# On the Geometry and Linear Convergence of Primal-Dual Dynamics

P. Bansode1, V. Chinde2, S. R. Wagh3, R. Pasumarthy4, and N. M. Singh3 *This
work was not supported by any organization1P. Bansode is with Customized
Energy Solutions, Pune, India<EMAIL_ADDRESS>Chinde is with National
Renewable Energy Laboratory, Colorado, USA. 3S. R. Wagh and N. M. Singh are
with Electrical Engineering Department, Veermata Jijabai Technological
Institute, Mumbai, India. 4R. Pasumarthy is with Electrical Engineering
Department, Indian Institute of Technology Madras, India

###### Abstract

The paper proposes a variational-inequality based primal-dual dynamic that has
a globally exponentially stable saddle-point solution when applied to solve
linear inequality constrained optimization problems. A Riemannian geometric
framework is proposed wherein we begin by framing the proposed dynamics in a
fiber-bundle setting endowed with a Riemannian metric that captures the
geometry of the gradient (of the Lagrangian function). A strongly monotone
gradient vector field is obtained by using the natural gradient adaptation on
the Riemannian manifold. The Lyapunov stability analysis proves that this
adaption leads to a globally exponentially stable saddle-point solution.
Further, with numeric simulations we show that the scaling a key parameter in
the Riemannian metric results in an accelerated convergence to the saddle-
point solution.

##  I INTRODUCTION

Saddle-node dynamics has remained a subject of substantial research for many
years [1, 2, 3, 4]. It iteratively seeks a solution to saddle-point problems
that arise in a number of disciplines including equilibrium theory, game
theory and optimization. However its application to constrained optimization
problems has gained wide interest over the recent years, especially in the
areas of the power networks [5, 6, 7, 8] and wireless networks [9, 10, 11]),
and building automation systems [12], etc. Within this domain, it is popularly
regarded as primal-dual dynamics.

Over the last decade, various formulations of the primal-dual dynamics have
been explored in connection with the constrained optimization problems. These
algorithms were studied with primary focus on convergence of iterates to the
saddle-point solution and its stability. These algorithms either use a
framework of hybrid dynamical systems [9, 13] or an augmented Lagrangian
technique that involves projections in the Lagrangian function [14, 15, 16].
The hybrid dynamical systems approach involves switching in the dual dynamics
to handle constraint violations. But the inherent discontinuities present in
the dual dynamics make it difficult to prove exponential stability of the
saddle-point solution. Thus so far, only globally asymptotic stability of the
saddle-point solution has been proven. The augmented Lagrangian approach is
based on augmentation of the penalty terms in the Lagrangian function [17].
The penalty term is designed in such a way that the resulting dual dynamics is
discontinuity-free, meaning that the rate of change of the dual variable does
not involve switching terms. The basis of these penalty functions is either a
projection or a proximal operator. So far, the latter approach has been
successful in proving the global exponential stability of the algorithm when
applied to solve linearly constrained optimization problems. In [18], it is
shown that the algorithm attains a semi-globally exponential convergence for a
more general convex inequality constrained optimization problem.

The algorithm that we propose in this paper uses a variational inequality
based projected dynamical systems framework and produces a globally
exponentially stable saddle-point solution for a linear inequality constrained
optimization problem. The projected dynamical systems have been widely used to
solve variational inequalities [19, 20, 21]. We model the saddle-point problem
as a variational inequality and then use the projected dynamical system that
singularly handles the constraints without involving discontinuities in the
dual dynamics (the detailed approach of the proposed dynamics is documented in
our online report [22], which is omitted from this paper due to avoid
repetition).

In contrast to the existing research on primal-dual dynamics, our algorithm
does not depend on the framework of hybrid dynamical systems or the augmented
Lagrangian techniques. We consider differential equations for solution
trajectories of variational inequality proposed in [23, Section 5.7.1] as a
basis for our algorithm, and equate the time derivative of primal-dual
variables to the terms for which the corresponding fixed point problem of the
variational inequality fails to be satisfied (proved in our online report [22,
Section 2.2]). Instead of tuning free-parameters in the augmented Lagrangian
function we start by studying the geometry of gradient vector field defining
the primal-dual dynamics. This enables us to carry out the required
alterations to the dynamics in the Euclidean space so as to ensure linear
convergence rates under suitable assumptions. We first analyze the vector
field of the proposed algorithm in the Euclidean space and prove that the
Euclidean geometry is not suitable for achieving the same. Then we lay down a
procedure to construct a suitable differential geometry for the cause. The
geometrization of the problem is based in a fiber bundle with a semi-
Riemannian metric imposed by the structure of the primal-dual dynamics. A
suitable alteration of the Geometry is achieved by a scaling of the connection
leading to a natural gradient in the Riemannian space. As the natural gradient
behaves like the Euclidean gradient in the sense that it achieves the steepest
descent direction for the gradient descent algorithm [24, 25], the obtained
natural gradient dynamics is substituted for the original Primal Dual dynamics
in the Euclidean space to ensure a faster convergence rate (in our case, the
contraction region is obtained in the Euclidean domain because we are using
the natural gradient as opposed to the contraction region being defined by a
Riemann metric). Adapting this into the proposed algorithm results in a
strongly monotone gradient vector field with steepest descent and ascent
directions along the primal and dual variables, respectively, then we
construct a Lyapunov function which shows that for a strongly monotone
gradient the proposed algorithm has an exponentially stable saddle-point
solution. Further with the help of numeric simulations, it is shown that by
appropriate scaling of a key parameter in the natural gradient which in turn
leads to the scaling of the connection as alluded to above, the convergence
rate of the proposed algorithm can be accelerated.

### Notations

The set $\mathbb{R}$ (respectively $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{>0}$)
is the set of real (respectively non-negative or positive) numbers. If
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is continuously differentiable in
$x\in\mathbb{R}^{n}$, then
$\nabla_{x}f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the gradient of $f$
with respect to $x$. $\|.\|$ denotes the Euclidean norm.

##  II Problem Formulation

Consider the following constrained optimization problem

$\begin{array}[]{cc}\mathrm{minimize}&f(x)\\\
\mathrm{subject\leavevmode\nobreak\ to}&x\in X\end{array}$ (1)

where

$X=\\{x\in\mathbb{R}^{n}|g_{i}(x)\leq 0,\forall^{m}_{i=1}\\},$ (2)

is the domain of the problem (1). The functions
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ are assumed to be continuously
differentiable $(\mathcal{C}^{2})$ with respect to $x$, with the following
assumptions:

###### Assumption 1

$\nabla f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is strongly monotone on
$X$, with $\mu>0$ such that the following holds:

$(x_{1}-x_{2})^{T}(\nabla f(x_{1})-\nabla
f(x_{2}))\geq\mu\|x_{1}-x_{2}\|^{2}.$

As a consequence of Assumptions 1, it is derived that the objective function
$f$ is strongly convex in $x$ with the modulus of convexity given by
$\frac{\mu}{2}$.

###### Assumption 2

The constraint function $g(x)=Ax-b$ is linear in $x$.

###### Assumption 3

There exists an $x\in\mathrm{relint}X$ such that
$g_{i}(x)<0,\forall^{m}_{i=1}$.

###### Assumption 4

Let matrix $\frac{\partial g}{\partial x}$ have full row rank $m\leq n$ and
$q_{1}I\leq\frac{\partial g}{\partial x}\frac{\partial g}{\partial x}^{T}\leq
q_{2}I$, where $I$ is an identity matrix and $q_{1},q_{2}$ are positive
constants.

Assumptions (1)-(3) ensure that $x$ is strictly feasible and strong duality
holds for the optimization problem (1).

Let $L:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ define the
Lagrangian function of the optimization problem (1) as given below

$L(x,\lambda)=f(x)+\lambda^{T}g(x).$ (3)

Let $\lambda_{i}$ be the Lagrange multipliers associated with $g_{i}(x)$, then
$\lambda\in\Lambda\subseteq\mathbb{R}^{m}_{+}=\\{\lambda\in\mathbb{R}^{m},\lambda_{i}\geq
0,\forall^{m}_{i=1}\\}$ defines the corresponding vectors of Lagrange
multipliers.

The Lagrangian function $L$ defined in (3) is $\mathbb{C}^{2}$-differentiable
convex-concave in $x$ and $\lambda$ respectively, i.e., $L(.,\lambda)$ is
convex for all $\lambda\in\Lambda$ and $L(x,.)$ is concave for all $x\in X$.
We say that $(x^{*},\lambda^{*})$ is a saddle-point if the following holds:

$\displaystyle L(x^{*},\lambda)\leq L(x^{*},\lambda^{*})\leq L(x,\lambda^{*})$
(4)

for all $x\in X$ and $\lambda\in\Lambda$.

If $x^{*}$ is the unique minimizer of $L$, then it must satisfy the Karush-
Kuhn-Tucker (KKT) conditions stated as follows.

$\displaystyle g_{i}(x^{*})$ $\displaystyle\leq 0,\leavevmode\nobreak\
\forall^{m}_{i=1}$ (5) $\displaystyle\lambda^{*}_{i}$ $\displaystyle\geq
0,\leavevmode\nobreak\ \forall^{m}_{i=1}$ (6)
$\displaystyle\lambda^{*}_{i}g_{i}(x^{*})$
$\displaystyle=0,\leavevmode\nobreak\ \forall^{m}_{i=1}$ (7)
$\displaystyle\nabla f(x^{*})+\lambda^{*T}\nabla g(x^{*})$ $\displaystyle=0.$
(8)

Let us define $z=(x,\lambda)\in\Omega=X\times\Lambda$, where $\Omega$ by
definition is a nonempty and closed convex subset of $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}_{\geq 0}$. Then $z^{*}=(x^{*},\lambda^{*})$ is the saddle
point solution of (3).

###### Remark 1

Since strong duality holds, the KKT conditions (5)-(8) are necessary and
sufficient to guarantee optimality of the problem (1), with $x^{*}$ as the
unique minimizer of (1) and $z^{*}$ as the unique saddle-point of (3).

Let $G:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n+m}$ define
the gradient map of (3) as given below:

$\displaystyle G(z)=\nabla_{z}L=\begin{bmatrix}-\nabla_{x}L(x,\lambda)\\\
\nabla_{\lambda}L(x,\lambda)\end{bmatrix}$ (9)

Taking gradient descent and gradient ascent along the direction of $x$ and
$\lambda$ variables, respectively, we propose an algorithm which in principal
uses the framework of projected dynamical systems (for details, please refer
to our online report [22, Section 2] or Definition IV.1, Proposition IV.1, and
equation (57) in the Appendix section). We designate the algorithm as
projected primal-dual dynamics, it is as shown below:

$\dot{z}=\beta\\{P_{\Omega}[z-\alpha G(z)]-z\\},$ (10)

where $\alpha,\beta>0$ are parameters adjusted to control stability and assure
convergence, we set $\alpha=\beta=1$ to avoid confusion. $P_{\Omega}$ is a
minimum norm projection operator of the form
$P_{\Omega}=\arg\min_{v\in\Omega}\|z-v\|$. Note that, there is no projection
taken w.r.t. the primal variable $x$ as it belongs to $\mathbb{R}^{n}$ but
only w.r.t. the dual variable $\lambda$ which is restricted to
$\mathbb{R}^{m}_{\geq 0}$. The following property always holds for projection
on $\Omega$,

$\displaystyle[u-P_{\Omega}(u)]^{T}[P_{\Omega}(z)-z]\geq 0,\forall
u\in\mathbb{R}^{n}\times\mathbb{R}^{m}_{\geq 0},\forall z\in\Omega$ (11)

In what follows, we assess the stability of the proposed algorithm.

###  II-A Stability analysis

Before proceeding to the stability analysis of the proposed dynamics (10), it
is worth noting the Definition IV.2-IV.3 and Proposition IV.3 on the
monotonicity property, stated in the Appendix of this paper.

###### Lemma II.1

If Assumptions (1)-(3) hold, then $G(z)$ is monotone such that
$[G(z_{1})-G(z_{2})]^{T}(z_{1}-z_{2})\geq 0$ for every pair of
$z_{1},z_{2}\in\Omega$.

###### Proof:

First we derive the Jacobian matrix of $G$ as

$\nabla G=\begin{bmatrix}\nabla^{2}f(x)+\lambda^{T}\nabla^{2}g(x)&\nabla
g(x)^{T}\\\ -\nabla g(x)&\mathbf{0}\end{bmatrix}$ (12)

It is know that $G$ is monotone if and only if the $\nabla G$ is positive
semidefinite (see, [26]), which implies that $\nabla G$ must be positive
semidefinite $\frac{1}{2}\nabla G+\frac{1}{2}\nabla G^{T}\geq 0,\forall
z\in\Omega,\forall t$. We verify this property by evaluating the symmetric
part of $\nabla G$:

$\displaystyle\frac{\nabla G+\nabla G^{T}}{2}$
$\displaystyle=\begin{bmatrix}\nabla^{2}f(x)+\lambda^{T}\nabla^{2}g(x)&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}\end{bmatrix}$ (13) $\displaystyle\geq{0},\forall
z\in\Omega,\forall t$ (14)

This proves that $\nabla G(z)$ a is positive semi-definite matrix. Thus $G(z)$
is a monotone map. ∎

###### Lemma II.2

Let ${G}(z)$ be continuously differentiable on an open convex subset of
$\mathbb{R}^{n+m}$. If Assumptions 1-3 hold and $G$ is monotone for all
$z\in\Omega$, then with $\alpha>0$, the projected PD dynamics (10) is Lyapunov
stable.

###### Proof:

For the Lagrangian function (3), the following inequalities always hold:
${L}(x^{*},\lambda^{*})-{L}(x^{*},\lambda)\geq 0$ and
${L}(x,\lambda^{*})-{L}(x^{*},\lambda^{*})\geq 0$. Let us define the Lyapunov
function as follows:

$\displaystyle V(z)$
$\displaystyle=({L}(x^{*},\lambda^{*})-{L}(x^{*},\lambda))+({L}(x,\lambda^{*})-{L}(x^{*},\lambda^{*}))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1}{2}\|z-z^{*}\|^{2}.$ (15)

The last term in (15) ensures that $V(z)\geq\frac{1}{2}\|z-z^{*}\|^{2},\forall
z\in\Omega$, thus also ensures the boundedness of the level sets of $V(z)$.

Differentiating $V(z)$ along the trajectories of (10) yields,

$\displaystyle\dot{V}(z)$ $\displaystyle=\nabla V(z)\dot{z}$
$\displaystyle=-[(\nabla{L}(x,\lambda^{*})-\nabla{L}(x^{*},\lambda))+z-z^{*}]^{T}(z-\tilde{z})$
$\displaystyle=-[G(z)+z-z^{*}]^{T}(z-\tilde{z})$ (16)

using (11) with $u=z-\alpha G(z)$ and $z=z^{*}$, we get the following, [27],

$[z-z^{*}+\alpha
G(z)]^{T}(z-\tilde{z})\geq\|z-\tilde{z}\|^{2}+\alpha(z-z^{*})^{T}{G}(z).$ (17)

Using (17) in (16) yields,

$\displaystyle\dot{V}(z)$
$\displaystyle\leq-(z-z^{*})^{T}{G}(z)-\|z-\tilde{z}\|^{2}$
$\displaystyle\leq-(z-z^{*})^{T}(G(z)-G(z^{*}))-\|z-\tilde{z}\|^{2}$ (18)

Using Lemma II.1 in (18), we get $\dot{V}(z)\leq 0$. This proves that the
projected dynamical system (10) is stable in the sense of Lyapunov. ∎

We see that the results of Lemma II.1 and II.2 are not sufficient to prove
exponential stability of the proposed dynamics (10). This however is overcome
by adapting the framework of natural gradient [24], which allows us to prove
that the gradient $G(z)$ is strongly monotone. This property is later
exploited to prove that the proposed dynamics is exponentially stable as
discussed in the subsequent section.

###  II-B Geometry of the Primal-Dual Dynamics

First we employ the framework of fiber bundles to understand the geometry of
the gradient map (9). Assume that the proposed dynamics is embedded into a
tuple $(X,\mathcal{M})$ with $X$ a manifold and $\mathcal{M}$ a fiber manifold
above $X$, with projection $\Pi:\mathcal{M}\rightarrow X$. $X$ is considered
as a state-space of primal variables while the fibers of $\mathcal{M}$ are
along the space of the dual variables, $\Lambda$. If $x\in
X\subset\mathbb{R}^{n}$, $\lambda\in\Lambda\subseteq\mathbb{R}^{m}_{\geq 0}$
then $\mathcal{M}=(X,\Lambda)$ with coordinates $(x,\lambda)$. The tangent
space of $\mathcal{M}$, denoted by $T_{\mathcal{M}}$ has coordinates
$(\dot{x},\dot{\lambda})$.

Consider the function

$\displaystyle Q(x,\lambda)=\lambda-g(x).$ (19)

and the implicit surface $\mathcal{M}=\\{x|\lambda-g(x)=0\\}$. With (19), if
$x\in X$ the primal dynamics in (10) on the manifold $\mathcal{M}$ reduces to:

$\displaystyle\dot{x}=-\nabla f(x)-\Big{(}\frac{\partial g}{\partial
x}\Big{)}^{T}g(x).$ (20)

It follows that (20) is exponentially stable.

###### Lemma II.3

The gradient dynamics (20) is exponentially stable.

###### Proof:

Assume $g(x)=Ax-b$ and consider the following Krasovskii-type Lyapunov
candidate function $V(x)=\frac{1}{2}\dot{x}^{T}\dot{x}$. Differentiating it
along the trajectories of (20), we get

$\displaystyle\dot{V}$
$\displaystyle=\dot{x}\Big{(}-\nabla^{2}f(x)\dot{x}-\Big{[}\frac{\partial^{2}g}{\partial^{2}x}\Big{]}^{T}g(x)\dot{x}-\Big{[}\frac{\partial
g}{\partial x}\Big{]}^{T}\Big{[}\frac{\partial g}{\partial
x}\Big{]}\dot{x}\Big{)}$ (21)
$\displaystyle\leq\dot{x}^{T}[-\nabla^{2}f(x)-k]\dot{x}$ (22)
$\displaystyle\leq-\gamma V(x)$ (23)

where $\gamma>0$ and $k$ is chosen as

$\displaystyle k>\sqrt{q_{2}}.$ (24)

∎

Global attractivity of the manifold $\mathcal{M}$ would ensure that the
proposed dynamics in (10) is globally exponentially stable but since it is
known that this is not the case we study the geometry of the problem and
identify conditions to improve the convergence rates to the fixed point of
(10).

####  II-B1 Constructing a Riemannian metric

Let $R$ define a semi-Riemannian metric on space $T_{\mathcal{M}}$ that endows
a semi-Riemannian structure to the fiber-bundle
$(x,\lambda)\in\mathbb{R}^{n\times m}$, as shown below:

$\displaystyle R$ $\displaystyle=\nabla Q^{T}(x)\nabla Q(x)$ (25)
$\displaystyle=\begin{bmatrix}\big{(}\frac{\partial g}{\partial
x}\big{)}^{T}\frac{\partial g}{\partial x}&\big{(}-\frac{\partial g}{\partial
x}\big{)}^{T}\\\ -\frac{\partial g}{\partial x}&I\end{bmatrix}$ (26)

$R$ is a semi metric on the space $X\times\Lambda$ with connection
$\big{(}\frac{\partial g}{\partial x}$. The metric $R$ can be made symmetric
positive definite by introducing in it the parameter $k$ such that $k$ scales
the connection term and for $k=I$ the original connection as defined by the
above semi-Riemann metric is obtained, i.e

$\displaystyle R=\begin{bmatrix}\big{(}\frac{\partial g}{\partial
x}\big{)}^{T}\frac{\partial g}{\partial x}&\big{(}-\frac{\partial g}{\partial
x}\big{)}^{T}\\\ -\frac{\partial g}{\partial x}&kI\end{bmatrix}.$ (27)

We are now in a position to develop an understanding of the geometry of the
proposed dynamics on a Riemannian manifold $(\mathcal{M},R)$. The following
definition will be useful in understanding the concept of a linear connection
with respect to the manifold $(\mathcal{M},R)$,[28].

###### Definition II.1

Let $\Pi:\mathcal{M}\rightarrow X$ be a smooth fiber bundle, a tangent vector
$\nu\in T_{p}\mathcal{M},\leavevmode\nobreak\ p\in\mathcal{M}$, is said to be
vertical if $\Pi_{x_{p}}(\nu)=0$. $V(p)$ denotes the set of all vertical
tangent vectors in $P$. A distribution $H$ on $\mathcal{M}$ is said to be
horizontal if $T_{p}\mathcal{M}=V_{p}\oplus H(p)$ for all $p\in\mathcal{M}$.

###### Remark 2

If $H$ is horizontal, it implies that for all $p\in\mathcal{M}$, $H(p)$ is a
linear subspace of $T_{p}\mathcal{M}$ with the following properties:

$\displaystyle\dim H(p)$ $\displaystyle=\dim X$ (28) $\displaystyle
H(p)\cap\mathcal{M}(p)$ $\displaystyle=\Phi$ (29)

$\Pi_{X_{p}}$ maps $H(p)$ isometrically onto $T_{\Pi(p)}X$.

A connection in the bundle $\mathcal{M}$ is due to a unique splitting scheme
of the tangent space $T_{p}\mathcal{M}$ into a horizontal and vertical space
as shown in Remark 4 in the Appendix section. The preferred direction of the
vertical vector is along the fibers of $\mathcal{M}$.

####  II-B2 Strongly monotone gradient of the Lagrangian

The natural gradient of $L$ at $z\in\mathcal{M}$ is a unique tangent vector
$\mathrm{grad}L$ given as

$\langle\mathrm{grad}L,v\rangle_{r}=D_{z}L(v),\forall v\in T_{z}\mathcal{M}.$
(30)

In the matrix notation, (30) implies the following

$\mathrm{grad}_{r}L=R^{-1}\nabla L^{T}$ (31)

where $\nabla L=G(z)$ is the gradient vector of $L$ on Euclidean space
$\mathbb{R}^{m+n}$.

Denote $G_{r}(z)=\mathrm{grad}_{r}L$, the linear map
$\mathbb{H}_{G_{r}(z)}:T_{z}\mathcal{M}\rightarrow\mathcal{M}$ assigned to
each point $z\in\mathcal{M}$ is defined by the Hessian of $L$, denoted by
$\mathbb{H}_{G_{r}}(z)v=\nabla_{v}G_{r}(z)=R^{-1}\nabla G(z),\forall v\in
T_{z}\mathcal{M}$.

The projection operator
$P^{r}_{\mathcal{M}}:\mathbb{R}^{n+m}\rightarrow\mathcal{M}$ defined as

$\displaystyle P^{r}_{\mathcal{M}}(z)=\arg\min_{v\in
T_{z}\mathcal{M}}\|z-v\|^{2}_{r}.$

Correspondingly, the projected PD dynamics on $\mathcal{M}$ is defined as
follows:

$\dot{z}=\beta\\{P^{r}_{\mathcal{M}}[z-\alpha G_{r}(z)]-z\\}.$ (32)

Replacing $g(x)$ by $Ax-b$ as defined in Assumption 2, we define the gradient
vector $G_{r}(z)\in T_{z}\mathcal{M}$ as follows:

$\displaystyle G_{r}(z)$ $\displaystyle=R^{-1}G(z),$
$\displaystyle=\begin{bmatrix}k\nabla f(x)-A^{T}Ax+kA^{T}\lambda+A^{T}b\\\
A\nabla f(x)-kAx+AA^{T}\lambda+kb\end{bmatrix}$ (33)

In the following section, it is proved that the gradient map (33) is strongly
monotone.

###### Proposition II.4

Consider the optimization problem (1) and let $(\mathcal{M},r)$ be a
$n+m$-dimensional smooth manifold. If Assumption 1 and 4 hold for the problem
(1), then with the linear map $R^{-1}:T_{z}\mathcal{M}\rightarrow
T_{z}\mathcal{M}$, the gradient vector $G_{r}(z)$ is strongly monotone.

###### Proof:

For $G_{r}(z)$ to be strongly monotone, $\nabla_{z}{G}_{r}$ must be positive
definite[26], i.e., for the symmetric part of $\nabla_{z}{G}_{r}$, i.e.
$\frac{1}{2}\nabla{G}_{r}+\frac{1}{2}\nabla{G}^{T}_{r}$, the following must
hold:

$\displaystyle\nabla{G}_{r}+\nabla{G}^{T}_{r}$ $\displaystyle=R^{-1}\nabla
G+\nabla G^{T}R^{-1},$ $\displaystyle\geq\nu\mathrm{I},\forall
z\in\mathcal{M},\forall t$ (34)

where $\nu>0$ is a constant, $\mathrm{I}$ is an identity matrix of appropriate
dimensions.

The Jacobian of $G_{r}(z)$, denoted by $\nabla G_{r}(z)$ is given below:

$\displaystyle\nabla G_{r}(z)=\begin{bmatrix}k\nabla^{2}f(x)-A^{T}A&kA^{T}\\\
A\nabla^{2}f(x)-kA&AA^{T}\end{bmatrix}$

The symmetric part of $\nabla G_{r}(z)$ is obtained as:

$\displaystyle\frac{\nabla{G}_{r}(z)+\nabla{G}^{T}_{r}(z)}{2}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}k\nabla^{2}f(x)-A^{T}A&\frac{1}{2}(A\nabla^{2}f(x))^{T}\\\
\frac{1}{2}A\nabla^{2}f(x)&AA^{T}\end{bmatrix}$ (35)

Let $\mathrm{M}=\nabla{G}_{r}(z)+\nabla{G}^{T}_{r}(z)-q_{1}\mathrm{I}>0$. Then

$\displaystyle\mathrm{M}$
$\displaystyle=\begin{bmatrix}2k\nabla^{2}f(x)-2A^{T}A-q_{1}\mathrm{I}&(A\nabla^{2}f(x))^{T}\\\
A\nabla^{2}f(x)&2AA^{T}-q_{1}\mathrm{I}\end{bmatrix}$
$\displaystyle\geq\begin{bmatrix}2k\nabla^{2}f(x)-2A^{T}A-q_{1}\mathrm{I}&(A\nabla^{2}f(x))^{T}\\\
A\nabla^{2}f(x)&AA^{T}\end{bmatrix}.$ (36)

Further let $\mathrm{S}=AA^{T}$, then the Schur compliment of the block
$\mathrm{S}$ of the matrix $\mathrm{M}$, denoted by
$\mathrm{S}_{\mathrm{Schur}}$ is derived as

$\displaystyle\mathrm{S}_{\mathrm{Schur}}$
$\displaystyle=2k\nabla^{2}f(x)-2A^{T}A-q_{1}I$
$\displaystyle-(A\nabla^{2}f(x))^{T}(AA^{T})^{-1}A\nabla^{2}f(x).$ (37)

Let $\mathrm{H}=\nabla^{2}f(x)$ for the notational simplicity. Note that in
(38), $2k\mathrm{H}>0,\forall k>0$, $2A^{T}A\geq 0$, $q_{1}I>0$, and
$0\leq\mathrm{H}A^{T}(AA^{T})^{-1}A\mathrm{H}\leq\mathrm{H}^{2}$. The last
terms is a consequence of $A^{T}(AA^{T})^{-1}A\leq I$. Rearranging (37) as
given below

$\displaystyle 2k\mathrm{H}$
$\displaystyle>2A^{T}A+q_{1}I+\mathrm{H}A^{T}(AA^{T})^{-1}A\mathrm{H}$
$\displaystyle 2k\mathrm{H}$ $\displaystyle>2A^{T}A+q_{1}I+\mathrm{H}^{2}$
(38)

allows to choose $k$ such that $\mathrm{S}_{\mathrm{Schur}}>0$. Post
multiplying (38) by $(2H)^{-1}$ yields the following:

$\displaystyle 2kI$
$\displaystyle>2A^{T}A(2H)^{-1}+q_{1}(2H)^{-1}+\mathrm{H}^{2}(2H)^{-1}$
$\displaystyle kI$
$\displaystyle>A^{T}AH^{-1}+\frac{1}{2}q_{1}H^{-1}+\frac{1}{2}\mathrm{H}.$
(39)

Applying Courant-Fischer theorem [29] to (39) yields the following:

$\lambda_{max}(kI)>\lambda_{max}(A^{T}AH^{-1}+\frac{1}{2}q_{1}H^{-1}+\frac{1}{2}\mathrm{H}).$
(40)

Since $\lambda_{max}(kI)=k$, (40) has the following form:

$k>\lambda_{max}(A^{T}AH^{-1}+\frac{1}{2}q_{1}H^{-1}+\frac{1}{2}\mathrm{H}).$
(41)

By choosing $k$ as given in (41) ensures that $\mathrm{S}_{\mathrm{Schur}}>0$.
But $k$ must also satisfy (24), thus $k$ must be chosen such that the
following holds:

$k>\max\\{\sqrt{q_{2}},\lambda_{max}(A^{T}AH^{-1}+\frac{1}{2}q_{1}H^{-1}+\frac{1}{2}\mathrm{H})\\}$
(42)

ensures that both (24) and (40) are met. If $k$ is chosen according to (42),
then $\mathrm{S}_{\mathrm{Schur}}>0$ holds such that there exists a
$\nu\geq\frac{q_{1}}{2}$ which implies that

$\langle\mathbb{H}_{G_{r}(z)}v,v\rangle_{r}\geq\nu\|v\|^{2}_{r},\forall v\in
T_{z}\mathcal{M}.$ (43)

Hence it follows that

$\langle{G}_{r}(z_{1})-{G}_{r}(z_{2}),z_{1}-z_{2}\rangle_{r}\geq\nu\|z_{1}-z_{2}\|^{2}_{r}.$
(44)

Hence it is proved that $G_{r}(z)$ is strongly monotone. ∎

####  II-B3 Exponential stability

Without loss of generality, let us define $G_{r}(z)$ similar to (9) as
follows:

$G_{r}(z)=\begin{bmatrix}\nabla^{r}_{x}{L}(x,\lambda)\\\
-\nabla^{r}_{\lambda}{L}(x,\lambda)\end{bmatrix},$ (45)

where ${L}(x,\lambda)$ would represent the modified Lagrangian function whose
gradient vector field is given by $G_{r}(z)$. Since, $G_{r}(z)$ is strongly
monotone on $\mathcal{M}$, (32) will converge to a unique saddle-point
solution $z^{*}$.

###### Theorem II.5

Let $G_{r}(z)$ be Lipschitz continuous on an open set including $\mathcal{M}$,
then inequality (44) and $\alpha>0$, imply that the system (32) with
$z(0)\in\mathcal{M}$ is globally exponentially stable at the unique solution
$z^{*}$ of (4).

###### Proof:

For each $z(0)\in\mathcal{M}$, there exists a unique solution $z(t)$ of (10),
that started from $z(0)$. If $[0,t_{f})$ is the maximal interval of $z(t)$,
then from Lemma IV.4, $z(t)\in\mathcal{M}$ for all $t\in[0,t_{f})$. Since
$G_{r}(z)$ is strongly monotone, the following holds:
$L(x^{*},\lambda^{*})-L(x^{*},\lambda)>0,L(x,\lambda^{*})-L(x^{*},\lambda^{*})>0$
Let us define the Lyapunov function for the dynamics (32) as follows:

$\displaystyle V_{1}(z)$
$\displaystyle=(L(x^{*},\lambda^{*})-L(x^{*},\lambda))+(L(x,\lambda^{*})-L(x^{*},\lambda^{*}))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1}{2}\|z-z^{*}\|^{2}_{r}.$ (46)

It is to be noted that $V_{1}(z)$ possesses a similar structure as that of
$V(z)$ defined in (15), it is also differentiable convex on $\mathcal{M}$,
with $V_{1}(z)\geq\frac{1}{2}\|z-z^{*}\|^{2}_{r},\forall z\in\mathcal{M}$,
thus bounding all level sets of $V_{1}(z)$.

Differentiating $V_{1}(z)$ along the trajectories of (32) yields:

$\displaystyle\dot{V}_{1}(z)$ $\displaystyle=\nabla V_{1}(z)\dot{z}$
$\displaystyle=-\langle\nabla L(x,\lambda^{*})-\nabla
L(x^{*},\lambda)+z-z^{*},z-\tilde{z}\rangle_{r}$ $\displaystyle=-\langle
G_{r}(z)+z-z^{*},z-\tilde{z}\rangle_{r}$ (47)

Substituting $u=z-\alpha G_{r}(z)$ and $z=z^{*}$ in (11), yields

$\langle z-z^{*}+\alpha
G_{r}(z),z-\tilde{z}\rangle_{r}\geq\|z-\tilde{z}\|^{2}_{r}+\langle\alpha(z-z^{*}),G_{r}(z)\rangle_{r}.$
(48)

Using (48) in (47) yields,

$\dot{V}_{1}(z)\leq-\langle\alpha(z-z^{*}),G_{r}(z)\rangle_{r}.$ (49)

If $k$ is chosen such that the condition (40) is satisfied then $G_{r}(z)$ is
strongly monotone. The strong monotonicity of $G_{r}(z)$ leads to the
following property of the Lagrangian function $L(z),\forall z\in\mathcal{M}$,

$\displaystyle\langle z-z^{*},G_{r}(z)\rangle_{r}$ $\displaystyle\geq
L(x,\lambda^{*})-L(x^{*},\lambda)+\frac{\nu}{2}\|z-z^{*}\|^{2}_{r},z\in\mathcal{M}.$
(50)

Using (50), (49) modifies to the following

$\displaystyle\dot{V}_{1}(z)$
$\displaystyle\leq-\langle\alpha(z-z^{*}),G_{r}(z)\rangle_{r},$
$\displaystyle\leq-\alpha\beta[L(x,\lambda^{*})-L(x^{*},\lambda)+\frac{\nu}{2}\|z-z^{*}\|^{2}_{r}],$
$\displaystyle\leq-\alpha\beta[(L(x^{*},\lambda^{*})-L(x^{*},\lambda))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+(L(x,\lambda^{*})-L(x^{*},\lambda^{*}))+\frac{\nu}{2}\|z-z^{*}\|^{2}_{r}].$
(51)

With $\alpha,\beta>0$, it can be shown that,

$\dot{V}_{1}(z)\leq-\beta\min\\{1,\alpha\nu\\}V(z).$

Thus, it is proved that the system (10) is exponentially stable at the unique
solution $z^{*}$ of (4). Therefor,

$\|z-z^{*}\|_{r}\leq ce^{-\beta\frac{\min\\{1,\alpha\nu\\}}{2}t}$

where $c=\sqrt{2V_{1}(z(0))}$.

Further, if $G_{r}(z)$ is Lipschitz continuous on $\mathcal{M}$, i.e.,
$\|{G}_{r}(z_{1})-{G}_{r}(z_{2})\|_{r}\leq\mathbf{L}\|z_{1}-z_{2}\|_{r},\forall
z_{1},z_{2}\in\mathcal{M}$, where $\mathbf{L}$ is a Lipschitz constant then by
using [27, Theorem 4] the global exponential stability of the projected PD
dynamics can be established as follows:

$\|z(t)-z^{*}\|_{r}\leq\|z(0)-z^{*}\|_{r}e^{\frac{-\alpha\beta(4\nu-\alpha\ell^{2})}{8}t},\forall
t\geq 0.$ (52)

If $\alpha<\frac{4\nu}{\mathbf{L}^{2}}$, it follows that the projected PD
dynamics (32) is globally exponentially stable. ∎

##  III Simulation Results

This section presents simulation studies of the projected PD dynamics (32). It
is known that the Euler discretization of the exponentially stable dynamical
system owns geometric rate of convergence [30] for sufficiently small step-
sizes. The projected PD dynamics (32) is Euler discretized with a step size
$s>0$ and the following discrete-time projected PD dynamics[20] is obtained.

$z(\tau+1)=\beta P^{r}_{\mathcal{M}}\\{z(\tau)-\alpha G_{r}(\tau)\\}.$ (53)

First example (Example 1) considers an optimization problem of the form (1)
with $m=5$ and $n=10$. The Hessian matrix is assumed to be $H=20{I}$ with $A$
and $b$ taken as Gaussian random matrix and vector respectively. The distance
to the primal optimizer $x^{*}$ for different values of parameter $k$ is shown
in Fig. 1, where
$\varrho=\max\\{\sqrt{q_{2}},\lambda_{max}(A^{T}AH^{-1}+\frac{1}{2}q_{1}H^{-1}+\frac{1}{2}\mathrm{H})\\}$.
It can be seen from the plot that the rate of convergence to the equilibrium
point accelerates as the value of $k$ is increased. It implies that increasing
the value of $k$ allows increasing the value of $\nu$, which further increases
the coefficient of the negative exponential term in (52).

In the second example, an $L_{2}$ regularized least squares problem is
considered with $m=30$ and $n=50$. The objective function is
$f(x)=\|Cx-d\|^{2}_{2}+\frac{\theta}{2}\|x\|^{2}_{2}$ with $\theta>0$,
constrained to $Ax\leq b$. Matrices $(C,A)\in\mathbb{R}^{m\times n}$, and
vectors $(d,b)\in\mathbb{R}^{m\times 1}$ are Gaussian random matrices and
vectors, respectively. Parameters $\alpha,\beta$ are chosen as unity and the
proposed dynamics (32) is simulated for $k=1000\max(\varrho)$. A sketch of the
error norm as a function of time is shown in Fig. 2. It can be seen that the
error norm $\|x_{i}-x^{*}_{i}\|^{2}$ has geometric rate of convergence.

Figure 1: Distance to the primal optimizer $x^{*}$ for different values of $k$
(Example 1). Figure 2: Distance to the primal optimizer $x^{*}$ ($L_{2}$
regularized least squares problem).

##  IV Conclusions and discussion

In this paper we proposed a Riemannian geometric framework with natural
gradient adaptation to achieve exponentially convergent projected primal-dual
dynamics when applied to linear inequality constrained optimization problems.
We began by framing the proposed dynamics in a fiber-bundle setting endowed
with a Riemannian metric $R$ that captures the geometry of the gradient
vector. The metric $R$ induced a unique decomposition of the target space into
a horizontal and vertical distributions. The natural gradient proved to be
strongly monotone on $\mathcal{M}$ leading to an exponentially stable saddle-
point solution. We further showed that the increasing values of $k$ result in
much steeper gradient that leads to an accelerated convergence to the saddle-
point solution.

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## Appendix

###### Definition IV.1

(The Variational Inequality Problem, [20])
For a closed convex set $X\in\mathbb{R}^{n}$ and vector function
$F:X\rightarrow\mathbb{R}^{n}$, the finite dimensional variational inequality
problem, $\mathrm{VI(F,X)}$, is to determine a vector $x^{*}\in X$ such that

$(x-x^{*})^{T}F(x^{*})\geq 0,\leavevmode\nobreak\ \forall x\in X.$ (54)

A variational inequality problem (54) is equivalent to a fixed point problem
given below:

###### Proposition IV.1

(A Fixed Point Problem,[20])
$x^{*}$ is a solution to $\mathrm{VI(F,X)}$ if and only if for any $\alpha>0$,
$x^{*}$ is a fixed point of the projection map:

$x^{*}=P_{X}(x^{*}-\alpha F(x^{*}))$ (55)

where

$P_{X}=\arg\min_{v\in X}\|x-v\|.$ (56)

###### Theorem IV.2

(Uniqueness of the Solution to Variational Inequality, [20])
Suppose that $F(x)$ is strongly monotone on $X$. Then there exists precisely
one solution $x^{*}$ to $\mathrm{VI(F,X)}$.

Consider the following globally projected dynamical system proposed in [19]:

$\dot{x}=\beta\\{P_{X}[x-\alpha F(x)]-x\\}$ (57)

where $\beta,\alpha$ are positive constants and
$P_{X}:\mathbb{R}^{n}\rightarrow X$ is a projection operator as defined in
(56).

###### Remark 3

([27])
$x^{*}$ is an equilibrium point of (57) if and only if $x^{*}$ is a solution
of the variational inequality problem (54).

From Remark 3,

$\dot{x}=0\implies x^{*}=P_{X}(x^{*}-\alpha F(x^{*})).$

###### Definition IV.2

(Monotone Map,[26])
A mapping $F$ is monotone on $X\subseteq\mathbb{R}^{n}$, if for every pair of
distinct points $x,y\in X$, we have

$(y-x)^{T}(F(y)-F(x))\geq 0.$

###### Definition IV.3

(Strongly Monotone Map,[26])
A mapping $F$ is strongly monotone on $X\subseteq\mathbb{R}^{n}$, if there
exists $\mu>0$ such that, for every pair of distinct points $x,y\in X$, we
have

$(y-x)^{T}(F(y)-F(x))\geq\mu\|x-y\|^{2}.$

The relation between monotonicity of $F$ and positive definiteness of its
Jacobian matrix

$\displaystyle\nabla F(x)=\Bigg{(}\frac{\partial F_{i}(x)}{\partial
x_{j}}\Bigg{)}_{i,j=1,2,\ldots,n},$

as given below.

###### Proposition IV.3

((Strongly) Positive Definite Jacobian of $F(x)$ implies (Strongly) Monotone
$F(x)$,[20])
Suppose that $F$ is continuously differentiable on $X$.

  1. 1.

If the Jacobian matrix $\nabla F(x)$ is positive semidefinite, i.e.,

$\displaystyle y^{T}\nabla F(x)y\geq 0,\forall y\in\mathbb{R}^{n},x\in X,$

then $F$ is monotone on $X$.

  2. 2.

If the Jacobian matrix $\nabla F(x)$ is positive definite, i.e.,

$\displaystyle y^{T}\nabla F(x)y>0,\forall y\in\mathbb{R}^{n},x\in X,$

then $F$ is strictly monotone on $X$.

  3. 3.

If $\nabla F(x)$ is strongly positive definite, i.e.,

$\displaystyle y^{T}\nabla F(x)y\geq\mu\|z\|^{2},\forall
y\in\mathbb{R}^{n},x\in X,$

then $F(x)$ is strongly monotone on $X$.

####  IV-1 Splitting of the tangent vector

Consider the tangent vector $(\dot{x},\dot{\lambda})\in\mathbb{R}^{n+m}$, with
the Euclidean metric $\mathbb{I}$, then

$v_{H}=(\dot{x},0)$ and $v_{V}=(0,\dot{\lambda})$ such that

$\displaystyle\langle
v_{H},v_{V}\rangle_{I}=\begin{bmatrix}\dot{x}&\mathrm{0}\end{bmatrix}\begin{bmatrix}I&\mathrm{0}\\\
\mathrm{0}&I\end{bmatrix}\begin{bmatrix}\mathrm{0}\\\
\dot{\lambda}\end{bmatrix}$ (58)

i.e., $v_{H}\perp v_{V}$ and $(\dot{x},\dot{\lambda})=v_{H}\oplus v_{V}=0$.

###### Remark 4

Given a tangent vector $v=(\dot{x},\dot{\lambda})\in T_{\mathcal{M}}$,
orthogonality of $v_{V}$ and $v_{H}$ is preserved under the new metric $R$.

###### Proof:

From (26), we observe that the splitting of the vectors is only dependent on
$m_{12}$ or $m_{21}$, and $m_{22}$ in any case. Let us rewrite $R$ as
$R=\begin{bmatrix}m_{11}&m_{12}\\\ m_{21}&m_{22}\end{bmatrix}$. Now, the
relevant splitting of the tangent vectors $(\dot{x},\dot{\lambda})$ confirming
the structure of $R$ defined in (27) is as follows:

$\displaystyle(\dot{x},\dot{\lambda})$ $\displaystyle=v_{H}\oplus v_{V}$ (59)
$\displaystyle=\Big{(}\dot{x},-{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}\oplus\Big{(}0,\dot{\lambda}+{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}$
(60)
$\displaystyle=\begin{bmatrix}\dot{x}&-{m^{-1}_{22}}{m_{21}}\dot{x}\end{bmatrix}\begin{bmatrix}m_{21}\Big{(}\dot{\lambda}+{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}\\\
m_{22}\Big{(}\dot{\lambda}+{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}\end{bmatrix}$
(61)
$\displaystyle=m_{21}\dot{x}\Big{(}\dot{\lambda}+{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}-m_{21}\dot{x}\Big{(}\dot{\lambda}+{m^{-1}_{22}}{m_{21}}\dot{x}\Big{)}$
(62) $\displaystyle=0$ (63)

∎

###### Remark 5

As the value of parameter $k$ increases the vertical component in $v_{H}$,
i.e., $-{m^{-1}_{22}}{m_{21}}\dot{x}$ decreases in value, which causes the
trajectories off-manifold $\mathcal{M}$ to approach $\mathcal{M}$ faster.

###### Lemma IV.4

[31] Assume that $F$ is locally Lipschitz continuous in a domain $D$ that
contains $X$. Then the solution $x(t)$ of (57) will approach exponentially the
feasible set $X$ when the initial point $x^{0}\notin X$. Moreover, if
$x^{0}\in X$, then $x(t)\in X$.


# Lebesgue and Vilenkin-Lebesgue points and a.e. Convergence of T means with
respect to Vilenkin systems of integrable functions

D. Baramidze, N. Gogolashvili and N. Nadirashvili  D. Baramidze, The
University of Georgia, School of science and technology, 77a Merab Kostava St,
Tbilisi 0128, Georgia<EMAIL_ADDRESS>N. Gogolashvili, The
University of Georgia, School of science and technology, 77a Merab Kostava St,
Tbilisi 0128, Georgia<EMAIL_ADDRESS>N. Nadirashvili, The
University of Georgia, School of science and technology, 77a Merab Kostava St,
Tbilisi 0128, Georgia<EMAIL_ADDRESS>
###### Abstract.

In this paper we derive converge of $T$ means of Vilenkin-Fourier series with
monotone coefficients of integrable functions in Lebesgue and Vilinkin-
Lebesgue points. Moreover, we also derive pointwise and norm convergence in
$L_{p}$ norms of such $T$ means.

2000 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Vilenkin systems, Vilenkin groups, $T$ means, Nörlund
means, a.e. convergence, Lebesgue points, Vilenkin-Lebesgue points.

##  1\. Introduction

The definitions and notations used in this introduction can be found in our
next Section.

It is well-known (see e.g. the books [32] and [37]) that there exists an
absolute constant $c_{p},$ depending only on $p,$ such that

$\left\|S_{n}f\right\|_{p}\leq c_{p}\left\|f\right\|_{p},\text{ \ when \
}p>1.$

On the other hand, (for details see [7, 8, 31, 43, 44, 45, 49]) boundedness
does not hold for $p=1.$ The analogue of Carleson’s theorem for Walsh system
was proved by Billard [3] for $p=2$ and by Sjölin [38] for $1<p<\infty$, while
for bounded Vilenkin systems by Gosselin [15]. For Walsh-Fourier series,
Schipp [33, 34, 37] gave a proof by using methods of martingale theory. A
similar proof for Vilenkin-Fourier series can be found in Schipp and Weisz
[35, 53] (see also [26] and [32]). In each proof, they show that the maximal
operator of the partial sums is bounded on $L_{p}$, i.e. there exists an
absolute constant $c_{p}$ such that

$\left\|S^{\ast}f\right\|_{p}\leq c_{p}\left\|f\right\|_{p},\text{ \ when \
}f\in L_{p},\text{ \ }p>1.$

Hence, if $f\in L_{p}(G_{m}),$ where $p>1,$ then $S_{n}f\to f,\ \ \text{a.e.
on }\ \ G_{m}.$ Stein [39] constructed the integrable function whose Vilenkin-
Fourier (Walsh-Fourier) series diverges almost everywhere. In [37] was proved
that there exists an integrable function whose Walsh-Fourier series diverges
everywhere. a.e convergence of subsequences of Vilenkin-Fourier series was
considered in [6], where was used methods of martingale Hardy spaces.

If we consider the following restricted maximal operator
$\widetilde{S}_{\\#}^{\ast}f:=\sup_{n\in\mathbb{N}}\left|S_{M_{n}}f\right|,$
we have weak $(1,1)$ type inequality for $f\in L_{1}(G_{m}).$ Hence, if $f\in
L_{1}(G_{m}),$ then $S_{M_{n}}f\to f,\ \text{a.e. on }\ G_{m}.$ Moreover, for
any integrable function it is known that a.e. point is Lebesgue point and for
any such point $x$ of integrable function $f$ we have that

(1) $S_{M_{n}}f(x)\to f(x),\ \text{as}\ n\to\infty,\ \text{for any Lebesgue
point}\ x\ \text{of }\ f\in L_{1}(G_{m}).$

In the one-dimensional case Yano [52] proved that

$\left\|\sigma_{n}f-f\right\|_{p}\rightarrow 0,\text{ \ \ \ when \ \ \ \
}n\rightarrow\infty,\text{ \ }(f\in L_{p}(G_{m}),\text{ \ }1\leq
p\leq\infty).$

If we consider the maximal operator of Féjer means

$\displaystyle\sigma^{\ast}f:=\sup_{n\in\mathbb{N}}\left|\sigma_{n}f\right|,$

then

$\displaystyle\lambda\mu\left\\{\sigma^{\ast}f>\lambda\right\\}\leq
c\left\|f\right\|_{1},\ \ \ f\in L_{1}(G_{m}),\ \ \lambda>0.$

This result can be found in Zygmund [56] for the trigonometric series, in
Schipp [36] and [12, 27, 28, 40, 41, 46, 47, 48]) for Walsh series and in Pál,
Simon [25] for bounded Vilenkin series (see also Weisz [54, 55]). The
boundedness does not hold from Lebesgue space $L_{1}(G_{m})$ to the space
$L_{1}(G_{m})$. The $\text{weak}-(1,1)$ type inequality follows that for any
$f\in L_{1}(G_{m}),$

$\sigma_{n}f(x)\to f(x),\ \ \ \text{a.e., as}\ \ \ n\to\infty.$

Moreover, in [11] (see also [10]) was proved that for any integrable function,
a.e. point is Vilenkin-Lebesgue points and for any such point $x$ of
integrable function $f$ we have that

$\sigma_{n}f(x)\to f(x),\ \ \text{as}\ \ n\to\infty.$

Móricz and Siddiqi [18] investigate approximation properties of some special
Nörlund means of Walsh-Fourier series of $L_{p}$ functions in norm. Similar
results for the two-dimensional case can be found in Nagy [19, 20], Nagy and
Tephnadze [21, 22, 23, 24], Gogolashvili and Tephnadze [13, 14] (see also [2],
[17]). Approximation properties of general summability methods can be found in
[4, 5]. Fridli, Manchanda and Siddiqi [9] improved and extended results of
Móricz and Siddiqi [18] to Martingale Hardy spaces. The a.e. convergence of
Nörlund means of Vilenkin-Fourier series with monotone coefficients of $f\in
L_{1}$ was proved in [29] (see also [30]). In [50] was proved that the maximal
operators of $T$ means $T^{\ast}$ defined by
$T^{\ast}f:=\sup_{n\in\mathbb{N}}\left|T_{n}f\right|$ either with non-
increasing coefficients, or non-decreasing sequence satisfying condition

(2) $\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \ }\
n\rightarrow\infty$

are bounded from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}$.
Moreover, there exists a martingale and such $T$ means for which boundedness
does not hold from the Hardy space $H_{p}$ to the space $L_{p}$ when $0<p\leq
1/2.$

One of the most well-known mean of $T$ means is the Riesz summability. In [42]
(see also [16]) it was proved that the maximal operator of Riesz logarithmic
means

$\displaystyle R^{\ast}f:=\sup_{n\in\mathbb{N}}\left|R_{n}f\right|$

is bounded from the Hardy space $H_{1/2}$ to the space $weak-L_{1/2}$ and is
not bounded from $H_{p}$ to the space $L_{p},$ for $0<p\leq 1/2.$ There was
also proved that Riesz summability has better properties than Fejér means.

In this paper we derive convergence of $T$ means of Vilenkin-Fourier series
with monotone coefficients of integrable functions in Lebesgue and Vilinkin-
Lebesgue points.

This paper is organized as follows: In order not to disturb our discussions
later on some definitions and notations are presented in Section 2. For the
proofs of the main results we need some auxiliary Lemmas, some of them are new
and of independent interest. These results are presented in Section 3. The
main results and some of its consequences and detailed proofs are given in
Section 4.

##  2\. Definitions and Notation

Denote by $\mathbb{N}_{+}$ the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$ Let $m:=(m_{0,}$ $m_{1},...)$ be a
sequence of the positive integers not less than 2. Denote by

$Z_{m_{k}}:=\\{0,1,...,m_{k}-1\\}$

the additive group of integers modulo $m_{k}$.

Define the Vilenkin group $G_{m}$ as the complete direct product of the groups
$Z_{m_{i}}$ with the product of the discrete topologies of $Z_{m_{j}}`$s (for
details see [51]). T In this paper we discuss bounded Vilenkin groups, i.e.
the case when $\sup_{n}m_{n}<\infty.$ The direct product $\mu$ of the measures
$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \ }(j\in Z_{m_{k}})$ is the Haar
measure on $G_{m}$ with $\mu\left(G_{m}\right)=1.$ The elements of $G_{m}$ are
represented by sequences

$x:=\left(x_{0},x_{1},...,x_{j},...\right),\ \left(x_{j}\in Z_{m_{j}}\right).$

It is easy to give a basis for the neighborhoods of $G_{m}:$

$I_{0}\left(x\right):=G_{m},\text{ \ }I_{n}(x):=\\{y\in G_{m}\mid
y_{0}=x_{0},...,y_{n-1}=x_{n-1}\\},\ \text{ where }\ x\in G_{m},\
n\in\mathbb{N}.$

If we define the so-called generalized number system based on $m$ in the
following way:

$M_{0}:=1,\ M_{k+1}:=m_{k}M_{k}\,\,\,\ \ (k\in\mathbb{N}),$

then every $n\in\mathbb{N}$ can be uniquely expressed as
$n=\sum_{j=0}^{\infty}n_{j}M_{j},$ where $n_{j}\in Z_{m_{j}}$
$(j\in\mathbb{N}_{+})$ and only a finite number of $n_{j}`$s differ from zero.

We introduce on $G_{m}$ an orthonormal system which is called the Vilenkin
system. First, we define complex-valued function
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions, by

$r_{k}\left(x\right):=\exp\left(2\pi ix_{k}/m_{k}\right),\text{
}\left(i^{2}=-1,x\in G_{m},\text{ }k\in\mathbb{N}\right).$

Next, we define the Vilenkin system $\,\,\,\psi:=(\psi_{n}:n\in\mathbb{N})$ on
$G_{m}$ by:

$\psi_{n}(x):=\prod\limits_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right),\,\,\ \
\,\left(n\in\mathbb{N}\right).$

Specifically, we call this system the Walsh-Paley system when $m\equiv 2.$

The norms (or quasi-norms) of the spaces $L_{p}(G_{m})$ and $weak-
L_{p}\left(G_{m}\right)$ $\left(0<p<\infty\right)$ are respectively defined by

$\left\|f\right\|_{p}^{p}:=\int_{G_{m}}\left|f\right|^{p}d\mu,\text{
}\left\|f\right\|_{weak-
L_{p}}^{p}:=\underset{\lambda>0}{\sup}\lambda^{p}\mu\left(f>\lambda\right)<+\infty.$

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)$
(see [51]).

Now, we introduce analogues of the usual definitions in Fourier-analysis. If
$f\in L_{1}\left(G_{m}\right)$ we can define Fourier coefficients, partial
sums and Dirichlet kernels with respect to the Vilenkin system in the usual
manner:

$\widehat{f}\left(n\right):=\int_{G_{m}}f\overline{\psi}_{n}d\mu,\ \ \ \ \ \ \
S_{n}f:=\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k},\text{ \ \
}D_{n}:=\sum_{k=0}^{n-1}\psi_{k\text{ }},\text{ \ \
}\left(n\in\mathbb{N}_{+}\right).$

Recall that

(3) $\displaystyle\int_{G_{m}}D_{n}(x)dx=1,$ (4) $\displaystyle
D_{M_{n}-j}(x)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{D}_{j}(x),\ j<M_{n}.$

The convolution of two functions $f,g\in L_{1}(G_{m})$ is defined by

$\left(f\ast
g\right)\left(x\right):=\int_{G_{m}}f\left(x-t\right)g\left(t\right)dt\text{ \
\ }\left(x\in G_{m}\right).$

It is easy to see that if $f\in L_{p}\left(G_{m}\right),$ $g\in
L_{1}\left(G_{m}\right)$ and $1\leq p<\infty.$ Then $f\ast g\in
L_{p}\left(G_{m}\right)$ and

(5) $\left\|f\ast g\right\|_{p}\leq\left\|f\right\|_{p}\left\|g\right\|_{1}.$

Let $\\{q_{k}:k\geq 0\\}$ be a sequence of nonnegative numbers. The $n$-th
Nörlund mean $t_{n}$ for a Fourier series of $f$ is defined by

(6) $t_{n}f=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}q_{n-k}S_{k}f,\ \
\ \text{where}\ \ \ Q_{n}:=\sum_{k=0}^{n-1}q_{k}.$

It is obvious that

$t_{n}f\left(x\right)=\underset{G_{m}}{\int}f\left(t\right)F_{n}\left(x-t\right)d\mu\left(t\right),\
\ \ \text{where}\ \ \
F_{n}:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}D_{k}$

is called $T$ kernel.

######  Proposition 1.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers.
Then, for any $n,N\in\mathbb{N_{+}}$,

(7) $\displaystyle\int_{G_{m}}F_{M_{n}}(x)d\mu(x)=1,$ (8)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F_{M_{n}}(x)\right|d\mu(x)\leq
c<\infty,$ (9) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F_{M_{n}}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$

Let $\\{q_{k}:k\geq 0\\}$ be a sequence of non-negative numbers. The $n$-th
$T$ means $T_{n}$ for a Fourier series of $f$ are defined by

(10) $T_{n}f:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}S_{k}f,\
\ \ \text{where}\ \ \ Q_{n}:=\sum_{k=0}^{n-1}q_{k}.$

It is obvious that

$T_{n}f\left(x\right)=\underset{G_{m}}{\int}f\left(t\right)F^{-1}_{n}\left(x-t\right)d\mu\left(t\right),\
\ \text{ where}\ \
F^{-1}_{n}:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}D_{k}$

is called the $T$ kernel.

We always assume that $\\{q_{k}:k\geq 0\\}$ is a sequence of non-negative
numbers and $q_{0}>0.$ Then the summability method (10) generated by
$\\{q_{k}:k\geq 0\\}$ is regular if and only if

$\lim_{n\rightarrow\infty}Q_{n}=\infty.$

It is easy to show that, for any real numbers $a_{1},\dots,a_{m},$
$b_{1},\dots,b_{m}$ and $a_{k}=A_{k}-A_{k-1},$ $k=n,\dots,m,$ we have so
called Abel transformation:

$\displaystyle\sum_{k=m}^{n}a_{k}b_{k}$ $\displaystyle=$ $\displaystyle
A_{n}b_{n}-A_{m-1}b_{m}+\sum_{k=m}^{n-1}A_{k}(b_{k}-b_{k+1}).$

For $a_{j}=A_{j}-A_{j-1},\ j=1,...,n,$ if we invoke Abel transformations

(11) $\displaystyle\overset{n-1}{\underset{j=1}{\sum}}a_{j}b_{j}$
$\displaystyle=$ $\displaystyle
A_{n-1}b_{n-1}-A_{0}b_{1}+\overset{n-2}{\underset{j=0}{\sum}}A_{j}(b_{j}-b_{j+1}),$

For $b_{j}=q_{j}$, $a_{j}=1$ and $A_{j}=j$ for any $j=0,1,...,n$ we get the
following identity:

(12) $\displaystyle Q_{n}$ $\displaystyle=$
$\displaystyle\overset{n-1}{\underset{j=0}{\sum}}q_{j}=q_{0}+\overset{n-1}{\underset{j=1}{\sum}}q_{j}=q_{0}+\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}{(n-1)},$

Moreover, if use $D_{0}=K_{0}=0\ \text{for any}\ x\in G_{m}$ and invoke Abel
transformations (11) for $b_{j}=q_{j}$, $a_{j}=D_{j}$ and $A_{j}=jK_{j}$ for
any $j=0,1,...,n-1$ we get identities:

(13) $\displaystyle\ \ \
F_{n}^{-1}=\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}q_{j}D_{j}=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)jK_{j}+q_{n-1}(n-1)K_{n-1}\right),$

Analogously, if use $S_{0}f=\sigma_{0}f=0,\ \text{for any}\ x\in G_{m}$ and
invoke Abel transformations (11) for $b_{j}=q_{j}$, $a_{j}=S_{j}$ and
$A_{j}=j\sigma_{j}$ for any $j=0,1,...,n-1$ we get identities:

(14) $\displaystyle
T_{n}f=\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}q_{j}S_{j}f=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)j\sigma_{j}f+q_{n-1}(n-1)\sigma_{n-1}f\right).$

If $q_{k}\equiv 1$ in (6) and (10) we respectively define the Fejér means
$\sigma_{n}$ and Fejér Kernels $K_{n}$ as follows:

$\sigma_{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f\,,\text{ \ \
}K_{n}:=\frac{1}{n}\sum_{k=1}^{n}D_{k}.$

It is well-known that (for details see [1])

(15) $n\left|K_{n}\right|\leq
c\sum_{l=0}^{\left|n\right|}M_{l}\left|K_{M_{l}}\right|$

and for any $n,N\in\mathbb{N_{+}}$,

(16) $\displaystyle\int_{G_{m}}K_{n}(x)d\mu(x)=1,$ (17)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|K_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (18) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|K_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$

The well-known example of Nörlund summability is the so-called
$\left(C,\alpha\right)$ mean (Cesàro means) for $0<\alpha<1,$ which are
defined by

$\sigma_{n}^{\alpha}f:=\frac{1}{A_{n}^{\alpha}}\overset{n}{\underset{k=1}{\sum}}A_{n-k}^{\alpha-1}S_{k}f,\
\ \ \text{where }\ \ \ A_{0}^{\alpha}:=0,\qquad
A_{n}^{\alpha}:=\frac{\left(\alpha+1\right)...\left(\alpha+n\right)}{n!}.$

We also consider the "inverse" $\left(C,\alpha\right)$ means, which is an
example of $T$ means:

$U_{n}^{\alpha}f:=\frac{1}{A_{n}^{\alpha}}\overset{n-1}{\underset{k=0}{\sum}}A_{k}^{\alpha-1}S_{k}f,\qquad
0<\alpha<1.$

Let $V_{n}^{\alpha}$ denote the $T$ mean, where $\left\\{q_{0}=0,\
q_{k}=k^{\alpha-1}:k\in\mathbb{N}_{+}\right\\},$ that is

$V_{n}^{\alpha}f:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=1}{\sum}}k^{\alpha-1}S_{k}f,\qquad
0<\alpha<1.$

The $n$-th Riesz logarithmic mean $R_{n}$ and the Nörlund logarithmic mean
$L_{n}$ are defined by

$R_{n}f:=\frac{1}{l_{n}}\sum_{k=1}^{n-1}\frac{S_{k}f}{k}\text{ \ \ \ and \ \ \
}L_{n}f:=\frac{1}{l_{n}}\sum_{k=1}^{n-1}\frac{S_{k}f}{n-k},\ \ \ \text{ where
}\ \ \ l_{n}:=\sum_{k=1}^{n-1}1/k.$

Up to now we have considered $T$ means in the case when the sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ is bounded but now we consider $T$ summabilities
with unbounded sequence $\\{q_{k}:k\in\mathbb{N}\\}$.

$B_{n}^{\alpha}$ defines the class of $T$ means with non-increasing
coefficients:

$B_{n}^{\alpha}f:=\frac{1}{Q_{n}}\sum_{k=1}^{n}\log^{\alpha}(k+1)S_{k}f.$

It is obvious that $\frac{n}{2}\log^{\alpha}\left(n/2\right)\leq Q_{n}\leq
n\log^{\alpha}n.$ It follows that

$\displaystyle\frac{1}{Q_{n}}=O\left(\frac{1}{n}\right)\rightarrow 0,\text{ \
as \ }n\rightarrow\infty\ \ \ \ \ \text{and}\ \ \ \ \
\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right)\rightarrow 0,\text{ \ as \
}n\rightarrow\infty.$

A point $x\in G_{m}$ is called a Lebesgue point of integrable function $f$ if

$\displaystyle\lim_{n\to
0}\frac{1}{|I_{n}(x)|}\int_{I_{n}(x)}\left|f(t)-f\left(x\right)\right|d\mu(t)=0.$

It is known that a.e. point $x\in G_{m}$ is a Lebesgue point of function $f$
and Fejér means $\sigma_{n}f$ of trigonometric Fourier series of $f\in
L_{1}(G_{m})$ converge to $f$ at each Lebesgue point. It is also known that if
$x\in G_{m}$ is point of continuity of function $f$ then it is Lebesgue point.

Let introduced the operator

$\displaystyle
W_{n}f(x):=\sum_{s=0}^{n-1}M_{s}\sum_{r_{s}=1}^{m_{s}-1}\int_{I_{n}(x-r_{s}e_{s})}\left|f(t)-f\left(x\right)\right|d\mu(t)$

A point $x\in G_{m}$ is a Vilenkin-Lebesgue point of $f\in L_{1}(G_{m}),$ if

$\lim_{n\rightarrow\infty}W_{n}f(x)=0.$

##  3\. Auxiliary lemmas

First we consider kernels of $T$ kernels with non-increasing sequences:

######  Lemma 1.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers,
satisfying the condition

(19) $\frac{q_{0}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \
}n\rightarrow\infty.$

Then, for some constant $c,$ we have

$\left|F^{-1}_{n}\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\}.$

###### Proof.

Let sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-increasing. Then, by using
(2) we get that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)\leq\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)+q_{n-1}\right)\leq\frac{q_{0}}{Q_{n}}\leq\frac{c}{n}.$

Hence, if we apply (15) and use the equalities (12) and (13) we immediately
get that

$\displaystyle\left|F^{-1}_{n}\right|\leq\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|\leq\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete by just combining the estimates above. ∎

######  Lemma 2.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers.
Then, for any $n,N\in\mathbb{N_{+}}$,

(20) $\displaystyle\int_{G_{m}}F^{-1}_{n}(x)d\mu(x)=1,$ (21)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F^{-1}_{n}(x)\right|d\mu(x)<\infty,$
(22) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F^{-1}_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty.$

###### Proof.

According to (3) we immediately get proof of (20). By using (17) combining
with (12) and (13) we get that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\int_{G_{m}}\left|K_{j}\right|d\mu+q_{n-1}(n-1)\int_{G_{m}}|K_{n-1}|d\mu\right)$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j+q_{n-1}(n-1)\right)<c<\infty.$

By using (18) and inequalities (12) and (13) we can conclude that

$\displaystyle\int_{G_{m}\backslash I_{N}}\left|F^{-1}_{n}\right|d\mu$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\int_{G_{m}\backslash
I_{N}}\left|K_{j}\right|d\mu+\frac{q_{n-1}(n-1)}{Q_{n}}\int_{G_{m}\backslash
I_{N}}|K_{n-1}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}+\frac{q_{n-1}(n-1)\alpha_{n-1}}{Q_{n}},$

where $\alpha_{n}\to 0,\ \ \text{as}\ \ n\to\infty.$ Since sequence is non-
increasing we can conclude that

$\frac{q_{n-1}(n-1)\alpha_{n-1}}{Q_{n}}\leq\alpha_{n-1}\to 0,\ \ \text{as}\ \
n\to\infty.$

On the other hand, $\alpha_{n}<A$ where $n\in\mathbb{N}$ and for any
$\varepsilon>0$ there exists $N_{0}\in\mathbb{N},$ such that
$\alpha_{n}<\varepsilon$ when $n>N_{0}.$ Hence,

$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}=\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}+\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}$

Since sequence is non-increasing, we can conclude that
$|q_{j}-q_{j+1}|<2q_{0},$

$\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}\leq\frac{2q_{0}N_{0}}{Q_{n}}\to
0,\ \ \ \text{as}\ \ \ n\to\infty$

and

$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}\leq\frac{\varepsilon}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j}-q_{j+1}\right)j\leq\frac{\varepsilon}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j<\varepsilon.$

The proof is complete. ∎

Next we consider kernels of $T$ means with non-decreasing sequences:

######  Lemma 3.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers,
satisfying the condition

(23) $\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \
}n\rightarrow\infty.$

Then for some constant $c,$

$\left|F^{-1}_{n}\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\}.$

###### Proof.

Since sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-decreasing if we apply
condition (23) we find that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}\right)=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)+q_{n-1}\right)\leq\frac{2q_{n-1}}{Q_{n}}\leq\frac{c}{n}.$

If we apply Abel transformation (13) combine with (15) and (3) we get that

$\displaystyle\left|F^{-1}_{n}\right|$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{j}-q_{j+1}\right|+q_{n-1}+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|\leq\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete. ∎

######  Lemma 4.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers,
satisfying the condition (23). Then, for some constant $c,$

(24) $\displaystyle\int_{G_{m}}F^{-1}_{n}(x)d\mu(x)=1,$ (25)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F^{-1}_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (26) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F^{-1}_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty.$

###### Proof.

If we compare estimation of $F_{n}$ in Lemma 2 with estimation of $F_{n}$ in
Lemma 3 we find that they are quite same. It follows that proof is analogical
to Lemma 2. ∎

Finally, we study some special subsequences of kernels of $T$ means:

######  Lemma 5.

Let $n\in\mathbb{N}$. Then

(27) $\displaystyle
F^{-1}_{M_{n}}(x)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{F}_{M_{n}}(x).$

###### Proof.

By using (4) we get that

$\displaystyle F^{-1}_{M_{n}}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}q_{k}D_{k}(x)=\frac{1}{Q_{M_{n}}}\overset{M_{n}}{\underset{k=1}{\sum}}q_{M_{n}-k}D_{M_{n}-k}(x)$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}}{\underset{k=1}{\sum}}q_{M_{n}-k}\left(D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{D}_{k}(x)\right)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{F}_{M_{n}}(x).$

The proof is complete. ∎

######  Corollary 1.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers. Then

(28) $\displaystyle\int_{G_{m}}F^{-1}_{M_{n}}(x)d\mu(x)=1,$ (29)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F^{-1}_{M_{n}}(x)\right|d\mu(x)\leq
c<\infty,$ (30) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F^{-1}_{M_{n}}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,\ \ \text{for any}\ \ N\in\mathbb{N_{+}}.$

###### Proof.

The proof is direct consequence of Proposition 1 and Lemma 5. ∎

##  4\. proof of main result

######  Theorem 1.

a) Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
increasing numbers. Then, for all $f\in L_{p}(G_{m})$,

$\|T_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty$

Let function $f\in L_{1}(G_{m})$ is continuous at a point $x.$ Then

${{T}_{n}}f(x)\to f(x),\ \ \ \text{as}\ \ \ n\to\infty.$

Moreover, for all Vilenkin-Lebesgue points of $f\in L_{p}(G_{m})$,

$\underset{n\rightarrow\infty}{\lim}T_{n}f(x)=f(x).$

b) Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
decreasing numbers satisfying the condition (23). Then, for all $f\in
L_{p}(G_{m})$,

$\|T_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty.$

Let function $f\in L_{1}(G_{m})$ is continuous at a point $x.$ Then

${{T}_{n}}f(x)\to f(x),\ \ \ \text{as}\ \ \ n\to\infty.$

Moreover, for all Vilenkin-Lebesgue points of $f\in L_{p}(G_{m})$,

$\underset{n\rightarrow\infty}{\lim}T_{n}f(x)=f(x).$

###### Proof.

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a non-increasing sequence. Lemma 2
immediately follows norm and pointwise convergence.

Suppose that $x$ is Vilenkin-Lebesgue point of function $f\in L_{p}(G_{m}).$
Then we have that
$\underset{n\rightarrow\infty}{\lim}|\sigma_{n}f(x)-f(x)|=0.$ Hence,

$\displaystyle|T_{n}f(x)-f(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j|\sigma_{j}f(x)-f(x)|+q_{n-1}(n-1)|\sigma_{n-1}f(x)-f(x)|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{j}-q_{j+1}\right)j\alpha_{j}+\frac{q_{n-1}(n-1)\alpha_{n-1}}{Q_{n}}:=I+II,\
\text{ where }\ \alpha_{n}\to 0,\ \text{as}\ n\to\infty.$

To prove $I\to 0,\ \text{as}\ n\to\infty$ and $II\to 0,\ \text{as}\
n\to\infty,$ we just have to analogous steps of Lemma 2. It follows that Part
a) is proved.

Let sequence is non-decreasing, satisfying condition (23). According to (25)
in Lemma 4 get norm and and pointwise convergence. To prove convergence in
Vilenkin-Lebesgue points we use estimation

$\displaystyle|T_{n}f(x)-f(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{j+1}-q_{j}\right)j\alpha_{j}+\frac{q_{n-1}(n-1)\alpha_{n}}{Q_{n}}=III+IV.$

where $\alpha_{n}\to 0,\ \text{as}\ n\to\infty.$ It is evident that

$IV\leq\frac{q_{n-1}(n-1)\alpha_{n}}{Q_{n}}\leq\alpha_{n}\to 0,\ \ \text{as}\
\ n\to\infty.$

On the other hand, for any $\varepsilon>0$ there exists $N_{0}\in\mathbb{N},$
such that $\alpha_{n}<\varepsilon/2$ when $n>N_{0}.$ We can write that

$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j\alpha_{j}=\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j\alpha_{j}+\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j+1}-q_{j}\right)j\alpha_{j}=III_{1}+III_{2}.$

Since sequence is non-decreasing, we can conclude that
$|q_{j+1}-q_{j}|<2q_{j+1}<2q_{n-1}.$ Hence,

$\displaystyle III_{1}\leq\frac{2q_{0}N_{0}}{Q_{n}}\to 0,\ \ \ \text{as}\ \ \
n\to\infty$

and

$\displaystyle III_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j+1}-q_{j}\right)j\alpha_{j}\leq\frac{\varepsilon(n-1)}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{j+1}-q_{j}\right)$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon(n-1)}{Q_{n}}\left(q_{n-1}-q_{n-N_{0}}\right)\leq\frac{2q_{n-1}\varepsilon(n-1)}{Q_{n}}<\varepsilon.$

The proof of part b) is also complete. ∎

######  Corollary 2.

Let $f\in L_{p},$ where $p\geq 1.$ Then

$\displaystyle R_{n}f$ $\displaystyle\rightarrow$ $\displaystyle f,\text{ \ \
\ a.e., \ \ \ \ as \ }n\rightarrow\infty,\ \ \ \ \ U_{n}^{\alpha}f\rightarrow
f,\text{ \ \ \ a.e., \ \ \ as \ }n\rightarrow\infty,\text{\ \ \ }$
$\displaystyle V_{n}^{\alpha}f$ $\displaystyle\rightarrow$ $\displaystyle
f,\text{ \ \ \ a.e., \ \ \ as \ }n\rightarrow\infty,\ \ \ \ \
B^{\alpha,\beta}_{n}f\rightarrow f,\text{ \ \ \ a.e., \ \ \ as \
}n\rightarrow\infty.$

######  Theorem 2.

Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing
numbers. Then, for all $f\in L_{p}(G_{m})$,

$\|T_{M_{n}}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty$

Moreover,

$\underset{n\rightarrow\infty}{\lim}T_{M_{n}}f(x)=f(x),\ \ \text{for all
Lebesgue points of}\ \ f\in L_{p}(G_{m}).$

###### Proof.

Corollary 1 immediately follows norm convergence. To prove a.e convergence we
use first identity in Lemma 5 to write that

$\displaystyle T_{M_{n}}f\left(x\right)$ $\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)F^{-1}_{n}\left(x-t\right)d\mu\left(t\right)$
$\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)D_{M_{n}}\left(x-t\right)d\mu\left(t\right)-\underset{G_{m}}{\int}f\left(t\right)\psi_{M_{n}-1}(x-t)\overline{F}_{M_{n}}(x-t)=I-II.$

By applying (1) we can conclude that $I=S_{M_{n}}f(x)\to f(x)$ for all
Lebesgue points of $f\in L_{p}(G_{m})$. By using
$\psi_{M_{n}-1}(x-t)=\psi_{M_{n}-1}(x)\overline{\psi}_{M_{n}-1}(t)$ we can
conclude that

$II=\psi_{M_{n}-1}(x)\underset{G_{m}}{\int}f\left(t\right)\overline{F}_{M_{n}}(x-t)\overline{\psi}_{M_{n}-1}(t)d(t)$

By combining (5) and Proposition 1 we find that function

$f\left(t\right)\overline{F}_{M_{n}}(x-t)\in L_{p}\ \ \text{ where}\ \ p\geq
1\ \ \text{for any }\ \ x\in G_{m},$

and $II$ is Fourier coefficients of integrable function. According to Riemann-
Lebesgue Lemma we get that $II\to 0\ \ \text{for any }\ \ x\in G_{m}.$ The
proof is complete. ∎

######  Corollary 3.

Let $f\in L_{1}(G_{m})$ is continuous at a point $x.$ Then $T_{n}f(x)\to
f(x),\ \ \text{as}\ \ n\to\infty.$

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