ensemble, we obtain the outcome $0$ in half of the cases and $1$ in the other
half. Denote by $\Lambda_{0+}$ the set of ontic states for which measurement
$A$ would definitely yield outcome $0$ and measurement $B$ would definitely
yield outcome $+$.111Cabbolet introduces the quantum state $\ket{0+}$ to
represent the ensemble $\Lambda_{0+}$. But such an ensemble cannot be prepared
quantum mechanically; so it is better to refer to it at the level of the
ontological model as $\Lambda_{0+}$. Similarly define $\Lambda_{0-}$,
$\Lambda_{1+}$ and $\Lambda_{1-}$ for the other outcome pairs. (Note that $A$
and $B$ need not be performed simultaneously for $\Lambda_{0+}$ to exist.)
Then, in our first ensemble the half of the ontic states is in $\Lambda_{0+}$
and the other half is in $\Lambda_{0-}$. In our second ensemble the half of
the ontic states is in $\Lambda_{0+}$ and the other half is in $\Lambda_{1+}$.

Now, consider again the four ensembles each consisting of independently
prepared pairs of systems corresponding to the above four quantum states:

$\displaystyle\ket{0}\otimes\ket{0}\qquad\ket{0}\otimes\ket{+}\qquad\ket{+}\otimes\ket{0}\qquad\ket{+}\otimes\ket{+}$

It follows that in each of the four ensembles the proportion of those pairs
which are in an ontic state
$(\lambda,\lambda^{\prime})\in\Lambda_{0+}\times\Lambda_{0+}$ will be $1/4$.

Cabbolet claims that for these pairs the entangled measurement represented by
the entangled states $\\{\ket{\xi_{i}}\\}$ will not give an outcome. This is
true but not new; it follows simply from the PBR theorem since $\Lambda_{0+}$
is in the intersection of the support of $\mu_{0}(\lambda)$ and
$\mu_{+}(\lambda)$. At this point, however, Cabbolet and PBR, as said before,
come to different conclusions. PBR argues that because the entangled
measurement would not give an outcome, therefore there is no $\psi$-epistemic
ontological model for QM. In contrast, Cabbolet argues that because the
entangled measurement would not give an outcome, therefore there are no such
entangled measurements. Both parties agree that $\psi$-epistemic ontological
models and entangled measurement represented by the states
$\\{\ket{\xi_{i}}\\}$ are incompatible but they differ in which assumption to
give up.

To decide the dispute in favour of Cabbolet, one would like to see a full-
fledged $\psi$-epistemic model of QM with the ontic regions $\Lambda_{0+}$,
$\Lambda_{0-}$, $\Lambda_{1+}$ and $\Lambda_{1-}$ embedded into the model.
However, we don’t have such a model (Leifer, 2014, 72) and neither Cabbolet
offers one. On the other hand, experimental techniques are rapidly evolving
today and entangled measurements are applied in experimental physics on a
daily basis. We stress here that the sole criterion of whether a measurement
qualifies as an entangled measurement represented by the states
$\\{\ket{\xi_{i}}\\}$ is that it should yield the outcomes in every quantum
state $\ket{\psi}$ with probability $|\\!\braket{\psi,\xi_{i}}\\!|^{2}$. The
existence of entangled measurement doesn’t in any sense depend on whether
certain ontological models of QM are possible or not. All this tips the
dispute in favor of PBR’s interpretation of the PBR theorem: one should give
up $\psi$-epistemicity and retain entangled measurements.

Now, I turn to the smaller problems (i) and (ii).

First, let’s see how Cabbolet realizes the entangled measurement
$\\{\ket{\xi_{i}}\\}$. He considers the following four measurement
instructions:

M1: Perform measurement $A$ on both ensembles;

M2: Perform measurement $A$ on the first ensembles and measurement $B$ on the
second;

M3: Perform measurement $B$ on the first ensembles and measurement $A$ on the
second;

M4: Perform measurement $B$ on both ensembles.

Cabbolet then associates each measurement Mi with the entangled state
$\ket{\xi_{i}}$. But this is no good. Each measurement Mi has four outcomes.
Which of the four outcomes of the four measurements should be associated with
which entangled state? Here is one way to make Cabbolet’s idea precise:

M${}_{1}^{\prime}$: Perform measurement $A$ on both ensembles and write $+1$
if the outcomes are opposite and write $-1$ if they are the same;

M${}_{2}^{\prime}$: Perform measurement $A$ on the first ensembles and
measurement $B$ on the second and write $+1$ if the outcomes are $(0,-)$ or
$(1,+)$ and write $-1$ if they are $(0,+)$ or $(1,-)$;

M${}_{3}^{\prime}$: Perform measurement $B$ on the first ensembles and
measurement $A$ on the second and write $+1$ if the outcomes are $(+,1)$ or
$(-,0)$ and write $-1$ if they are $(+,0)$ or $(-,1)$;

M${}_{4}^{\prime}$: Perform measurement $B$ on both ensembles and write $+1$
if the outcomes are opposite and write $-1$ if they are the same.

So in fact Cabbolet associates not the measurement M${}_{i}^{\prime}$ but the
outcome $+1$ of M${}_{i}^{\prime}$ with the entangled state $\ket{\xi_{i}}$.
Then he shows that for the pairs in ontic state
$(\lambda,\lambda^{\prime})\in\Lambda_{0+}\times\Lambda_{0+}$ none of the four
M${}_{i}^{\prime}$ will provide outcome $+1$. Thus, the entangled measurement
$\\{\ket{\xi_{i}}\\}$ has no outcome.

But why does the outcome $+1$ of M${}_{i}^{\prime}$ correspond to the
entangled state $\ket{\xi_{i}}$? Cabolet’s answer is that because if we
prepare the two batches of bolts not independently but putting the pairs of
bolts in pairs of envelopes in a certain way specified in the paper, then all
the measurements M${}_{i}^{\prime}$ will give outcome $+1$. I am unable to see
how this fact justifies that the outcome $+1$ of the measurement
M${}_{i}^{\prime}$ corresponds to the entangled state $\ket{\xi_{i}}$. In the
formalism of QM, the outcome $+1$ of the measurement M${}_{1}^{\prime}$
corresponds to coarse-graining over those pairs of outcomes of a measurement
with basis vectors

$\displaystyle\big{\\{}\ket{0}\otimes\ket{0},\ket{0}\otimes\ket{1},\ket{1}\otimes\ket{0},\ket{1}\otimes\ket{1}\big{\\}}$

for which the product is $+1$; while $\ket{\xi_{i}}$ is the entangled basis
vector

$\displaystyle\frac{1}{\sqrt{2}}\big{(}\ket{0}\otimes\ket{1}+\ket{1}\otimes\ket{0}\big{)}$

They are definitely not the same.

More generally, entangled measurements, like the Bell state measurement for
example, are typically complicated global measurements which cannot be reduced
to local measurements of the two parties. Measuring a photon pair in a Bell
basis requires recombining the two beams by a beam splitter and not just
performing polarization measurements on the two distant photons [7]. Anyway,
we can easily check whether M${}_{1}^{\prime}$-M${}_{4}^{\prime}$ is an
entangled measurement. If it is, then the outcome statistics in every quantum
state needs to be calculated by the Born rule, that is taking the square of
the product of the four entangled state $\ket{\xi_{i}}$ with the quantum
state. Pick, for example, the quantum state (preparation)
$\ket{0}\otimes\ket{1}$. What is the probability to obtain the outcome $+1$
for the measurements M${}_{1}^{\prime}$ in this state? The probability is $1$
since measuring $A$ on both sides, we always get opposite outcomes. But this
is not the same as $|\\!\bra{\xi_{1}}(\ket{0}\otimes\ket{1})\\!|^{2}$ which is
$1/2$. So the $+1$ outcomes of M${}_{1}^{\prime}$-M${}_{4}^{\prime}$ cannot
correspond to the states $\\{\ket{\xi_{i}}\\}$ of an entangled measurement.

But regardless of M${}_{1}^{\prime}$-M${}_{4}^{\prime}$, can we not show
directly that if we perform a "true" entangled measurement
$\\{\ket{\xi_{i}}\\}$ on a pair in state
$(\lambda,\lambda^{\prime})\in\Lambda_{0+}\times\Lambda_{0+}$, then we will
get no outcome? Yes, we can. This fact follows directly from the PBR theorem.
But again, to show that the entangled measurement $\\{\ket{\xi_{i}}\\}$ does
not yield an outcome and hence it is nonexisting, we should first know that QM
has a $\psi$-epistemic model with some
$(\lambda,\lambda^{\prime})\in\Lambda_{0+}\times\Lambda_{0+}$. But we don’t
know that.

In sum, Cabbolet devises an entangled measurement and shows that this
measurement does not yield an outcome for a $1/4$ of the cases, therefore it
is nonexistent. Next, he concludes that there exist no entangled measurements
in general. But Cabbolet’s "entangled measurement" (i) is not the entangled
measurement $\\{\ket{\xi_{i}}\\}$ and (ii) from its nonexistence it does not
follow that there are no entangled measurements in general. To show that there
are no entangled measurements one should first give a full-fledged
$\psi$-epistemic ontological models for QM and use the PBR theorem in a
reversed way. Until we have such model, it is more reasonable to draw the
standard conclusion from the incompatibility of the PBR, namely that it rules
out $\psi$-epistemic ontological models for QM.

## References

  * [1] Cabbolet, M. (2023). On the Nonreality of the PBR Theorem, Found. Phys., 53: 64.
  * [2] Fine, A. (1996). The Shaky Game, Einstein, Realism and the Quantum Theory, Chicago: University of Chicago Press.
  * [3] Harrigan, N. Spekkens, R. (2010). Einstein, Incompleteness, and the Epistmic View of Quantum States, Found. Phys., 40, 125–157
  * [4] Leifer, M. (2014). Is the Quantum State Real? An Extended Review of $\Psi$-ontology Theorems, Quanta 3.1, 67–155.
  * [5] Pusey M, Barrett J, Rudolph T. (2012). On the reality of the quantum state, Nature Physics 8, 475–478.
  * [6] Spekkens, R. W. (2005). Contextuality for preparations transformations and unsharp measurements, Phys. Rev. A, 71, 052108.
  * [7] Weihs, G., and Zeilinger, A. (2001). Photon statistics at beam splitters: an essential tool in quantum information and teleportation, in: Jan Perina (ed), Coherence and Statistics of Photons and Atoms, (John Wiley and Son, Inc, New York) 262-288.


# On Nörlund summability of Taylor series in
weighted Dirichlet spaces

A. Bërdëllima∗ and N. L. Braha†

###### Abstract.

In this note we show that the Taylor series of a function in a weighted
Dirichlet space is (generalized) Nörlund summable, provided that the sequence
determining the Nörlund operator is non-decreasing and has finite upper growth
rate. In particular the Taylor series is Nörlund summable for all
$\alpha>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The
inequality $\alpha>1/2$ is sharp. On the other hand if the Taylor series is
Nörlund summable and the partial sums of the determining sequence enjoy a
certain growth condition then the determining sequence has finite lower growth
rate. An analogue result is derived for a non-increasing sequence that is
uniformly bounded away from zero.

∗<EMAIL_ADDRESS>$\dagger$<EMAIL_ADDRESS>
##  1\. Introduction

A topic of interest in complex analysis is the approximation of a holomorphic
function by its Taylor series. It is known that for certain spaces of
holomorphic functions defined on the unit disk $\mathbb{D}$, the Taylor series
of a function $f$ converges in norm of the space to $f$ itself. Such results
hold in particular for the (general) Hardy space $H^{p}$ for any
$1<p<+\infty$, or the Dirichlet space $\mathcal{D}$, the space of all
holomorphic functions $f$ defined on $\mathbb{D}$ for which the Dirichlet
integral

(1) $\mathcal{D}(f)=\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\,dA(z)$

is finite, where $dA$ is the normalized area measure on $\mathbb{D}$. On the
other hand there are spaces of holomorphic functions where approximation in
norm by Taylor series fails. Such examples include for instance the Hardy
space $H^{1}$ or the disk algebra $A(\mathbb{D})$ as witnessed by the
classical example of du Bois-Raymond [4]. However by a result of [5, 6, Fejér]
if instead the Cesàro sum is considered then Cesàro sums of Taylor series of a
function converge in norm to the function itself in both aforementioned
spaces. Later [12, Riesz] refined the result of Fejér for the generalized
Cesáro sums for functions in $A(\mathbb{D})$. Recently as an application of
their work about Hadamard multipliers on weighted Dirichlet spaces [8,
Mashreghi et al.], Mashreghi–Ransford consider generalized Cesàro summability
of Taylor series for functions in a weighted Dirichlet space [7]. A weighted
Dirichlet space $\mathcal{D}_{\omega}$ is similar to the Dirichlet space
$\mathcal{D}$, except for (1) is replaced by

(2)
$\mathcal{D}_{\omega}(f)=\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\,\omega(z)\,dA(z),$

where $\omega$ is a superharmonic function on $\mathbb{D}$. These spaces were
studied by [1, Aleman], which are a generalization of the case where $\omega$
is a harmonic function. The latter were originally introduced by [10, Richter]
and later studied by [11, Richter–Sundberg]. In view of these recent
developments we consider a different summability method, that of (generalized)
Nörlund sums in $\mathcal{D}_{\omega}$. We show convergence results for the
Nörlund method and we also provide rates of convergence. Differently from
Cesàro method, the growth rate of the sequence that determines Nörlund’s
operator plays a pivotal role. In particular we prove that when this sequence
is non-decreasing and enjoys finite upper growth rate, then the Taylor series
of a holomorphic function in $\mathcal{D}_{\omega}$ is Nörlund summable in the
norm of $\mathcal{D}_{\omega}$ (Theorem 3.1(i)) and that this rate of
convergence is of the order $O(n^{-1/2})$ (Theorem 3.2). Moreover under a
certain growth condition for partial sums of the determining sequence of
Nörlund operator, we get that this sequence has finite lower growth rate,
whenever Taylor series are Nörlund summable (Theorem 3.1(ii)). Like in the
case of generalized Cesàro method, the lower bound $\alpha>1/2$, the parameter
in the generalized Nörlund method, is sharp (Theorem 3.1(ii)). An analogue
result (Theorem 3.3) is derived for non-increasing sequences that are
uniformly bounded away from zero.

##  2\. Preliminary results

###  2.1. Nörlund sums

Let $f(z)$ be a holomorphic function of the complex variable $z$ and denote by
$f(z)=\displaystyle\sum_{k=0}^{\infty}a_{k}z^{k}$ its formal power series
expansion. Let

(3) $s_{k}[f](z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{k}z^{k},\quad
k\in\mathbb{N}_{0}$

be the $k$-th degree Taylor polynomial of $f$. Given a sequence of non-
negative numbers $(p_{n})_{n\in\mathbb{N}_{0}}$ such that
$P_{n}=p_{0}+p_{1}+\cdots+p_{n}>0$ for all $n\in\mathbb{N}_{0}$, the Nörlund
operator $(N,(p_{n})_{n\in\mathbb{N}})$ acts on a sequence
$x=(x_{n})_{n\in\mathbb{N}_{0}}$ by the formula

(4) $(Nx)_{n}:=\frac{1}{P_{n}}\sum_{k=0}^{n}p_{n-k}x_{k},\quad
n\in\mathbb{N}_{0}.$

For more on Nörlund method and other summability methods we refer to [2,
Boos]. We define Nörlund sums of a holomorphic function $f$ as follows

(5) $N_{n}[f](z):=\frac{1}{P_{n}}\sum_{k=0}^{n}p_{n-k}s_{k}[f](z),\quad
n\in\mathbb{N}_{0}.$

######  Lemma 2.1.

The following holds

(6) $N_{n}[f](z)=\frac{1}{P_{n}}\sum_{k=0}^{n}P_{n-k}a_{k}z^{k},\quad
n\in\mathbb{N}_{0}.$

###### Proof.

From (5) we have

$\displaystyle N_{n}[f](z)=\frac{1}{P_{n}}\sum_{k=0}^{n}$ $\displaystyle
p_{n-k}s_{k}[f](z)=\frac{1}{P_{n}}\sum_{k=0}^{n}p_{n-k}\sum_{j=0}^{k}a_{j}z^{j}$
$\displaystyle=\frac{1}{P_{n}}((P_{n}a_{0})+(P_{n-1}a_{1}z)+\cdots+(P_{0}a_{n}z^{n}))=\frac{1}{P_{n}}\sum_{k=0}^{n}P_{n-k}a_{k}z^{k}.$

∎

In view of Lemma 2.1 we define

(7)
$N^{\alpha}_{n}[f](z):=\frac{1}{P_{n}^{\alpha}}\sum_{k=0}^{n}P^{\alpha}_{n-k}a_{k}z^{k},\quad
f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$

to be the generalized Nörlund sum $N_{n}^{\alpha}[f]$ with parameter
$\alpha>0$ of a function $f$. It is evident that $N^{\alpha}_{n}$ is a linear
operator acting on the space of all holomorphic functions and its image is a
certain subset of polynomials of degree $\leq n$ defined on the complex plane
$\mathbb{C}$.

###  2.2. Superharmonic functions and weighted Dirichlet spaces

Let $U\subseteq\mathbb{C}$ be an open set. A function
$\omega:U\to[-\infty,+\infty)$ is subharmonic on $U$ if it is upper
semicontinuous and given $z\in U$ there exists $R>0$ such that

(8)
$\omega(z)\leq\frac{1}{2\pi}\int^{2\pi}_{0}\omega(z+re^{i\theta})\,d\theta\quad(0\leq
r<R).$

A function $\omega:U\to(-\infty,+\infty]$ is superharmonic on $U$ if $-\omega$
is subharmonic on $U$. In particular a harmonic function is both sub- and
superharmonic, see e.g. [9, Ransford §2.2]. Denote by
$\mathbb{D}=\\{z\in\mathbb{C}\,:\,|z|<1\\}$ the unit disk in $\mathbb{C}$, let
$\omega>0$ be superharmonic, and $f$ holomorphic on $\mathbb{D}$. Define

(9)
$\mathcal{D}_{\omega}(f):=\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\omega(z)\,dA(z),$

where $dA$ denotes the normalized area measure on $\mathbb{D}$. The weighted
Dirichlet space $\mathcal{D}_{\omega}$ is the set of all holomorphic functions
$f$ on $\mathbb{D}$ such that $\mathcal{D}_{\omega}(f)<+\infty$. When equipped
with the inner product

(10) $\langle
f,g\rangle_{\mathcal{D}_{\omega}}:=f(0)\,\overline{g(0)}+\int_{\mathbb{D}}f^{\prime}(z)\overline{g^{\prime}(z)}\,\omega(z)\,dA(z),$

$\mathcal{D}_{\omega}$ becomes a Hilbert space. This induces the norm

(11)
$\|f\|^{2}_{\mathcal{D}_{\omega}}:=|f(0)|^{2}+\mathcal{D}_{\omega}(f),\quad
f\in\mathcal{D}_{\omega}.$

If $\omega$ is a positive superharmonic function, then there exists a unique
positive finite Borel measure $\mu_{\omega}$ on $\overline{\mathbb{D}}$, such
that for all $z\in\mathbb{D}$ it holds

(12)
$\omega(z)=\int_{\mathbb{D}}\log\Big{|}\frac{1-\overline{\zeta}\,z}{\zeta-z}\Big{|}\,\frac{2}{1-|\zeta|^{2}}\,d\mu_{\omega}(\zeta)+\int_{\partial{\mathbb{D}}}\frac{1-|z|^{2}}{|\zeta-z|^{2}}\,d\mu_{\omega}(\zeta).$

When $\mu_{\omega}=\delta_{\zeta}$ (the Dirac mass at $\zeta$), we write
$\mathcal{D}_{\zeta}$ for $\mathcal{D}_{\omega}$, see e.g. [9, Theorem 4.5.1].
Note that one of the integrals above vanishes depending on
$\zeta\in\mathbb{D}$ or $\zeta\in\partial\mathbb{D}$. From the properties of
Dirac delta function $\delta_{\zeta}$ for a given holomorphic function $f$ we
then have

(13)
$\mathcal{D}_{\zeta}(f)=\left\\{\begin{array}[]{ccc}\displaystyle\int_{\mathbb{D}}\log\Big{|}\frac{1-\overline{\zeta}\,z}{\zeta-z}\Big{|}\,\frac{2}{1-|\zeta|^{2}}\,|f^{\prime}(z)|^{2}dA(z),&\zeta\in\mathbb{D}\\\\[10.00002pt]
\displaystyle\int_{\mathbb{D}}\frac{1-|z|^{2}}{|\zeta-z|^{2}}\,|f^{\prime}(z)|^{2}\,dA(z),&\zeta\in\partial\mathbb{D}.\end{array}\right.$

By virtue of Fubini’s Theorem one can recover $\mathcal{D}_{\omega}(f)$ from
$\mathcal{D}_{\zeta}(f)$ from the integral formula
$\mathcal{D}_{\omega}(f)=\displaystyle\int_{\overline{\mathbb{D}}}\mathcal{D}_{\zeta}(f)\,d\mu_{\omega}(\zeta)$.
Often $\mathcal{D}_{\zeta}$ is called a local Dirichlet space. If $H^{2}$
denotes the Hardy space, the space of holomorphic functions $f$ on
$\mathbb{D}$ such that
$\|f\|_{H^{2}}^{2}=\displaystyle\sum_{k=0}^{\infty}|a_{k}|^{2}<\infty$, then
we have the characterization due to [1, Aleman §4.1], see e.g. [8, Theorem
2.1]:

######  Lemma 2.2.

Let $\zeta\in\overline{\mathbb{D}}$, then $f\in\mathcal{D}_{\zeta}$ if and
only if there exists $g\in H^{2}$ and $a\in\mathbb{C}$ such that
$f(z)=a+(z-\zeta)\,g(z)$ for all $z\in\mathbb{D}$, and in this case
$D_{\zeta}(f)=\|g\|_{H^{2}}$.

Given a holomorphic function $f\in\mathcal{D}_{\omega}$ we say that the Taylor
series of $f$ is (Nörlund) $(N,\alpha)$–summable, if
$\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\to 0$ as $n\to+\infty$.

###  2.3. Hadamard product

Let $f(z)=\displaystyle\sum_{k=0}^{\infty}a_{k}z^{k}$ and
$g(z)=\displaystyle\sum_{k=0}^{\infty}b_{k}z^{k}$ be two formal power series.
The Hadamard product of $f$ and $g$ is defined as

(14) $(f\ast g)(z):=\sum_{k=0}^{+\infty}a_{k}b_{k}z^{k}.$

It is clear that Hadamard product is commutative and associative operation.
Moreover if $f$ or $g$ is a polynomial then $(f\ast g)$ is a polynomial too,
and if both $f$ and $g$ are holomorphic then so is $f\ast g$. If $h$ is a
power series with the property that $h\ast f\in\mathcal{D}_{\omega}$ whenever
$f\in\mathcal{D}_{\omega}$, then $h$ is a Hadamard multiplier.

######  Proposition 2.1.

$h$ is a Hadamard multiplier for any positive superharmonic weight $\omega$,
whenever $h$ is a polynomial.

###### Proof.

Let $h=b_{0}+b_{1}z+\cdots+b_{n}z^{n}$ be a polynomial then so is $h\ast f$
and in particular $h\ast f$ is holomorphic on $\mathbb{D}$. It only remains to
show that $\mathcal{D}_{\omega}(h\ast f)<+\infty$. Note that by definition of
$\mathcal{D}_{\omega}$ we have

$\displaystyle\mathcal{D}_{\omega}(h\ast f)$
$\displaystyle=\int_{\mathbb{D}}|(h\ast
f)^{\prime}(z)|^{2}\,\omega(z)\,dA(z)=\int_{\mathbb{D}}|\sum_{k=1}^{n}a_{k}b_{k}z^{k}|^{2}\,\omega(z)\,dA(z)$
$\displaystyle\leq\int_{\mathbb{D}}(\sum_{k=1}^{n}|a_{k}||b_{k}||z|^{k})^{2}\omega(z)\,dA(z)\leq
C_{n}\int_{\mathbb{D}}\omega(z)\,dA(z)$

where $C_{n}:=(\displaystyle\sum_{k=1}^{n}|a_{k}||b_{k}|)^{2}$ is a certain
positive constant depending on $n\in\mathbb{N}_{0}$. From the general theory
of harmonic functions, e.g. see [9, Theorem 2.5.1 §2.5], if $\omega$ is
positive and superharmonic on $\mathbb{D}$ then $\omega$ is integrable on
$\mathbb{D}$. Consequently $\mathcal{D}_{\omega}(h\ast f)<+\infty$. ∎

Given $n\in\mathbb{N}$ define

(15)
$h^{\alpha}_{n}(z)=\frac{1}{P_{n}^{\alpha}}\sum_{k=0}^{n}P_{n-k}^{\alpha}z^{k}.$

In view of Hadamard product we can then write
$N^{\alpha}_{n}[f](z)=(h^{\alpha}_{n}\ast f)(z)$ for every
$n\in\mathbb{N}_{0}$.

######  Remark 2.1.

Since $h^{\alpha}_{n}$ in (15) is a polynomial for every $n\in\mathbb{N}_{0}$
then $h^{\alpha}_{n}$ is a Hadamard multiplier for any positive superharmonic
weight $\omega$.

Define the matrix $T^{\alpha}_{n}$ as

(16)
$T^{\alpha}_{n}:=\begin{pmatrix}c_{1}&(c_{2}-c_{1})&(c_{3}-c_{2})&\cdots\\\
0&c_{2}&(c_{3}-c_{2})&\cdots\\\ 0&0&c_{3}&\cdots\\\
\cdots&\cdots&\cdots&\cdots\\\
\end{pmatrix},\quad\text{where}\;c_{k}:=\left\\{\begin{array}[]{ll}\displaystyle\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}&1\leq
k\leq n\\\\[10.00002pt] 0&k>n.\end{array}\right.$

Since $h^{\alpha}_{n}$ is a polynomial for each $n\in\mathbb{N}_{0}$, then by
[7, Theorem 2.1] we have:

######  Corollary 2.1.

For every superharmonic weight $\omega$ it holds that
$\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}\leq\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}$,
with equality if $\omega(z)=(1-|z|^{2})/|1-z|^{2}$.

As a consequence of [7, Theorem 2.2 (i)] we have the lemma:

######  Lemma 2.3.

For each $n\in\mathbb{N}$ it holds that

(17)
$\|T_{n}^{\alpha}\|^{2}_{\ell^{2}\to\ell^{2}}\leq(n+1)\sum_{k=1}^{n}\Big{|}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-\Big{(}\frac{P_{n-k-1}}{P_{n}}\Big{)}^{\alpha}\Big{|}^{2}.$

###  2.4. Growth rate

The upper growth rate $\rho$ of the sequence $(p_{n})_{n\in\mathbb{N}_{0}}$ is
defined as

(18)
$\rho:=\limsup_{n\to+\infty}\rho_{n},\quad\rho_{n}:=\frac{n\,p_{n}}{P_{n}}.$

If $\rho<+\infty$ we say that the sequence $(p_{n})_{n\in\mathbb{N}_{0}}$ has
finite upper growth rate. Similarly one can define the lower growth rate if
limsup is replaced by liminf. Evidently a sequence with finite upper growth
rate has finite lower growth rate. For instance the sequence $p_{n}=n^{k}$ for
any $k\in\mathbb{N}$ has finite growth rate. An example of a sequence with
unbounded growth rate is $p_{n}=r^{n}$ for any $r>1$ or the sequence
$p_{n}=\ln(n+1)$ for $n\in\mathbb{N}$. Note that the quantity $\rho_{n}$
appears in [3] and plays a crucial role in determining the $\ell_{p}$ norm of
the Nörlund operator. The following lemma is necessary for our main results;
it ties together the importance of growth rate of the determining sequence
$(p_{n})_{n\in\mathbb{N}_{0}}$ with convergence of Nörlund method.

######  Lemma 2.4.

Let $(p_{n})_{n\in\mathbb{N}_{0}}$ have finite upper growth rate, then for any
polynomial $f$ it holds that
$\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\to 0$ as $n\to+\infty$.

###### Proof.

Let $f(z)=a_{0}+a_{1}z+\cdots+a_{m}z^{m}$ for some $m\in\mathbb{N}_{0}$, then

$N^{\alpha}_{n}[f](z)=\frac{1}{P_{n}^{\alpha}}\sum_{k=0}^{m}P_{n-k}^{\alpha}a_{k}z^{k}$

implies

$\displaystyle\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}$
$\displaystyle=\int_{\mathbb{D}}|N_{n}^{\alpha}[f]^{\prime}(z)-f^{\prime}(z)|^{2}\,\omega(z)\,dA(z)$
$\displaystyle=\int_{\mathbb{D}}|\sum_{k=1}^{m}\Big{(}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-1\Big{)}ka_{k}z^{k-1}|^{2}\,\omega(z)\,dA(z)\leq
C^{\alpha}_{m,n}\int_{\mathbb{D}}\omega(z)\,dA(z)$

where

(19)
$C^{\alpha}_{m,n}=\Big{(}\sum_{k=1}^{m}\Big{(}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-1\Big{)}k|a_{k}|\Big{)}^{2}.$

We claim that $\lim_{n\to+\infty}P_{n-k}/P_{n}=1$ for every
$k\in\\{1,2,\cdots,m\\}$. We proceed by strong induction. For $k=1$ we have

$\limsup_{n\to+\infty}\frac{P_{n-1}}{P_{n}}=1-\liminf_{n\to+\infty}\frac{p_{n}}{P_{n}}=1-\lim_{n\to+\infty}\frac{1}{n}\cdot\liminf_{n\to+\infty}\frac{np_{n}}{P_{n}}=1-0\cdot\liminf_{n\to+\infty}\frac{np_{n}}{P_{n}}=1.$

Note that finite upper growth rate implies finite lower growth rate and
therefore the last equality is justified. Similarly

$\liminf_{n\to+\infty}\frac{P_{n-1}}{P_{n}}=1-\limsup_{n\to+\infty}\frac{p_{n}}{P_{n}}=1-\lim_{n\to+\infty}\frac{1}{n}\cdot\limsup_{n\to+\infty}\frac{np_{n}}{P_{n}}=1-0\cdot\limsup_{n\to+\infty}\frac{np_{n}}{P_{n}}=1.$

Consequently $\lim_{n\to+\infty}P_{n-1}/P_{n}=1$. Now let the claim holds for
all $k\in\\{2,\cdots,m-1\\}$ and set $k=m$. Then we obtain

$\displaystyle\limsup_{n\to+\infty}\frac{P_{n-m}}{P_{n}}$
$\displaystyle=\limsup_{n\to+\infty}\frac{P_{n-m}}{P_{n-m+1}}\frac{P_{n-m+1}}{P_{n}}=\limsup_{n\to+\infty}\frac{P_{n-m}}{P_{n-m+1}}\cdot\lim_{n\to+\infty}\frac{P_{n-m+1}}{P_{n}}=\limsup_{n\to+\infty}\frac{P_{n-m}}{P_{n-m+1}}$
$\displaystyle=1-\liminf_{n\to+\infty}\frac{p_{n-m+1}}{P_{n-m+1}}=1-\lim_{n\to+\infty}\frac{1}{n-m+1}\cdot\liminf_{n\to+\infty}\frac{(n-m+1)p_{n-m+1}}{P_{n-m+1}}$
$\displaystyle=1-0\cdot\liminf_{n\to+\infty}\frac{(n-m+1)p_{n-m+1}}{P_{n-m+1}}=1.$

Similarly one computes $\liminf_{n\to+\infty}P_{n-m}/P_{n}=1$, hence
$\lim_{n\to+\infty}P_{n-m}/P_{n}=1$. Then it follows

$\displaystyle\lim_{n\to+\infty}C^{\alpha}_{m,n}=\lim_{n\to+\infty}\Big{(}\sum_{k=1}^{m}\Big{(}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-1\Big{)}k|a_{k}|\Big{)}^{2}=\Big{(}\sum_{k=1}^{m}\Big{(}\lim_{n\to+\infty}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-1\Big{)}k|a_{k}|\Big{)}^{2}=0.$

Consequently
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$. This
completes the proof. ∎

##  3\. Nörlund summability of Taylor series

######  Theorem 3.1.

Let $(N,(p_{n})_{n\in\mathbb{N}_{0}})$ be the Nörlund operator such that
$(p_{n})_{n\in\mathbb{N}_{0}}$ is non-decreasing and let $\alpha>1/2$. Then
the followings are true:

  1. (i)

If $(p_{n})_{n\in\mathbb{N}_{0}}$ has finite upper growth, then
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$ for every
$f\in\mathcal{D}_{\omega}$.

  2. (ii)

If additionally the growth condition

(20) $\liminf_{n\to+\infty}\frac{P_{n-1}}{P_{2n}}>0$

is satisfied, then
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$ for every
$f\in\mathcal{D}_{\omega}$ implies that $(p_{n})_{n\in\mathbb{N}_{0}}$ has
finite lower growth rate. Moreover the inequality $\alpha>1/2$ is sharp.

###### Proof.

  1. (i)

Let $\rho=\limsup_{n\to+\infty}\rho_{n}<+\infty$. By Lemma 2.3 we have

$\displaystyle\|T^{\alpha}_{n}\|^{2}_{\ell^{2}\to\ell^{2}}\leq(n+1)\sum_{k=1}^{n}\Big{|}\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}-\Big{(}\frac{P_{n-k-1}}{P_{n}}\Big{)}^{\alpha}\Big{|}^{2}=\frac{n+1}{P_{n}^{2\alpha}}\sum_{k=1}^{n}|P_{n-k}^{\alpha}-P_{n-k-1}^{\alpha}|^{2}.$

By Mean Value Theorem, e.g. see [13, Theorem 8, pp.178-9] for each
$k=1,2,\cdots,n$ there is $\xi_{k}\in(P_{n-k-1},P_{n-k})$ such that
$|P_{n-k}^{\alpha}-P_{n-k-1}^{\alpha}|=|P_{n-k}-P_{n-k-1}|\,\alpha\xi^{\alpha-1}_{k}$
and consequently we get the upper estimate

$\displaystyle\|T^{\alpha}_{n}\|^{2}_{\ell^{2}\to\ell^{2}}$
$\displaystyle\leq\alpha^{2}\frac{n+1}{P_{n}^{2\alpha}}\sum_{k=1}^{n}|P_{n-k}-P_{n-k-1}|^{2}P^{2\alpha-2}_{n-k}$
$\displaystyle\leq\alpha^{2}\frac{n+1}{P_{n}^{2\alpha}}\sum_{k=1}^{n}(n-k+1)^{2\alpha-2}\,p^{2\alpha}_{n-k}$
$\displaystyle=\alpha^{2}\frac{n+1}{P_{n}^{2\alpha}}\sum_{k=1}^{n}k^{2\alpha-2}\,p^{2\alpha}_{n-k}\leq\left\\{\begin{array}[]{ll}\displaystyle\alpha^{2}\frac{n+1}{n}\rho_{n}^{2\alpha}&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\frac{\alpha^{2}}{2\alpha-1}\frac{(n+1)(n^{2\alpha-1}+2\alpha-2)}{n^{2\alpha}}\rho_{n}^{2\alpha}&1/2<\alpha<1.\end{array}\right.$

For every $\varepsilon>0$ there exists $n(\varepsilon)\in\mathbb{N}_{0}$ such
that

$\displaystyle\|T^{\alpha}_{n}\|^{2}_{\ell^{2}\to\ell^{2}}<\left\\{\begin{array}[]{ll}\displaystyle\alpha^{2}\,\rho^{2\alpha}+\varepsilon&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\frac{\alpha^{2}}{2\alpha-1}\,\rho^{2\alpha}+\varepsilon&1/2<\alpha<1.\end{array}\right.\quad\text{for
all}\;n\geq n(\varepsilon).$

Now let $f\in\mathcal{D}_{\omega}$, since polynomials are dense in
$\mathcal{D}_{\omega}$ then there exists a sequence of polynomials
$(f_{k})_{k\in\mathbb{N}}\subset\mathcal{D}_{\omega}$ such that
$\|f_{k}-f\|_{\mathcal{D}_{\omega}}\to 0$ as $k\to+\infty$. Moreover

$\displaystyle\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}$
$\displaystyle\leq\|N^{\alpha}_{n}[f]-N^{\alpha}_{n}[f_{k}]\|_{\mathcal{D}_{\omega}}+\|N^{\alpha}_{n}[f_{k}]-f_{k}\|_{\mathcal{D}_{\omega}}+\|f_{k}-f\|_{\mathcal{D}_{\omega}}$
$\displaystyle\leq(\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}+1)\|f_{k}-f\|_{\mathcal{D}_{\omega}}+\|N^{\alpha}_{n}[f_{k}]-f_{k}\|_{\mathcal{D}_{\omega}}.$

By Lemma 2.4 $\|N^{\alpha}_{n}[f_{k}]-f_{k}\|_{\mathcal{D}_{\omega}}\to 0$ as
$n\to+\infty$. By Corollary 2.1 we have that

$\limsup_{n\to+\infty}\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}\leq\left\\{\begin{array}[]{ll}\displaystyle\alpha\,\rho^{\alpha}&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\frac{\alpha}{(2\alpha-1)^{1/2}}\,\rho^{\alpha}&1/2<\alpha<1,\end{array}\right.,$

consequently by letting $k\to+\infty$ the first term in the last inequality
vanishes.

  2. (ii)

Now suppose that $(p_{n})_{n\in\mathbb{N}_{0}}$ has unbounded lower growth
rate. Let

$\omega(z)=\frac{1-|z|^{2}}{|1-z|^{2}},\quad z\in\mathbb{D},$

then by Corollary 2.1 we have
$\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}=\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}$.
By [7, Theorem 2.2 (ii)], but instead applying it to the $(n-m+1)\times m$
submatrix

$A=\begin{pmatrix}(c_{2}-c_{1})&(c_{3}-c_{2})&\cdots&(c_{n-m+1}-c_{n-m})\\\
(c_{2}-c_{1})&(c_{3}-c_{2})&\cdots&(c_{n-m+1}-c_{n-m})\\\
\cdots&\cdots&\cdots&\cdots\\\
(c_{2}-c_{1})&(c_{3}-c_{2})&\cdots&(c_{n-m+1}-c_{n-m})\end{pmatrix},\;\text{where}\;c_{k}:=\left\\{\begin{array}[]{ll}\displaystyle\Big{(}\frac{P_{n-k}}{P_{n}}\Big{)}^{\alpha}&1\leq
k\leq n\\\\[10.00002pt] 0&k>n\end{array}\right.$

we get for any $1\leq m\leq n$

$\displaystyle\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}^{2}\geq\|A\|^{2}$
$\displaystyle=\|AA^{*}\|=\frac{n-m+1}{P_{n}^{2\alpha}}\sum_{k=1}^{m}|P_{n-k}^{\alpha}-P_{n-k-1}^{\alpha}|^{2}$
$\displaystyle=\alpha^{2}\frac{n-m+1}{P_{n}^{2\alpha}}\sum_{k=1}^{m}|P_{n-k}-P_{n-k-1}|^{2}\,\xi_{k}^{2\alpha-2},$

where $\xi_{k}\in(P_{n-k-1},P_{n-k})$. From monotonicity of the sequence
$(p_{n})_{n\in\mathbb{N}_{0}}$ and the recurrence relation
$P_{n-k}=P_{n-k-1}+p_{n-k}$ we obtain the lower bound

$\displaystyle\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}^{2}$
$\displaystyle\geq\alpha^{2}\frac{n-m+1}{P_{n}^{2\alpha}}\sum_{k=1}^{m}p_{n-k}^{2}\,P_{n-k-1}^{2\alpha-2}$
$\displaystyle\geq\alpha^{2}\frac{n-m+1}{P_{n}^{2\alpha}}\sum_{k=1}^{m}(m-k+1)^{2\alpha-2}\,p_{n-m-1}^{2\alpha}=\alpha^{2}(n-m+1)\Big{(}\frac{p_{n-m-1}}{P_{n}}\Big{)}^{2\alpha}\sum_{k=1}^{m}k^{2\alpha-2}.$

For $\alpha>1/2$ the last finite sum can be estimated by the elementary
integrals

(21)
$\sum_{k=1}^{m}k^{2\alpha-2}\geq\left\\{\begin{array}[]{ll}\displaystyle\int^{m}_{0}t^{2\alpha-2}\,dt=\frac{m^{2\alpha-1}}{2\alpha-1}&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\int^{m+1}_{1}t^{2\alpha-2}\,dt=\frac{(m+1)^{2\alpha-1}-1}{2\alpha-1}&1/2<\alpha<1,\end{array}\right.$

implying finally the lower bound for the operator norm as follows

$\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}^{2}\geq\left\\{\begin{array}[]{ll}\displaystyle\frac{\alpha^{2}}{2\alpha-1}\frac{n-m+1}{m}\Big{(}\frac{m\,p_{n-m-1}}{P_{n}}\Big{)}^{2\alpha}&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\frac{\alpha^{2}}{2\alpha-1}\frac{(n-m+1)((m+1)^{2\alpha-1}-1)}{m^{2\alpha}}\Big{(}\frac{m\,p_{n-m-1}}{P_{n}}\Big{)}^{2\alpha}&1/2<\alpha<1.\end{array}\right.$

Restricting to even indices $2n$ and taking $m=n$ yields

$\displaystyle\|T^{\alpha}_{2n}\|_{\ell^{2}\to\ell^{2}}^{2}$
$\displaystyle\geq\left\\{\begin{array}[]{ll}\displaystyle\frac{\alpha^{2}}{2\alpha-1}\frac{n+1}{n}\Big{(}\frac{n\,p_{n-1}}{P_{n-1}}\Big{)}^{2\alpha}\Big{(}\frac{P_{n-1}}{P_{2n}}\Big{)}^{2\alpha}&\alpha\geq
1\\\\[10.00002pt]
\displaystyle\frac{\alpha^{2}}{2\alpha-1}\frac{(n+1)((n+1)^{2\alpha-1}-1)}{n^{2\alpha}}\Big{(}\frac{n\,p_{n-1}}{P_{n-1}}\Big{)}^{2\alpha}\Big{(}\frac{P_{n-1}}{P_{2n}}\Big{)}^{2\alpha}&1/2<\alpha<1.\end{array}\right.$

Passing in lower limit for any $\alpha>1/2$ yields

$\displaystyle\liminf_{n\to+\infty}\|T^{\alpha}_{2n}\|_{\ell^{2}\to\ell^{2}}^{2}$
$\displaystyle\geq\frac{\alpha^{2}}{2\alpha-1}\liminf_{n\to+\infty}\Big{[}\Big{(}\frac{n\,p_{n-1}}{P_{n-1}}\Big{)}^{2\alpha}\Big{(}\frac{P_{n-1}}{P_{2n}}\Big{)}^{2\alpha}\Big{]}$
$\displaystyle\geq\frac{\alpha^{2}}{2\alpha-1}\liminf_{n\to+\infty}\Big{(}\frac{n\,p_{n-1}}{P_{n-1}}\Big{)}^{2\alpha}\cdot\liminf_{n\to+\infty}\Big{(}\frac{P_{n-1}}{P_{2n}}\Big{)}^{2\alpha}.$

By growth condition (20) we have that
$\beta=\liminf_{n\to+\infty}P_{n-1}/P_{2n}>0$. Moreover $\beta\leq 1$, since
$P_{n-1}\leq P_{2n}$. Then we obtain

$\displaystyle\liminf_{n\to+\infty}\|T^{\alpha}_{2n}\|_{\ell^{2}\to\ell^{2}}^{2}$
$\displaystyle\geq\beta^{2\alpha}\,\frac{\alpha^{2}}{2\alpha-1}\liminf_{n\to+\infty}\Big{(}\frac{n\,p_{n-1}}{P_{n-1}}\Big{)}^{2\alpha},$

which tends to infinity if the sequence $(p_{n})_{n\in\mathbb{N}_{0}}$ has
unbounded lower growth rate for any $\alpha>1/2$. Consequently from the
inequalities

$\limsup_{n\to+\infty}\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}\geq\limsup_{n\to+\infty}\|T^{\alpha}_{2n}\|_{\ell^{2}\to\ell^{2}}\geq\liminf_{n\to+\infty}\|T^{\alpha}_{2n}\|_{\ell^{2}\to\ell^{2}}$

and
$\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}=\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}$
it follows that
$\limsup_{n\to+\infty}\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}=+\infty$
and so
$\sup_{n\in\mathbb{N}_{0}}\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}=+\infty$.
By virtue of uniform boundedness principle (Banach–Steinhaus Theorem, e.g. see
[14, Theorem 1, p.68]), there exists $f\in\mathcal{D}_{\omega}$ satisfying
$\sup_{n\in\mathbb{N}_{0}}\|N^{\alpha}_{n}[f]\|_{\mathcal{D}_{\omega}}=+\infty$,
i.e. $\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\neq 0$.
Now let $\alpha=1/2$, then

$\sum_{k=1}^{n}k^{-1}\geq\int^{n+1}_{1}t^{-1}\,dt=\ln(n+1).$

On the other hand note that $P_{n}\leq(n+1)p_{n}$ implies $\rho_{n}\geq
n/(n+1)$ for all $n\in\mathbb{N}$. Thus we obtain

$\liminf_{n\to+\infty}\|T^{1/2}_{2n}\|_{\ell^{2}\to\ell^{2}}^{2}\geq\beta\,\liminf_{n\to+\infty}(\ln(n+1)\rho_{n})\geq\beta\,\liminf_{n\to+\infty}\ln(n+1)=+\infty.$

Again by uniform boundedness principle the claim follows.

∎

######  Example 3.1.

Let $p_{n}=1$ for all $n\in\mathbb{N}_{0}$, then $P_{n}=n+1$ and
$\lim_{n\to+\infty}\rho_{n}=1$. Moreover $p_{n}\leq p_{n+1}$, conditions of
Theorem 3.1 are fulfilled, thus
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$.

######  Example 3.2.

Let $p_{n}=n$ for $n\in\mathbb{N}$, then $P_{n}=n(n-1)/2$ and
$\lim_{n\to+\infty}\rho_{n}=2$. Moreover $p_{n}<p_{n+1}$, conditions of
Theorem 3.1 are fulfilled, thus
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$.

Another aspect we would like to touch is the rate at which $N_{n}^{\alpha}[f]$
converges towards $f$ as $n\to+\infty$. The next result gives essentially a
quantitative statement of Theorem 3.1.

######  Theorem 3.2.

Let $(N,(p_{n})_{n\in\mathbb{N}_{0}})$ be the Nörlund operator with
$(p_{n})_{n\in\mathbb{N}_{0}}$ non-decreasing and $\alpha>1/2$. If
$(p_{n})_{n\in\mathbb{N}_{0}}$ has finite upper growth, then
$\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=O(n^{-1/2})$ for all
$f\in\mathcal{D}_{\omega}$.

###### Proof.

Let $\rho<+\infty$ be the upper growth rate of $(p_{n})_{n\in\mathbb{N}_{0}}$.
We consider only the case when $\alpha\geq 1$. The situation $1/2<\alpha<1$ is
treated analogously by just replacing the corresponding operator norms from
Theorem 3.1. It suffices to prove that there is some positive constant $C$,
possibly depending on $f,\omega,\rho$ and $\alpha$ such that
$\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\leq C/n^{1/2}$ for all
sufficiently large $n$. For every $k\in\mathbb{N}$ there is a polynomial
$f_{k}$ with $\|f-f_{k}\|_{\mathcal{D}_{\omega}}<1/k^{5}$. On the other hand
we have the estimate

(22)
$\displaystyle\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\leq\|N^{\alpha}_{n}[f]-N^{\alpha}_{n}[f_{k}]\|_{\mathcal{D}_{\omega}}+\|N^{\alpha}_{n}[f_{k}]-f_{k}\|_{\mathcal{D}_{\omega}}+\|f_{k}-f\|_{\mathcal{D}_{\omega}}.$

For the first term we have

$\|N^{\alpha}_{n}[f]-N^{\alpha}_{n}[f_{k}]\|_{\mathcal{D}_{\omega}}\leq\|N^{\alpha}_{n}\|_{\mathcal{D}_{\omega}\to\mathcal{D}_{\omega}}\,\|f-f_{k}\|_{\mathcal{D}_{\omega}}\leq
2\alpha\rho^{\alpha}\,\frac{1}{k^{5}},\quad\text{for sufficiently large}\;n.$

Next we get an upper estimate for the middle term in (22). Take $n\geq k$ then

$\displaystyle\|N_{n}^{\alpha}[f_{k}]-f_{k}\|^{2}_{\mathcal{D}_{\omega}}$
$\displaystyle=\int_{\mathbb{D}}|N^{\alpha}_{n}[f_{k}]^{\prime}(z)-f_{k}^{\prime}(z)|^{2}\,\omega(z)\,dA(z)$
$\displaystyle=\int_{\mathbb{D}}\Big{|}\sum_{j=1}^{k}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{)}ja^{(k)}_{j}z^{j-1}\Big{|}^{2}\,\omega(z)\,dA(z)$
$\displaystyle\leq\Big{(}\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}j|a^{(k)}_{j}|\Big{)}^{2}\,\int_{\mathbb{D}}\omega(z)\,dA(z)=C(n,\alpha,f_{k})\cdot
C(\omega).$

We need only then to show that the positive constant $C(n,\alpha,f_{k})$
depends only on $\alpha,f$ for sufficiently large $k$ (and also $n$). Note
that $C(\omega)>0$ since $\omega$ is a positive superharmonic function on
$\mathbb{D}$. Next we find an upper estimate for the first constant above

$\displaystyle
C(n,\alpha,f_{k})=\Big{(}\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}j|a_{j}^{(k)}|\Big{)}^{2}$
$\displaystyle\leq\Big{(}\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}^{2}\Big{)}\cdot\Big{(}\sum_{j=1}^{k}j^{2}|a_{j}^{(k)}|^{2}\Big{)}$
$\displaystyle\leq
k^{2}\|f_{k}\|^{2}_{H^{2}}\,\Big{(}\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}^{2}\Big{)}.$

By the Mean Value Theorem for every $j\in\\{1,2,\cdots,k\\}$ there is
$\xi_{j}\in(P_{n-j},P_{n})$ such that
$|P_{n-j}^{\alpha}-P_{n}^{\alpha}|=\alpha\xi_{j}^{\alpha-1}\,|P_{n-j}-P_{n}|$.
The last sum can be estimated from above by

$\displaystyle\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}^{2}$
$\displaystyle\leq\frac{1}{P_{n}^{2\alpha}}\sum_{j=1}^{k}\alpha^{2}|P_{n-j}-P_{n}|^{2}\,P^{2\alpha-2}_{n}\leq\alpha^{2}\frac{p_{n}^{2}}{P^{2}_{n}}\sum_{j=1}^{k}j^{2}\leq\alpha^{2}\frac{k^{3}}{n^{2}}\rho_{n}^{2}.$

Therefore for sufficiently large $n$ we get the estimate

$\sum_{j=1}^{k}\Big{|}\Big{(}\frac{P_{n-j}}{P_{n}}\Big{)}^{\alpha}-1\Big{|}^{2}\leq
2\alpha^{2}\rho^{2}\frac{k^{3}}{n^{2}},$

consequently $C(n,\alpha,f_{k})\leq
2\alpha^{2}\rho^{2}\,\|f_{k}\|^{2}_{H^{2}}k^{5}/n^{2}$. Convergence
$\|f_{k}-f\|_{\mathcal{D}_{\omega}}$ implies
$\mathcal{D}_{\zeta}(f_{k})\to\mathcal{D}_{\zeta}(f)\to 0$ as $k\to+\infty$
for $\mu$-a.a. $\zeta\in\overline{\mathbb{D}}$. For
$\zeta\in\overline{\mathbb{D}}$, then by Lemma 2.2, for every $k\in\mathbb{N}$
there is $g_{k}\in H^{2}$ such that $f_{k}(z)=a^{(k)}+(z-\zeta)\,g_{k}(z)$ for
some $a^{(k)}\in\mathbb{C}$ and
$\mathcal{D}_{\zeta}(f_{k})=\|g_{k}\|^{2}_{H^{2}}$. Similarly for
$f\in\mathcal{D}_{\zeta}$ there is $g\in H^{2}$ with $f(z)=a+(z-\zeta)\,g(z)$
for some $a\in\mathbb{C}$ and $\mathcal{D}_{\zeta}(f)=\|g\|^{2}_{H^{2}}$.
Taking $\zeta\in\overline{\mathbb{D}}$ such that
$\mathcal{D}_{\zeta}(f_{k})\to\mathcal{D}_{\zeta}(f)$ implies
$\|g_{k}\|_{H^{2}}\to\|g\|_{H^{2}}$. On the other hand if we write
$g_{k}(z)=b_{0}^{(k)}+b_{1}^{(k)}z+\cdots+b_{k-1}^{(k)}z^{k-1}$ for
$k\in\mathbb{N}_{0}$, then we have

$\|f_{k}\|^{2}_{H^{2}}\leq
2(|a^{(k)}|^{2}+|\zeta|^{2}\,\sum_{j=0}^{k-1}|b^{(k)}_{j}|^{2}+\sum_{j=0}^{k-1}|b_{j}^{(k)}|^{2})=2(|a^{(k)}|^{2}+(1+|\zeta|^{2})\,\|g_{k}\|^{2}_{H^{2}}).$

Moreover $\|f_{k}-f\|_{\mathcal{D}_{\omega}}\to 0$ yields in particular that
$|f_{k}(0)-f(0)|=|a^{(k)}_{0}-a_{0}|\to 0$ as $k\to+\infty$, which together
with $\sup_{k\in\mathbb{N}_{0}}|b^{(k)}_{0}|<+\infty$ (since
$\|g_{k}\|_{H^{2}}\to\|g\|_{H^{2}}$ as $k\to+\infty$), give
$\limsup_{k\to+\infty}|a^{(k)}|=|a^{*}|$ for a certain $a^{*}\in\mathbb{C}$.
Consequently $\limsup_{k\to+\infty}\|f_{k}\|^{2}_{H^{2}}\leq
2(|a^{*}|^{2}+(1+|\zeta|^{2})\|g\|^{2}_{H^{2}})\leq
2(|a^{*}|^{2}+2\|g\|^{2}_{H^{2}})$. Let
$C(f)=2(|a^{*}|^{2}+2\|g\|^{2}_{H^{2}})=2(|a^{*}|^{2}+2\mathcal{D}_{\zeta}(f))$
then for sufficiently large $k$ (and hence for sufficiently large $n$) we have
$C(n,\alpha,f_{k})\leq 4\alpha^{2}\rho^{2}C(f)k^{5}/n^{2}$. Taking $n=k^{5}$
we can write $C(n,\alpha,f_{k})\leq 4\alpha^{2}\rho^{2}C(f)/n$ and so

$\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}\leq(2\alpha\rho^{\alpha}+2\alpha\rho(C(f)C(\omega))^{1/2}+1)\frac{1}{n^{1/2}},\quad\text{for
all sufficiently large}\;n.$

∎

We close this section with an analogue result when the sequence
$(p_{n})_{n\in\mathbb{N}_{0}}$ is non-increasing, provided that
$\liminf_{n\to+\infty}p_{n}>0$. This last condition is essential in getting
uniform upper bound for the operator norm
$\|T^{\alpha}_{n}\|_{\ell^{2}\to\ell^{2}}$. In this situation, unlike for non-
decreasing sequences, the assumption of finite upper growth rate is redundant,
since from non-increasing property it holds that $P_{n}\geq(n+1)p_{n}$ for all
$n\in\mathbb{N}_{0}$, consequently $\rho_{n}\leq n/(n+1)$ and in particular
$\limsup_{n\to+\infty}\rho_{n}\leq 1$. We state the next result without proof.

######  Theorem 3.3.

Let $(N,(p_{n})_{n\in\mathbb{N}_{0}})$ be the Nörlund operator with
$(p_{n})_{n\in\mathbb{N}_{0}}$ non-increasing and satisfying
$\liminf_{n\to+\infty}p_{n}>0$, and let $\alpha>1/2$. Then
$\lim_{n\to+\infty}\|N^{\alpha}_{n}[f]-f\|_{\mathcal{D}_{\omega}}=0$ for every
$f\in\mathcal{D}_{\omega}$. Moreover the convergence rate is of the order
$O(n^{-1/2})$.

######  Question 3.1.

Is $\alpha=1/2$ a sharp bound in this case? Can the convergence rate in
Theorem 3.2 and Theorem 3.3 be improved?

######  Question 3.2.

Is generalized Nörlund method monotone? In other words does
$N^{\alpha}_{n}[f]\to f$ imply $N^{\beta}_{n}[f]\to f$ when $\alpha\leq\beta$?

### Discussion

The sequence $(p_{n})_{n\in\mathbb{N}_{0}}$ is assumed to be monotone. In fact
this condition could be relaxed to eventually monotone, that is there is some
$n_{0}\in\mathbb{N}_{0}$ such that $(p_{n})_{n\geq n_{0}}$ is monotone. Indeed
this would not affect the analysis since both the upper and lower estimates
for the operator norm $\|T^{\alpha}_{n}\|^{2}_{\ell^{2}\to\ell^{2}}$ would
only differ by an additive term (sum of $P_{k}$–like terms up to index $n_{0}$
divided by $p_{n}^{2\alpha}$) which in the limit would be bounded by a uniform
positive constant. The situation for a general sequence
$(p_{n})_{n\in\mathbb{N}_{0}}$, not necessarily monotone, remains also to be
investigated.

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[∗]Berlin 10709, Germany.

[$\dagger$]University of Prishtina,
Faculty of Natural Science and Mathematics,
Department of Mathematics and Computer Sciences,
Av. Mother Teresa, Nr.5, 10000 Prishtina, Kosovo.