# Fejér-type positive operator based on Takenaka–Malmquist system on unit
circle

F.G. Abdullayev  Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic;
Mersin University, Turkey<EMAIL_ADDRESS>and  V.V. Savchuk
Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine
<EMAIL_ADDRESS>
###### Abstract.

Let $\varphi=\\{\varphi_{k}\\}_{k=-\infty}^{\infty}$ denote the extended
Takenaka–Malmquist system on unit circle $\mathbb{T}$ and let
$\sigma_{n,\varphi}(f),$ $f\in L^{1}(\mathbb{T})$, be the Fejér-type operator
based on $\varphi$, introduced by V. N. Rusak. We give the convergence
criteria for $\sigma_{n,\varphi}(f)$ in Banach space
$X(\mathbb{T}):=L^{p}(\mathbb{T})\vee C(\mathbb{T})$, $p\geq 1$. Also we prove
the Voronovskaya-type theorem for $\sigma_{n,\varphi}(f)$ on class of
holomorphic functions representable by Cauchy-type integrals with bounded
densities.

###### Key words and phrases:

Holomorphic functions, Takenaka–Malmquist system, Fejér type operator,
Blaschke product, Frostman condition.

###### 2020 Mathematics Subject Classification:

Primary 30C10, 30E10; Secondary 41A20, 41A36, 42C05

Research supported by the Kyrgyz-Turkish Manas University (Bishkek/Kyrgyz
Republic), Project No. KTMU-BAP-20I9.FBE,06.

Keywords and Phrases: Holomorphic functions, Takenaka–Malmquist system, Fejér
type operator, Blaschke product, Frostman condition.

2000 MS Classification: 30C10, 30E10, 41A20, 41A36, 42C05

##  1\. Introduction

Let $C(\mathbb{T})$ denote a Banach space of continuous functions on the unit
circle $\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$ equipped with the norm
$\|f\|_{C(\mathbb{T})}=\max_{t\in\mathbb{T}}|f(t)|.$

Let a function $f\in C(\mathbb{T})$ and let

$f(\mathrm{e}^{\mathrm{i}x})\sim\sum_{k\in\mathbb{Z}}c_{k}\mathrm{e}^{\mathrm{i}kx}$

be its trigonometric Fourier series. The Cesàro $(C,1)$ means of $f$ is
defined by

(1.1)
$\sigma_{0}(f)=f~{}\mbox{and}~{}\sigma_{n}(f)(\mathrm{e}^{\mathrm{i}x}):=\frac{1}{n}\sum_{k=1}^{n}S_{k}(f)(\mathrm{e}^{\mathrm{i}x}),~{}\mbox{if}~{}n=1,2,\ldots,$

where $S_{k}(f)(z):=\sum_{j=-k+1}^{k-1}c_{j}\mathrm{e}^{\mathrm{i}jx}$. The
trigonometric polynomials $\sigma_{n}(f)$ also are called the Fejér means of
$f$.

The famous L. Fejér theorem says:

$(i)$ for each $n=0,1,\ldots$, $\sigma_{n}(f)\geq 0$ on $\mathbb{T}$ if $f\geq
0$ on $\mathbb{T}$;
$(ii)$ for each $n=0,1,\ldots$ and $f\in C(\mathbb{T})$,
$\|\sigma_{n}(f)\|_{C(\mathbb{T})}\leq\|f\|_{C(\mathbb{T})}$;
$(iii)$ if $f\in C(\mathbb{T})$, then $\sigma_{n}(f)$ converge uniformly to
$f$ on $\mathbb{T}$ as $n\to\infty$.

The classical proof is based on the representation of $\sigma_{n}(f)$ by
convolution

(1.2) $\displaystyle\sigma_{n}(f)(\mathrm{e}^{\mathrm{i}x})$ $\displaystyle=$
$\displaystyle\left(f*K_{n}\right)(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\mathrm{e}^{\mathrm{i}t})K_{n}(x-t)dt,$

where

$K_{n}(t):=\frac{1}{n}\left(\frac{\sin\frac{nt}{2}}{\sin\frac{t}{2}}\right)^{2}$

is the approximative identity (Fejér kernel).

If the Fourier series of $f$ is of the power series type, i. e.

$f(\mathrm{e}^{\mathrm{i}x})\sim\sum_{k=0}^{\infty}c_{k}\mathrm{e}^{\mathrm{i}kx},$

then, it is readily verified that

(1.3) $\displaystyle\sigma_{n}(f)(z)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\left(1-\frac{k}{n}\right)c_{k}z^{k}$
$\displaystyle=$ $\displaystyle
S_{n}(f)(z)-\frac{z}{n}S_{n}^{\prime}(f)(z),\quad z\in\mathbb{D},$

where $f(z)=\sum_{k=0}^{\infty}c_{k}z^{k}$ is holomorphic function in the unit
disc $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ and $S_{n}(f)(z)$ its partial
Taylor’s sums. In this case we have the following assertion:

$(iv)$ $n(f(z)-\sigma_{n}(f)(z))/z$ converge to $f^{\prime}(z)$ uniformly on
compact sets of $\mathbb{D}$.

The assertion $(iv)$ is a Voronovskaja type theorem for $\sigma_{n}(f)$ in
holomorphic case. See [6] and [15] for further generalization.

In this paper we are interested in extending of $(i)$–$(iv)$ on Fejér type
means of Fourier series expansion based on Takenaka–Malmquist orthonormal
system.

For more precisely formulation of our goals, we need some notations and some
well known facts.

Let $\mathcal{H}$ denot the set of all functions holomorphic in $\mathbb{D}$
and let $dm$ be the normalized Lebesgue measure on $\mathbb{T}$. For $1\leq
p\leq\infty$, the Hardy space $H^{p}$ consist of those $f\in\mathcal{H}$ for
which

$+\infty>\|f\|_{p}:=\begin{cases}\displaystyle\sup_{\rho\in[0,1)}\left(\int_{\mathbb{T}}|f(\rho
t)|^{p}dm(t)\right)^{1/p},\hfill&1\leq
p<\infty,\cr\sup_{z\in\mathbb{D}}|f(z)|,\hfill&p=\infty.\end{cases}$

It is well known that each function $f$ from the space $H^{p}$ has non-
tangential limits almost everywhere on the circle $\mathbb{T}$ and, moreover,
these limit values form a measurable function belonging to
$L^{p}(\mathbb{T})$. This function we will also denote by $f$ and note that

$\displaystyle\|f\|_{L^{p}(\mathbb{T})}$ $\displaystyle:=$
$\displaystyle\left(\int_{\mathbb{T}}|f|^{p}dm\right)^{1/p}$ $\displaystyle=$
$\displaystyle\|f\|_{p},\quad 1\leq p<\infty,$

and

$\|f\|_{L^{\infty}(\mathbb{T})}=\mathrm{ess}\sup_{t\in\mathbb{T}}|f(t)|.$

For a given system of points ${\bf a}:=\\{a_{k}\\}_{k=0}^{\infty},$
$a_{k}\in\mathbb{D}$ (the points $a_{k}$ are enumerated taking into account
their multiplicity), the Takenaka–Malmquist system
$\varphi=\\{\varphi_{k}\\}_{k=0}^{\infty}$ is defined by

(1.4)
$\varphi_{k}(z):=\frac{\sqrt{1-|a_{k}|^{2}}}{1-z\overline{a}_{k}}B_{k}(z),~{}k\in\mathbb{Z}_{+},$

where

$B_{k}(z):=\begin{cases}1,\hfill&\mbox{if}~{}k=0,\cr\displaystyle\prod_{j=0}^{k-1}\frac{z-a_{j}}{1-z\overline{a}_{j}},\hfill&\mbox{if}~{}k\in\mathbb{N},\end{cases}$

is Blaschke product for the disc $\mathbb{D}$ of degree $k$.

It is well known that $\varphi$ is an orthonormal and complete system in
$H^{2}$ if and only if $B_{n}(z)\to 0$ as $n\to\infty$ uniformly on compact
sets in $\mathbb{D}$. The last condition is equivalent to (see [1], p. 195)

(1.5) $\sum_{k=0}^{\infty}(1-|a_{k}|)=+\infty.$

So, if (1.5) is fulfilled, then for any function $f\in H^{2}$,

$\lim_{n\to\infty}\left\|f-\sum_{k=0}^{n}\langle
f,\varphi_{k}\rangle\varphi_{k}\right\|_{2}=0.$

Consequently, we have that for any $f\in H^{2}$ the Fourier series

(1.6) $\sum_{k=0}^{\infty}\langle f,\varphi_{k}\rangle\varphi_{k},$

where

(1.7) $\langle
f,\varphi_{k}\rangle:=\int_{\mathbb{T}}f\overline{\varphi_{k}}dm,\quad
k=0,1,\ldots,$

convergence to $f$ uniformly in $\mathbb{D}$ as well as in $H^{2}$–metric on
$\mathbb{T}$, provided (1.5).

Actually, we can extend the concept of Fourier series (1.6) on the space of
Cauchy transforms

$\mathcal{K}:=\left\\{f(z)=\int_{\mathbb{T}}\frac{d\mu(t)}{1-\overline{t}z}:z\in\mathbb{D},\mu\in
M\right\\},$

where $M$ is the space of finite complex Borel measures on $\mathbb{T}$. For
this one, we set

$\langle
f,\varphi_{k}\rangle:=\int_{\mathbb{T}}\overline{\varphi_{k}}d\mu,\quad
k=0,1,\ldots.$

In fact, we have that for $f\in\mathcal{K}$

$f(z)=\sum_{k=0}^{\infty}\langle f,\varphi_{k}\rangle\varphi_{k}(z),\quad
z\in\mathbb{D},$

provided $\varphi$ is complete Takenaka—Malmquist system. This follows
immediately from definition of $\mathcal{K}$ and the identity [3] (an analog
of the Christoffel–Darboux formula)

(1.8)
$\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi_{k}(t)}=\frac{1-B_{n}(z)\overline{B_{n}(t)}}{1-z\overline{t}},\quad
n\in\mathbb{N},~{}z,t\in\mathbb{C}.$

When a function $f$ is only defined on $\mathbb{T}$ and $f\in
L^{1}(\mathbb{T})$, then we will define $\langle f,\varphi_{k}\rangle$ for
every $k\in\mathbb{Z}$ in same manner as (1.7), where
$\varphi_{-k}(t)=\overline{t\varphi_{k-1}(t)}$ for $k\in\mathbb{N}$. In such
case we associate a function $f$ with the Fourier series

(1.9) $f\sim\sum_{k\in\mathbb{Z}}\langle f,\varphi_{k}\rangle\varphi_{k},$

based on extended Takenaka–Malmquist system
$\varphi=\\{\varphi_{k}\\}_{k\in\mathbb{Z}}$.

Through the paper, we let by $TMS$ denote the set of all Takenaka–Malmquist
systems (including extended).

The equations (1.1), (1.2) and (1.3) hint at three possible ways to generalize
of Fejér means based on $TMS$.

The first one is the Cesàro $(C,1)$ means of the series (1.9), that is

$\displaystyle\sigma_{n}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{n}S_{k}(f)$ $\displaystyle=$
$\displaystyle\sum_{k=-n+1}^{n-1}\left(1-\frac{|k|}{n}\right)\langle{f,\varphi_{k}}\rangle\varphi_{k},\quad
n=1,2,\ldots,$

where $S_{k}(f):=\sum_{j=-k+1}^{k-1}\langle{f,\varphi_{j}}\rangle\varphi_{j}$,
$k\in\mathbb{N}$, are the partial sums of the series in (1.9).

Unfortunately, in such case, the assertions $(i),(ii)$ and $(iv)$ are not
true.

For example, let $e_{0}(z):=1$ and $0<a_{k}<1$, $k=0,1,\ldots$. Then it
follows from (1.8), that

$e_{0}(t)\sim\sum_{k=0}^{\infty}\overline{\varphi_{k}(0)}\varphi_{k}(t),\quad
t\in\mathbb{T},$

provided (1.5) is satisfied.

Consequently,

$S_{k}(e_{0})(z)=1-\overline{B_{k}(0)}B_{k}(z),$

and therefore, by (1), we get

$\sigma_{n}(e_{0})(z)=1-\frac{1}{n}\sum_{k=1}^{n}\overline{B_{k}(0)}B_{k}(z).$

It follows that

$\displaystyle\|\sigma_{n}(e_{0})\|_{C(\mathbb{T})}$ $\displaystyle=$
$\displaystyle\left\|1-\frac{1}{n}\sum_{k=1}^{n}\overline{B_{k}(0)}B_{k}\right\|_{C(\mathbb{T})}$
$\displaystyle=$ $\displaystyle 1+\frac{1}{n}\sum_{k=1}^{n}B_{k}(0)B_{k}(-1)$
$\displaystyle=$ $\displaystyle
1+\frac{1}{n}\sum_{k=1}^{n}\left(\prod_{j=0}^{k-1}a_{j}\right)$
$\displaystyle>$ $\displaystyle\|e_{0}\|_{C(\mathbb{T})}.$

So, $(i)$ and $(ii)$ are not true in general.

Analogously, if we consider $e_{0}$ as a holomorphic function in $\mathbb{D}$,
we obtain that for all $z\in(-1,0]$ and for each natural $n$

$\displaystyle\left|n(e_{0}(z)-\sigma_{n}(e_{0})(z))-ze_{0}^{\prime}(z)\right|$
$\displaystyle=$ $\displaystyle\left|\sum_{k=1}^{n}B_{k}(0)B_{k}(z)\right|$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\left(\prod_{j=0}^{k-1}a_{j}\frac{|z|+a_{j}}{1+|z|a_{j}}\right)$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{n}\frac{1}{2^{k}}\left(\prod_{j=0}^{k-1}a_{j}\right)^{2}$
$\displaystyle>$ $\displaystyle 0.$

This gives that $(iv)$ is not true for all holomorphic functions.

The assertion $(iii)$ can be rescued at least for holomorphic functions. More
precisely, it was proved in [18] that $\sigma_{n}(f)$ converge uniformly to
the function $f$ in $\overline{\mathbb{D}}$ if the boundary function
$f(\mathrm{e}^{\mathrm{i}x})$ is continuous on $\mathbb{T}$, provided $a_{k}$
are all in a compact subset of $\mathbb{D}$ and satisfy certain (mild)
condition on the distribution.

In this paper we consider the other two cases of generalization of Fejér
means. Namely, we consider the operators $\sigma_{n,\varphi}$ and
$\sigma^{+}_{n,\varphi}$ defined on $L^{1}(\mathbb{T})$ and $\mathcal{K}$
respectively as follows:

$\sigma_{n,\varphi}(f)(\mathrm{e}^{\mathrm{i}x}):=\begin{cases}f(\mathrm{e}^{\mathrm{i}x}),\hfill&\mbox{if}~{}n=0,\cr\displaystyle\frac{1}{4\pi\gamma_{n}(x)}\int_{-\pi}^{\pi}f(\mathrm{e}^{\mathrm{i}y})\frac{\displaystyle\sin^{2}\left(\int_{x}^{y}\gamma_{n}(t)dt\right)}{\displaystyle\sin^{2}\frac{y-x}{2}}dy,\hfill&\mbox{if}~{}n\in\mathbb{N}\end{cases},\quad
x\in\mathbb{R},$

where

$\gamma_{n}(t):=\frac{1}{2}\sum_{k=0}^{n-1}\frac{1-|a_{k}|^{2}}{1-2|a_{k}|\cos(t-\arg
a_{k})+|a_{k}|^{2}},$

and

$\sigma^{+}_{n,\varphi}(f)(z):=\begin{cases}f(z),\hfil&\mbox{if}~{}n=0,\cr\displaystyle
S_{n}(f)(z)-\frac{B_{n}(z)}{B_{n}^{\prime}(z)}S^{\prime}_{n}(f)(z),\hfill&\mbox{if}~{}n\in\mathbb{N}\end{cases},\quad
z\in\mathbb{T},$

where

$S_{n}(f)=\sum_{k=0}^{n-1}\langle f,\varphi_{k}\rangle\varphi_{k}.$

The operator $\sigma_{n,\varphi}$ was introduced by V. Rusak [13]. We mention
[14, 10, 11, 12, 9] as general references for the approximation properties of
$\sigma_{n,\varphi}$ and their analogous.

As far as we know, the operator $\sigma^{+}_{n,\varphi}$ (based on TMS) is
considered here for the first time.

In case $a_{k}=0$, $k\in\mathbb{Z}_{+}$, the restrictions to $H^{1}$ of the
operators $\sigma_{n,\varphi}(f)$ and $\sigma^{+}_{n,\varphi}(f)$ coincide
with the usual Fejér means $\sigma_{n}(f)$ of trigonometric Fourier series of
$f\in H^{1}$ on the circle $\mathbb{T}$. Actually, as we will show later,
$\sigma_{n,\varphi}(f)=\sigma^{+}_{n,\varphi}(f)$ for all $n\in\mathbb{Z}_{+}$
and $\varphi\in TMS$, provided $f\in H^{1}$.

It should be remarked also that for $f\in\mathcal{K}$ the functions
$\sigma^{+}_{n,\varphi}(f)$, $n\in\mathbb{Z}_{+}$, are holomorphic in
$\mathbb{D}$, provided that $B_{n}/B^{\prime}_{n}$, $n\in\mathbb{Z}_{+},$ are
holomorphic in $\mathbb{D}$, and are meromorphic in $\mathbb{D}$ otherwise.
But in any case, we can consider the functions

$\delta_{n,\varphi}(f):=\begin{cases}0,\hfill&\mbox{if}~{}n=0,\cr\displaystyle\frac{B^{\prime}_{n}}{B_{n}}\left(f-\sigma^{+}_{n,\varphi}(f)\right),\hfill&\mbox{if}~{}n\in\mathbb{N},\end{cases}$

as the error functions of weighted approximation in $\mathbb{D}$ to $f\in
H^{1}$ by $\sigma^{+}_{n,\varphi}(f)$. In view of (1.2), it is natural to
conjecture that $\delta_{n,\varphi}(f)(z)\to f^{\prime}(z)$ uniformly on
compact sets in $\mathbb{D}$ as $n\to\infty$. We will show that this is really
true if (1.5) is satisfied.

Given these remarks, it is reasonable to studies the operators
$\sigma_{n,\varphi}(f)$ and $\sigma^{+}_{n,\varphi}(f)$ separately on the unit
circle $\mathbb{T}$ and in the unit disc $\mathbb{D}$ respectively.

Our goals are to solve the following two main problems.

######  Problem 1.1.

Let $X(\mathbb{T})$ be one of the space $C(\mathbb{T})$ and $L(\mathbb{T})$,
$1\leq p<\infty$. Find necessary and sufficient condition on $\varphi\in TMS$
in order that $\sigma_{n,\varphi}(f)$ converges in metric $X(\mathbb{T})$ to
$f$ for every function $f\in X(\mathbb{T})$.

######  Problem 1.2.

Let $K$ be a class of holomorphic functions in $\mathbb{D}$, $\varphi\in TMS$,
$n\in\mathbb{N}$ and $z\in\mathbb{D}$. Find the quantity

$\sup_{f\in K}|\delta_{n,\varphi}(f)(z)-f^{\prime}(z)|.$

##  2\. Main results

In this section we only state our main theorems and some corollaries, and the
proofs of theorems will be given in Section 3.

Our main result in part of uniform convergence of $\sigma_{n,\varphi}$ for
$C(\mathbb{T})$ is connected with the well–known Frostman theorem [5](see also
[8], p. 117).

Frostman’s theorem. Let $B$ be a infinite Blaschke product with zero-sequence
$\\{a_{k}\\}_{k=0}^{\infty}$, and $t\in\mathbb{T}$. Then $B$ has an angular
derivative in the sence of Carathéodory at $t$ (for definition see [8], p. 62)
if and only if

$\sum_{k=0}^{\infty}\frac{1-|a_{k}|^{2}}{|1-t\overline{a}_{k}|^{2}}<\infty.$

It is shown in [16] (p.185) that there is an infinite Blaschke product that
has an angular derivative at no point of $\mathbb{T}$ or, equivalently, there
is a sequance $\\{a_{k}\\}_{k=0}^{\infty}$ such that

$\sum_{k=0}^{\infty}(1-|a_{k}|)<\infty,$

yet

(2.1)
$\sum_{k=0}^{\infty}\frac{1-|a_{k}|^{2}}{|1-t\overline{a}_{k}|^{2}}=\infty\quad\mbox{for
all}~{}t\in\mathbb{T}.$

######  Theorem 2.1.

Suppose that $\varphi\in TMS$ and $X(\mathbb{T})$ is one of the space
$C(\mathbb{T})$ and $L^{p}(\mathbb{T})$, $1\leq p<\infty$. In order that
$\|f-\sigma_{n,\varphi}(f)\|_{X(\mathbb{T})}\to 0$ as $n\to\infty$ for every
$f\in X(\mathbb{T})$ it is necessary and sufficient that

(2.2)
$\lim_{n\to\infty}\left\|\frac{1}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})}=0.$

######  Corollary 2.1.

Let $\varphi\in TMS$. Then for every $f\in C(\mathbb{T})$,
$\sigma_{n,\varphi}(f)$ converges to $f$ uniformly on $\mathbb{T}$ if and only
if (2.1) is fulfilled.

Indeed, in case $X(\mathbb{T})=C(\mathbb{T})$ the norm in the left hand side
of (2.2) becomes

(2.3)
$\left\|\frac{1}{B^{\prime}_{n}}\right\|_{C(\mathbb{T})}=\frac{1}{\displaystyle\min_{t\in\mathbb{T}}\sum_{k=0}^{n-1}\frac{1-|a_{k}|^{2}}{|1-\overline{t}a_{k}|^{2}}}.$

######  Corollary 2.2.

Let $\varphi\in TMS$ be such that (2.1) is fulfilled. Then for every $f\in
L^{p}(\mathbb{T})$, $1\leq p<\infty$, we have
$\|f-\sigma_{n,\varphi}(f)\|_{p}\to 0$ as $n\to\infty$.

Indeed, similarly to (2.3), we get

(2.4)
$\left\|\frac{1-\overline{B^{\prime}_{n}(0)}B_{n}}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})}\leq\frac{2}{\displaystyle\min_{t\in\mathbb{T}}\sum_{k=0}^{n-1}\frac{1-|a_{k}|^{2}}{|1-\overline{t}a_{k}|^{2}}}$

and result follows.

######  Corollary 2.3.

Suppose that $\varphi\in TMS$ and $X(\mathbb{T})$ is one of the space
$C(\mathbb{T})$ and $L^{p}(\mathbb{T})$, $1\leq p<\infty$. If $\varphi$ is
complete, then for every $f\in X(\mathbb{T})$, we have
$\|f-\sigma_{n,\varphi}(f)\|_{X(\mathbb{T})}\to 0$ as $n\to\infty$.

This assertion readily follows from the estimates (2.3), (3.2) and (2.4).

Let us consider the class $\mathcal{K}^{+}$ consisting of those holomorphic
functions in $\mathbb{D}$, which can be represented by Cauchy type integral

(2.5)
$f(z)=K(\mu)(z):=\int_{\mathbb{T}}\frac{\mu(t)}{1-\overline{t}z}dm(t),\quad
z\in\mathbb{D},$

with $\mu\in L^{\infty}(\mathbb{T})$,
$\|\mu\|_{\infty}:=\mathrm{ess~{}sup}_{t\in\mathbb{T}}|\mu(t)|\leq 1.$

The following assertion gives solution of Problem 1.2.

######  Theorem 2.2.

Let $\varphi\in TMS$. Then for every $z\in\mathbb{D}$ and $n\in\mathbb{Z}_{+}$
we have

(2.6)
$\max_{f\in\mathcal{K}^{+}}\left|\delta_{n,\varphi}(f)(z)-f^{\prime}(z)\right|=\frac{|B_{n}(z)|}{1-|z|^{2}}.$

For given $z\in\mathbb{D}$ such that $B_{n}(z)\not=0$, and for given
$n\in\mathbb{Z}_{+}$ maximum is attained only for the functions

(2.7)
$f_{*}(t)=\mathrm{e}^{\mathrm{i}\theta}B_{n}(t)\frac{t-z}{1-t\overline{z}},\quad\theta\in\mathbb{R}.$

As immediate consequence of Theorem 2.2, we have the following assertion.

######  Corollary 2.4.

Let $f\in\mathcal{K}^{+}$, $\varphi\in TMS$ and let ${\bf
a}=\\{a_{j}\\}_{j=0}^{\infty}$ be as in (1.4). Then for every $n\in\mathbb{N}$
we have

(2.8) $\delta_{n,\varphi}(f)(a_{j})=f^{\prime}(a_{j}),\quad j=0,1,\ldots,n-1.$

Moreover, if (1.5) is satisfied, then uniformly on closed sets of $\mathbb{D}$

$\lim_{n\to\infty}\delta_{n,\varphi}(f)(z)=f^{\prime}(z),$

and, therefore, for given $z\in\mathbb{D}$ the equality

$\lim_{n\to\infty}\delta_{n,\varphi}(f)(z)=0$

hold true if and only if $f^{\prime}(z)=0$.

Now, let us consider the behavior of $|f(z)-\sigma^{+}_{n,\varphi}(f)(z)|$ for
$z\in\overline{\mathbb{D}}$.

We let by $\mathcal{S}$ denote the Schur class consisting of holomorphic
functions $f$ which satisfy $\sup_{z\in\mathbb{D}}|f(z)|\leq 1$ and let
$A(\mathbb{D})$ denote the disk algebra of holomorphic functions in
$\mathbb{D}$ that extend continuously to $\overline{\mathbb{D}}$ equipped with
the norm $\|f\|_{A(\mathbb{D})}=\max_{z\in\overline{\mathbb{D}}}|f(z)|$.

######  Theorem 2.3.

Suppose $\varphi\in TMS$, $z\in\mathbb{D}$ and $n\in\mathbb{N}$. Then we have

$\displaystyle\left|\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\right|\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}}$
$\displaystyle\leq$
$\displaystyle\sup_{f\in\mathcal{S}}|f(z)-\sigma_{n,\varphi}^{+}(f)(z)|$
$\displaystyle\leq$
$\displaystyle\left|\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\right|\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}}+|B_{n}(z)|^{2},$

provided $z\in\mathbb{D}$ and $|B^{\prime}_{n}(z)|>0$.

Moreover, we have that for all $n\in\mathbb{N}$

$\displaystyle\sup_{f\in\mathcal{S}}\|f-\sigma^{+}_{n,\varphi}(f)\|_{\infty}$
$\displaystyle=$ $\displaystyle\sup_{f\in\mathcal{S}\cap
A(\mathbb{D})}\|f-\sigma^{+}_{n,\varphi}(f)\|_{A(\mathbb{D})}$
$\displaystyle=$ $\displaystyle 2.$

In view of this result, it is interesting to obtain the lower estimate of
$|f-\sigma^{+}_{n,\varphi}(f)|$ for an individual function $f$. For this goal
we recall that the classical Fejér means $\sigma_{n}(f)$ considered as a
method of approximation on space $C(\mathbb{T})$ are saturated and its
saturation order is $O(n^{-1})$ (see for example [17], p. 79). In particular,
this means that the relation $\|f-\sigma_{n}(f)\|_{C(\mathbb{T})}=o(n^{-1})$
as $n\to\infty$ holds true only if $f=\mathrm{const}$. The next corollary
gives an analog of this assertion for $\sigma_{n,\varphi}$.

######  Corollary 2.5.

Suppose $f\in H^{\infty}$, $\varphi\in TMS$ and let ${\bf
a}=\\{a_{j}\\}_{j=0}^{\infty}$ be as in (1.4). Then for every $n\in\mathbb{N}$
we have

$\|f-\sigma^{+}_{n,\varphi}(f)\|_{L^{\infty}(\mathbb{T})}\geq\frac{1}{n}\max_{0\leq
j\leq n-1}\left((1-|a_{j}|^{2})\left|f^{\prime}(a_{j})\right|\right).$

Moreover, if (1.5) is satisfied and if

$\|f-\sigma_{n,\varphi}(f)\|_{L^{\infty}(\mathbb{T})}=o\left(\frac{1}{n}\right),\quad
n\to\infty,$

then $f=\mathrm{const}$.

For proving we fix $n\in\mathbb{N}$ and consider the function

$F:=\frac{f-S_{n}(f)}{B_{n}}.$

Since $f(a_{j})-S_{n}(f)(a_{j})=0$ for $j=0,\ldots,n-1$, we have (see [7], p.
53) that $F$ belongs to $H^{\infty}$ space. On the other side,
$\delta_{n,\varphi}(f)=B^{\prime}_{n}F+S^{\prime}_{n}(f)$. Therefore
$\delta_{n,\varphi}(f)\in H^{\infty}$ and, consequently, we can apply the
following well-known inequality (see [7], p. 85)

$(1-|z|^{2})|\delta_{n,\varphi}(f)(z)|\leq\|\delta_{n,\varphi}(f)\|_{1},\quad
z\in\mathbb{D}.$

Hence, from Corollary 2.4 we get

$\displaystyle(1-|a_{j}|^{2})|f^{\prime}(a_{j})|$ $\displaystyle=$
$\displaystyle(1-|a_{j}|^{2})|\delta_{n,\varphi}(f)(a_{j})|$
$\displaystyle\leq$ $\displaystyle\|\delta_{n,\varphi}(f)\|_{1}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{B^{\prime}_{n}}{B_{n}}(f-\sigma_{n,\varphi}(f))\right\|_{1}$
$\displaystyle\leq$
$\displaystyle\|B^{\prime}_{n}\|_{1}\|f-\sigma_{n,\varphi}(f)\|_{L^{\infty}(\mathbb{T})}$
$\displaystyle=$ $\displaystyle
n\|f-\sigma_{n,\varphi}(f)\|_{L^{\infty}(\mathbb{T})}.$

Here we used the following identity

$\displaystyle\|B^{\prime}_{n}\|_{1}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n-1}\int_{\mathbb{T}}\frac{1-|a_{j}|^{2}}{|1-t\overline{a}_{j}|^{2}}dm(t)$
$\displaystyle=$ $\displaystyle n.$

The second part of assertion follows from the inequality (as above)

$(1-|z|^{2})|\delta_{n,\varphi}(f)(z)|\leq
n\|f-\sigma_{n,\varphi}(f)\|_{1},\quad\forall z\in\mathbb{D}$

by Corollary 2.4.

##  3\. Proofs

Our approach in proving the main results is based on the following lemmas
concerning the some properties of operators $\sigma_{n,\varphi}$ and
$\sigma^{+}_{n,\varphi}$.

###  3.1. Auxiliary lemmas

All lemmas in this subsection are new and of independent interest.

We begin with the integral representation for $\delta_{n,\varphi}$ from which
follows that $\delta_{n,\varphi}(f)$ is holomorphic in $\mathbb{D}$.

######  Lemma 3.1.

Suppose that $\varphi\in TMS$ and $f\in\mathcal{K}$. Then the function
$\delta_{n,\varphi}(f)$ is holomorphic in $\mathbb{D}$.

Moreover, if $f$ is the Cauchy transform of $\mu\in M$, then for all
$n\in\mathbb{Z}_{+}$ and $z\in\mathbb{D}$,

(3.1) $\displaystyle\delta_{n,\varphi}(f)(z)$ $\displaystyle=$ $\displaystyle
f^{\prime}(z)-B_{n}(z)\int_{\mathbb{T}}\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\frac{\overline{B_{n}(t)}}{|1-\overline{t}z|^{2}}d\mu(t)$
(3.2) $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\frac{1-\overline{B_{n}(t)}B_{n}(z)}{|1-\overline{t}z|^{2}}d\mu(t).$

###### Proof.

Applying (1.8) we get

(3.3)
$\frac{B^{\prime}_{n}(z)}{B_{n}(z)}\left(f(z)-S_{n}(f)(z)\right)=B^{\prime}_{n}(z)\int_{\mathbb{T}}\frac{\overline{B_{n}(t)}}{1-\overline{t}z}d\mu(t)$

and

(3.4)
$f^{\prime}(z)-S^{\prime}_{n}(f)(z)=B^{\prime}_{n}(z)\int_{\mathbb{T}}\frac{\overline{B_{n}(t)}}{1-\overline{t}z}d\mu(t)+B_{n}(z)\int_{\mathbb{T}}\frac{\overline{B_{n}(t)t}}{(1-\overline{t}z)^{2}}d\mu(t).$

By subtracting (3.4) from (3.3) we get

$\delta_{n,\varphi}(f)(z)=f^{\prime}(z)-B_{n}(z)\int_{\mathbb{T}}\frac{\overline{B_{n}(t)t}}{(1-\overline{t}z)^{2}}d\mu(t).$

This means that the function $\delta_{n,\varphi}(f)$ is holomorphic in
$\mathbb{D}$.

In order to prove (3.1) and (3.2), it remains to note that

$\frac{\overline{t}}{(1-\overline{t}z)^{2}}=\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\frac{1}{|1-\overline{t}z|^{2}},\quad
t\in\mathbb{T}.$

∎

######  Lemma 3.2.

Let $\varphi\in TMS$ and $e_{0}(t)=1$ for $t\in\mathbb{T}$. Then for all
$n\in\mathbb{Z}_{+}$ we have $\sigma^{+}_{n,\varphi}(e_{0})=e_{0}.$

###### Proof.

By the Lemma 3.1 (see (3.2) for $d\mu=e_{0}dm$), according to the Poisson
integral formula, applying to the holomorphic function
$t\mapsto(t-z)(1-B_{n}(t)\overline{B_{n}(z)})/(1-t\overline{z})$, we have

$\displaystyle\overline{B^{\prime}_{n}(z)}\overline{\left(\frac{e_{0}(z)-\sigma^{+}_{n,\varphi}(e_{0})(z)}{B_{n}(z)}\right)}$
$\displaystyle=$ $\displaystyle\overline{\delta_{n,\varphi}(e_{0})(z)}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\frac{t-z}{1-t\overline{z}}\frac{1-B_{n}(t)\overline{B_{n}(z)}}{|1-\overline{t}z|^{2}}dm(t)$
$\displaystyle=$ $\displaystyle 0,~{}\forall z\in\mathbb{D}.$

The result follows, since $B^{\prime}_{n}$ has precisely $n-1$ zeros in
$\mathbb{D}$ (see [8], p. 41). ∎

######  Lemma 3.3.

Let $B$ be a finite Blacshke product for the disc $\mathbb{D}$ and let $f\in
H^{1}$. Then for every $z\in\mathbb{T}$ we have

$\lim_{\rho\to 1-}\left(1-|B(\rho z)|^{2}\right)f(\rho z)=0.$

###### Proof.

Since

$\lim_{\rho\to 1-}\frac{1-|B(\rho z)|^{2}}{1-|\rho z|^{2}}=|B^{\prime}(z)|>0$

for every $z\in\mathbb{T}$ (see [8], p. 45), it is enough to prove the lemma
in the case $B(z)=z$.

By Cauchy integral formula, applying to the function $t\mapsto
f(t)(1-\rho\overline{z}t)$, we have

$\left(1-\rho^{2}\right)f(\rho
z)=\int_{\mathbb{T}}f(t)\frac{1-\rho\overline{z}t}{1-\rho
z\overline{t}}dm(t).$

Since

$\lim_{\rho\to 1-}\frac{1-\rho\overline{z}t}{1-\rho
z\overline{t}}=\begin{cases}-\overline{z}t,\hfill&\mbox{if}~{}t\not=z,\\\
1,\hfill&\mbox{if}~{}t=z,\end{cases}$

by Lebesgue’s dominated convergence theorem we get

$\displaystyle\lim_{\rho\to 1-}\left(1-\rho^{2}\right)f(\rho z)$
$\displaystyle=$ $\displaystyle\lim_{\rho\to
1-}\int_{\mathbb{T}}f(t)\frac{1-\rho\overline{z}t}{1-\rho z\overline{t}}dm(t)$
$\displaystyle=$ $\displaystyle-\overline{z}\int_{\mathbb{T}}f(t)tdm(t)$
$\displaystyle=$ $\displaystyle 0.$

∎

For $\varphi\in TMS$ we define the kernels $\mathcal{F}_{n,\varphi}(t,z)$,
$n\in\mathbb{Z}_{+}$, as the functions on $\mathbb{T}^{2}$ by

$\mathcal{F}_{n,\varphi}(t,z):=\begin{cases}1,\hfil&\mbox{if}~{}n=0,\cr\displaystyle
2\mathrm{Re}\sum_{k=0}^{n-1}\left(\varphi_{k}(z)-\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\varphi^{\prime}_{k}(z)\right)\overline{\varphi_{k}(t)}-1,\hfill&\mbox{if}~{}n\in\mathbb{N}.\end{cases}$

Since $B^{\prime}_{n}(z)\not=0$ for $z\in\mathbb{T}$ and $n\in\mathbb{N}$ (see
[8], p. 40), the kernels $\mathcal{F}_{n,\varphi}(t,z)$ are well defined.

We call $\mathcal{F}_{n,\varphi}$ the Fejér–type kernel of order $n$ based on
the system $\varphi\in TMS$. The reason for this is that in case
$\varphi_{k}(t)=t^{k}$, $\mathcal{F}_{n,\varphi}$ coincide with the Fejér
kernel

$F_{n}(\overline{t}z):=\begin{cases}1,\hfill&\mbox{if}~{}n=0,\cr\displaystyle
2\mathrm{Re}\sum_{k=0}^{n-1}\left(1-\frac{k}{n}\right)z^{k}\overline{t}^{k}-1,\hfill&\mbox{if}~{}n\in\mathbb{N}.\end{cases}$

In addition, as we will show in the following Lemma 3.4,
$\mathcal{F}_{n,\varphi}$ is positive kernel with

(3.5) $\int_{\mathbb{T}}\mathcal{F}_{n,\varphi}(t,z)dm(t)=1,\quad
z\in\mathbb{T}.$

######  Lemma 3.4.

Suppose that $\varphi\in TMS$, $f\in H^{1}$ and $n\in\mathbb{N}$. Then for
every $z\in\mathbb{T}$ we have

(3.6) $\displaystyle\sigma^{+}_{n,\varphi}(f)(z)$ $\displaystyle=$
$\displaystyle
c_{n}(z)\int_{\mathbb{T}}f(t)\left|\frac{B_{n}(t)-B_{n}(z)}{t-z}\right|^{2}dm(t)$
(3.7) $\displaystyle=$ $\displaystyle\sigma_{n,\varphi}(f)(z),$

where

(3.8)
$c_{n}(z):=\frac{1}{\sum_{k=0}^{n-1}|\varphi_{k}(z)|^{2}}=\frac{1}{|B^{\prime}_{n}(z)|}.$

Moreover, if $(t,z)\in\mathbb{T}^{2}$ and $x=\arg z,$ $y=\arg t$ we have

$\displaystyle\mathcal{F}_{n,\varphi}(t,z)$ $\displaystyle=$ $\displaystyle
c_{n}(z)\left|\frac{B_{n}(t)-B_{n}(z)}{t-z}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\gamma_{n}(x)}\cdot\frac{\displaystyle\sin^{2}\left(\int_{x}^{y}\gamma_{n}(t)dt\right)}{\displaystyle\sin^{2}\frac{y-x}{2}}.$

###### Proof.

Fix $z\in\mathbb{D}$ and consider the holomorphic function
$F_{f,z}(t):=f(t)(1-\overline{B_{n}(z)}B_{n}(t))$. Observe that

(3.10) $\sigma^{+}_{n,\varphi}(F_{f,z})=\sigma^{+}_{n,\varphi}(f).$

This follows easily from the equalities

$\displaystyle S_{n}(fB_{n})(z)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)B_{n}(t)\frac{1-B_{n}(z)\overline{B_{n}(t)}}{1-\overline{t}z}dm(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)(B_{n}(t)-B_{n}(z))\frac{1}{1-\overline{t}z}dm(t)$
$\displaystyle=$ $\displaystyle 0,\quad\forall~{}z\in\mathbb{D}.$

Thus applying (3.2) to $F_{f,z}$ and taken into account (3.10), we get

(3.11) $\displaystyle\frac{B^{\prime}_{n}(z)}{B_{n}(z)}\left((1\right.$
$\displaystyle-$
$\displaystyle\left.|B_{n}(z)|^{2})f(z)-\sigma^{+}_{n,\varphi}(f)(z)\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\left|\frac{1-\overline{B_{n}(t)}B_{n}(z)}{1-\overline{t}z}\right|^{2}dm(t)$
(3.12) $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\left|\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi_{k}(t)}\right|^{2}dm(t).$

Since for $(t,z)\in\mathbb{T}^{2}$,

$\lim_{\rho\to 1-}\frac{\overline{t}-\rho\overline{z}}{1-\overline{t}\rho
z}=\begin{cases}-\overline{z},\hfill&\mbox{if}~{}t\not=z,\\\
\overline{z},\hfill&\mbox{if}~{}t=z,\end{cases}$

we can go to the limit as $|z|\to 1$ in both side of (3.11) and (3.12). As a
result, by Lemma 3.3 and by Lebesgue’s dominated convergence theorem we get

$\displaystyle\frac{zB^{\prime}_{n}(z)}{B_{n}(z)}\sigma^{+}_{n,\varphi}(f)(z)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\left|\frac{B_{n}(t)-B_{n}(z)}{t-z}\right|^{2}dm(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\left|\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi_{k}(t)}\right|^{2}dm(t),\quad\forall
z\in\mathbb{T}.$

By Lemma 3.2 and the Parseval identity, the last equalities implies

$\displaystyle\frac{zB^{\prime}_{n}(z)}{B_{n}(z)}$ $\displaystyle=$
$\displaystyle\frac{zB^{\prime}_{n}(z)}{B_{n}(z)}\sigma^{+}_{n,\varphi}(e_{0}(z))$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\left|\frac{1-B_{n}(z)\overline{B_{n}(t)}}{1-z\overline{t}}\right|^{2}dm(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\left|\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi_{k}(t)}\right|^{2}dm(t)$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n-1}|\varphi_{k}(z)|^{2}.$

Since $zB^{\prime}_{n}(z)/B_{n}(z)=|B^{\prime}_{n}(z)|$ (see [8], p. 40),
(3.6) follows.

In order to prove (3.7) and (3.4), we fix $z\in\mathbb{T}$ and consider the
holomorphic function

$g(t)=\sum_{k=0}^{n-1}\overline{\left(\varphi_{k}(z)-\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\varphi^{\prime}_{k}(z)\right)}\varphi_{k}(t),\quad
t\in\mathbb{D}.$

Differentiating (1.8) with respect to $z$ and then taking the conjugate, we
obtaine the formula

$\sum_{k=0}^{n-1}\overline{\varphi^{\prime}_{k}(z)}\varphi_{k}(t)=-\frac{\overline{B^{\prime}_{n}(z)}B_{n}(t)}{1-\overline{z}t}+\frac{(1-\overline{B_{n}(z)}B_{n}(t))t}{(1-\overline{z}t)^{2}}.$

This formula and (1.8) together give us

$\displaystyle g(0)$ $\displaystyle=$ $\displaystyle
1-\overline{B_{n}(z)}B_{n}(0)+\overline{\frac{B_{n}(z)}{B^{\prime}_{n}(z)}B^{\prime}_{n}(z)}B_{n}(0)$
$\displaystyle=$ $\displaystyle 1,$

and consequently,

$\displaystyle\int_{\mathbb{T}}f\cdot(g-1)dm$ $\displaystyle=$ $\displaystyle
f(0)(g(0)-1)$ $\displaystyle=$ $\displaystyle 0,\quad\forall~{}f\in H^{1}.$

Therefore,

$\displaystyle\sigma^{+}_{n,\varphi}(f)(z)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\overline{g(t)}dm(t)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\left(\overline{g(t)}+g(t)-1\right)dm(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(t)\mathcal{F}_{n,\varphi}(t,z)dm(t)$

and (3.7) is proved.

Now let us consider the difference

$\Delta_{n,z}(t):=\mathcal{F}_{n,\varphi}(t,z)-c_{n}(z)\left|\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi_{k}(t)}\right|^{2}$

as the function defined on $\mathbb{T}$. From (3.6) and (3.7) we see that
$\Delta_{n,z}$ satisfy

$\int_{\mathbb{T}}f\Delta_{n,z}dm=0$

for every $f\in H^{\infty}$. This means that $\Delta_{n,z}$ is the
nontangential limits of some function from $H^{1}$, say $\Delta_{n,z}$, with
$\Delta_{n,z}(0)=0$ (see [7], p. 85). But $\mathrm{Im}\Delta_{n,z}(t)=0$. Thus
by Schwarz integral formula, $\Delta_{n,z}(t)=0$ for every
$t\in\overline{\mathbb{D}}$. ∎

######  Lemma 3.5.

Suppose that $X(\mathbb{T})$ is one of space $L^{p}(\mathbb{T})$ and
$C(\mathbb{T})$ and that $X_{+}(\mathbb{D})$ is one of space $H^{p}$ and
$A(\mathbb{D})$. Then for any $\varphi\in TMS$ and for $n\in\mathbb{N}$ we
have

$\|\sigma_{n,\varphi}\|_{X(\mathbb{T})\rightarrow
X(\mathbb{T})}=\|\sigma^{+}_{n,\varphi}\|_{X_{+}(\mathbb{D})\rightarrow
X_{+}(\mathbb{D})}=1.$

###### Proof.

The result follows immediately from Lemma 3.2 and Lemma 3.4. ∎

######  Lemma 3.6.

Let $\varphi\in TMS$, $\alpha\in\mathbb{D}$ be fixed and

$w_{\alpha}(z):=\frac{z-\alpha}{1-z\overline{\alpha}}.$

Then for every $z\in\mathbb{D}$ and $n\in\mathbb{N}$ we have

(3.13) $\sigma_{n,\varphi}^{+}(w_{\alpha})(z)=\begin{cases}\displaystyle
z-\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\left(1-\overline{B^{\prime}_{n}(0)}B_{n}(z)\right),\hfill&\mbox{if}~{}\alpha=0,\\\
\displaystyle\frac{z-\alpha}{1-z\overline{\alpha}}-\frac{1-|\alpha|^{2}}{(1-z\overline{\alpha})^{2}}\frac{B_{n}(z)}{B_{n}^{\prime}(z)}\left(1-\overline{B_{n}(\alpha)}B_{n}(z)\right),\hfill&\mbox{if}~{}\alpha\not=0.\end{cases}$

###### Proof.

Let us consider case $\alpha=0$. Differentiating (1.8) with respect to
$\overline{t}$ and then letting $t=0$ we get

(3.14)
$\sum_{k=0}^{n-1}\varphi_{k}(z)\overline{\varphi^{\prime}_{k}(0)}=z-\left(\overline{B^{\prime}_{n}(0)}+\overline{B_{n}(0)}z\right)B_{n}(z).$

Differentiating this equality with respect to $z$ gives

(3.15)
$\sum_{k=0}^{n-1}\varphi^{\prime}_{k}(z)\overline{\varphi^{\prime}_{k}(0)}=1-\overline{B_{n}(0)}B_{n}(z)-\left(\overline{B^{\prime}_{n}(0)}+\overline{B_{n}(0)}z\right)B^{\prime}_{n}(z).$

From (3.14) we also see that

$\displaystyle\langle w_{0},\varphi_{j}\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\left(\sum_{k=0}^{n-1}\varphi_{k}(t)\overline{\varphi^{\prime}_{k}(0)}+\left(\overline{B^{\prime}_{n}(0)}+\overline{B_{n}(0)}t\right)B_{n}(t)\right)\overline{\varphi_{j}(t)}dm(t)$
$\displaystyle=$
$\displaystyle\overline{\varphi^{\prime}_{j}(0)}+\int_{\mathbb{T}}\left(\overline{B^{\prime}_{n}(0)}+\overline{B_{n}(0)}t\right)\prod_{l=j}^{n-1}\frac{t-a_{l}}{1-t\overline{a}_{l}}\frac{\sqrt{1-|a_{j}|^{2}}}{1-\overline{t}a_{j}}dm(t)$
$\displaystyle=$ $\displaystyle\overline{\varphi^{\prime}_{j}(0)},\quad
j=0,1,\ldots,n-1.$

Therefore, the sums in the left-hand side of (3.14) and (3.15) are
$S_{n}(w_{0})(z)$ and $S^{\prime}_{n}(w_{0})(z)$ respectively.

After some calculations, we obtain

$\displaystyle\sigma^{+}_{n,\varphi}(w_{0})(z)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\overline{\varphi^{\prime}_{k}(0)}\varphi_{k}(z)-\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\sum_{k=0}^{n-1}\overline{\varphi^{\prime}_{k}(0)}\varphi^{\prime}_{k}(z)$
$\displaystyle=$ $\displaystyle
z-\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\left(1-\overline{B^{\prime}_{n}(0)}B_{n}(z)\right).$

Now let us consider case $\alpha\not=0$. According to (1.8) and to the
identity

$w_{\alpha}(z)=-\frac{1}{\overline{\alpha}}+\frac{1-|\alpha|^{2}}{\overline{\alpha}}\frac{1}{1-t\overline{\alpha}},$

we have, similarly to the above,

$\displaystyle S_{n}\left(w_{\alpha}\right)(z)$ $\displaystyle=$
$\displaystyle-\frac{1}{\overline{\alpha}}S_{N}\left(e_{0}\right)(z)+\frac{1-|\alpha|^{2}}{\overline{\alpha}}S_{n}\left(\frac{1}{1-\cdot\overline{\alpha}}\right)(z)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{\alpha}}\sum_{k=0}^{n-1}\left(\left(1-|\alpha|^{2}\right)\overline{\varphi_{k}(\alpha)}-\overline{\varphi_{k}(0)}\right)\varphi_{k}(z)$
$\displaystyle=$
$\displaystyle\frac{1-|\alpha|^{2}}{\overline{\alpha}}\frac{1-\overline{B_{n}(\alpha)}B_{n}(z)}{1-z\overline{\alpha}}-\frac{1}{\overline{\alpha}}\left(1-\overline{B_{n}(0)}B_{n}(z)\right)$

and

$\displaystyle S_{n}^{\prime}\left(w_{\alpha}\right)(z)$ $\displaystyle=$
$\displaystyle\frac{1-|\alpha|^{2}}{\overline{\alpha}}\left(\frac{-\overline{B_{n}(\alpha)}B_{n}^{\prime}(z)}{1-z\overline{\alpha}}+\overline{\alpha}\frac{1-\overline{B_{n}(\alpha)}B_{n}(z)}{(1-z\overline{\alpha})^{2}}\right)$
$\displaystyle+\frac{1}{\overline{\alpha}}\overline{B_{n}(0)}B_{n}^{\prime}(z).$

The result follows from the above with a little algebra. ∎

######  Lemma 3.7.

Let $\varphi\in TMS$, $z\in\overline{\mathbb{D}}$ and $n\in\mathbb{Z}_{+}$.
Then we have

(3.16)
$\max_{f\in\mathcal{S}}\left|\delta_{n,\varphi}(f)(z)-B_{n}^{\prime}(z)\overline{B_{n}(z)}f(z)\right|=\begin{cases}\displaystyle\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}},\hfill&\mbox{if}~{}z\in\mathbb{D},\cr|B^{\prime}_{n}(z)|,\hfill&\mbox{if}~{}z\in\mathbb{T}.\end{cases}$

For given $z\in\overline{\mathbb{D}}$ and $n\in\mathbb{Z}_{+}$ maximum is
attained only for the functions

$f(t)=\mathrm{e}^{\mathrm{i}\theta}\begin{cases}\displaystyle\frac{t-z}{1-t\overline{z}},\hfill&\mbox{if}~{}z\in\mathbb{D},\\\
1,\hfill&\mbox{if}~{}z\in\mathbb{T},\end{cases}\quad\theta\in\mathbb{R}.$

###### Proof.

Fix $z\in\mathbb{D}$ and consider the presentations (3.11) and (3.12). It
follows that

$\displaystyle\left|\delta_{n,\varphi}(f)(z)-B_{n}^{\prime}(z)\overline{B_{n}(z)}f(z)\right|$
$\displaystyle=$
$\displaystyle\left|\frac{B^{\prime}_{n}(z)}{B_{n}(z)}\left((1-|B_{n}(z)|^{2})f(z)-\sigma^{+}_{n,\varphi}(f)(z)\right)\right|$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{T}}\left|f(t)\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\right|\left|\frac{1-\overline{B_{n}(t)}B_{n}(z)}{1-\overline{t}z}\right|^{2}dm(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\left|\frac{1-\overline{B_{n}(t)}B_{n}(z)}{1-\overline{t}z}\right|^{2}dm(t)$
$\displaystyle=$ $\displaystyle\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}}.$

Equalities hold if and only if

$f(t)=\mathrm{e}^{\mathrm{i}\theta}\frac{t-z}{1-t\overline{z}}$

for some $\theta\in\mathbb{R}$.

Now, let us fix $z\in\mathbb{T}$. Then we have

$\left|\delta_{n,\varphi}(f)(z)-B_{n}^{\prime}(z)\overline{B_{n}(z)}f(z)\right|=\left|B^{\prime}_{n}(z)\right||\sigma^{+}_{n,\varphi}(f)(z)|,$

and the result follows by Lemma 3.5. ∎

###  3.2. Proof of Theorem 2.1

First of all, note that by Lemma 3.6

$\left\|w_{0}-\sigma^{+}_{n,\varphi}(w_{0})\right\|_{X(\mathbb{T})}=\left\|\frac{1-\overline{B_{n}^{\prime}(0)}B_{n}}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})},$

where $w_{0}(z)=z$.

To estimate the left side of the last equation we observe that

$\prod_{k=0}^{n-1}|a_{k}|^{2}\leq\left|1-\overline{B^{\prime}_{n}(0)}B_{n}(z)\right|\leq
2$

for $z\in\mathbb{T}$. Here we used the well known estimate
$|B^{\prime}_{n}(0)|\leq 1-|B_{n}(0)|^{2}$.

Therefore

(3.17)
$\prod_{k=0}^{n-1}|a_{k}|^{2}\left\|\frac{1}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})}\leq\left\|w_{0}-\sigma^{+}_{n,\varphi}(w_{0})\right\|_{X(\mathbb{T})}\leq
2\left\|\frac{1}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})}.$

To prove the necessity of (2.2), suppose that $\prod_{k=0}^{n}|a_{k}|^{2}\to
0$ as $n\to\infty$, yet
$\lim_{n\to\infty}\|1/B^{\prime}_{n}\|_{X(\mathbb{T})}>0$. In this case,
$a_{k}$ must satisfies the condition (1.5). But

$\displaystyle\frac{1}{2}\sum_{k=0}^{n}(1-|a_{k}|)$ $\displaystyle\leq$
$\displaystyle\min_{t\in\mathbb{T}}\sum_{k=0}^{n}\frac{1-|a|^{2}}{|1-\overline{t}a_{k}|^{2}}$
$\displaystyle\leq$ $\displaystyle|B^{\prime}_{n}(t)|$

for $t\in\mathbb{T}$. Therefore

$\left\|\frac{1}{B^{\prime}_{n}}\right\|_{X(\mathbb{T})}\leq\frac{2}{\sum_{k=0}^{n}(1-|a_{k}|)}\to
0,$

as $n\to\infty$. This gives a contradiction. The necessity of (2.2) follows.

To prove the sufficiency, we require the following Curtis’s generalization of
Korovkin’s theorem [2] (see also [4]).

Curtis’s theorem Let $T_{n}$ be a uniformly bounded sequence of positive
operators in $X(\mathbb{T})$. Then
$\lim_{n\to\infty}\|f-T_{n}(f)\|_{X(\mathbb{T})}$ for each $f\in
X(\mathbb{T})$ provided that
$\lim_{n\to\infty}\|e_{k}-T_{n}(e_{k})\|_{X(\mathbb{T})}=e_{k}$ for $k=0,1$,
where $e_{k}(t)=t^{k}$.

To apply this theorem, we note that an analogues of the assertions $(i)$ and
$(ii)$ for the operator $\sigma_{n,\varphi}$ in $X(\mathbb{T})$ follow by the
definition of $\sigma_{n,\varphi}$ and Lemma 3.5:

$(i)^{\prime}$ for each $n=0,1,\ldots$, $\sigma_{n,\varphi}(f)\geq 0$ on
$\mathbb{T}$ if $f\geq 0$ on $\mathbb{T}$;

$(ii)^{\prime}$ for each $n=0,1,\ldots$,
$\|\sigma_{n,\varphi}(f)\|_{X(\mathbb{T})}\leq\|f\|_{X(\mathbb{T})}$.

The assertions $(i)^{\prime}$ and $(ii)^{\prime}$ together with Lemma 3.2 and
Curtis’s theorem, where we take $T_{n}=\sigma_{n,\varphi}$, proves the
sufficiency of Theorem 2.1 because
$\lim_{n\to\infty}\|e_{1}-\sigma_{n,\varphi}(e_{1})\|_{X(\mathbb{T})}=0$
according to (3.17).

###  3.3. Proof of Theorem 2.2

It follows from (3.1) that

$\displaystyle\left|\delta_{n,\varphi}(f)(z)-f^{\prime}(z)\right|$
$\displaystyle\leq$
$\displaystyle|B_{n}(z)|\int_{\mathbb{T}}\left|\mu(t)\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\overline{B_{n}(t)}\right|\frac{1}{|1-\overline{t}z|^{2}}dm(t)$
$\displaystyle=$
$\displaystyle|B_{n}(z)|\int_{\mathbb{T}}|\mu(t)|\frac{dm(t)}{|1-\overline{t}z|^{2}}$
$\displaystyle\leq$ $\displaystyle\frac{|B_{n}(z)|}{1-|z|^{2}}$

for every functions $f\in\mathcal{K}^{+}$ and for every $z\in\mathbb{D}$.

If $B_{n}(z)\not=0$, then equalities hold throughout this chain of relation if
and only if

$|\mu(t)|=1$

and

$\arg\left(\mu(t)\frac{\overline{t}-\overline{z}}{1-\overline{t}z}\overline{B_{n}(t)}\right)=\mathrm{const}$

almost everywhere on $\mathbb{T}$.

But this conditions are equivalent to the condition

$\mu(t)=\mathrm{e}^{\mathrm{i}\theta}B_{n}(t)\frac{t-z}{1-t\overline{z}}\quad\mbox{a.e.
on}~{}\mathbb{T}$

for some $\theta\in\mathbb{R}$. So, the only the functions

$f_{*}(t)=K(\mu)(t)=\mathrm{e}^{\mathrm{i}\theta}B_{n}(t)\frac{t-z}{1-t\overline{z}},\quad\theta\in\mathbb{R},$

are extremal.

###  3.4. Proof of Theorem 2.3

Let us note that the assertion is trivial in case
$z\in\\{a_{0},\ldots,a_{n-1}\\}$. Indeed, in this case we have
$f(a_{j})-\sigma_{n,\varphi}^{+}(f)(a_{j})=B_{n}(z)=0$, therefore (2.3)
becomes equality.

Fix
$z\in\mathbb{D}\setminus\left(\\{a_{0},\ldots,a_{n-1}\\}\cup\\{z\in\mathbb{D}:B^{\prime}_{n}(z)=0\\}\right)$.
Then it follows from Lemma 3.7 that for arbitrary function $f\in\mathcal{S}$

$\displaystyle|f(z)-\sigma^{+}_{n,\varphi}(f)(z)|$ $\displaystyle\leq$
$\displaystyle|(1-|B_{n}(z)|^{2})f(z)-\sigma^{+}_{n,\varphi}(f)(z)|+|B^{2}_{n}(z)f(z)|$
$\displaystyle=$
$\displaystyle\left|\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\right||\delta_{n,\varphi}(f)(z)-B_{n}^{\prime}(z)\overline{B_{n}(z)}f(z)|+|B^{2}_{n}(z)f(z)|$
$\displaystyle\leq$
$\displaystyle\left|\frac{B_{n}(z)}{B^{\prime}_{n}(z)}\right|\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}}+|B_{n}(z)|^{2}.$

On the other side, applying (3.11) and (3.12) to the function
$f(t)=(t-z)/(1-t\overline{z})$, we get

$\displaystyle\frac{B^{\prime}_{n}(z)}{B_{n}(z)}\left(f(z)-\sigma^{+}_{n,\varphi}(f)(z)\right)$
$\displaystyle=$
$\displaystyle-\frac{B^{\prime}_{n}(z)}{B_{n}(z)}\sigma^{+}_{n,\varphi}(f)(z)$
$\displaystyle=$ $\displaystyle\frac{1-|B_{n}(z)|^{2}}{1-|z|^{2}}$

and (2.3) follows.

Now fix $z\in\mathbb{T}$. Then by Lemma 3.5 we get

$\displaystyle\sup_{f\in\mathcal{S}}|f(z)-\sigma_{n,\varphi}^{+}(f)(z)|$
$\displaystyle\leq$
$\displaystyle\sup_{f\in\mathcal{S}}|f(z)|+\sup_{f\in\mathcal{S}}|\sigma_{n,\varphi}^{+}(f)(z)|$
$\displaystyle\leq$ $\displaystyle 2.$

In order to prove the lower estimate, consider the sequence
$\\{f_{\lambda}\\}_{\quad 0<\lambda<1}$ of functions

$f_{\lambda}(t)=\overline{z}\frac{t-\lambda z}{1-t\lambda\overline{z}}.$

It is clear that $f_{\lambda}\in\mathcal{S}$ and $|f_{\lambda}(t)|=1$ for all
$t\in\mathbb{T}$. According to (1.8) and to the identity

$\frac{t-\omega}{1-t\overline{\omega}}=-\frac{1}{\overline{\omega}}+\frac{1}{\overline{\omega}}(1-|\omega|^{2})\frac{1}{1-t\overline{\omega}},\quad
0<|\omega|<1,$

we get

$\displaystyle S_{n}\left(f_{\lambda}\right)(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\lambda}S_{N}\left(e_{0}\right)(z)+\frac{1}{\lambda}(1-\lambda^{2})S_{n}\left(\frac{1}{1-t\lambda\overline{z}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{k=0}^{n-1}\left(\left(1-\lambda^{2}\right)\overline{\varphi_{k}(\lambda
z)}-\overline{\varphi_{k}(0)}\right)\varphi_{k}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}(1-\lambda^{2})\frac{1-\overline{B_{n}(\lambda
z)}B_{n}(t)}{1-t\lambda\overline{z}}-\frac{1}{\lambda}\left(1-\overline{B_{n}(0)}B_{n}(t)\right)$

and, consequence,

$S_{n}^{\prime}\left(f_{\lambda}\right)(t)=\frac{1}{\lambda}(1-\lambda^{2})\left(\frac{-\overline{B_{n}(\lambda
z)}B_{n}^{\prime}(t)}{1-t\lambda\overline{z}}+\lambda\overline{z}\frac{1-\overline{B_{n}(\lambda
z)}B_{n}(t)}{(1-t\lambda\overline{z})^{2}}\right)+\frac{1}{\lambda}\overline{B_{n}(0)}B_{n}^{\prime}(t).$

With a little algebra it follows that

$f_{\lambda}(z)-\sigma_{n,\varphi}^{+}\left(f_{\lambda}\right)(z)=\overline{z}\frac{1+\lambda}{1-\lambda}\frac{B_{n}(z)}{B_{n}^{\prime}(z)}\left(1-\overline{B_{n}(\lambda
z)}B_{n}(z)\right).$

Therefore

$\displaystyle\left|f_{\lambda}(z)-\sigma_{n,\varphi}^{+}\left(f_{\lambda}\right)(z)\right|$
$\displaystyle=$
$\displaystyle\frac{1+\lambda}{1-\lambda}\frac{1}{|B_{n}^{\prime}(z)|}\left|1-\overline{B_{n}(\lambda
z)}B_{n}(z)\right|$ $\displaystyle\geq$
$\displaystyle\frac{(1+\lambda)^{2}}{1+|B_{n}(\lambda
z)|}\frac{1-|B_{n}(\lambda z)|^{2}}{|B_{n}^{\prime}(z)|(1-\lambda^{2})}.$

Since

$\lim_{\lambda\to 1}\frac{1-|B_{n}(\lambda
z)|^{2}}{1-\lambda^{2}}=|B_{n}^{\prime}(z)|,$

the result follows from the above relation by letting $\lambda\to 1$.

## References

  * [1] Burckel R. B. An Introduction to Classical Complex Analysis, Vol. 1. Birkhäuser, 1979, 570 pp.
  * [2] Curtis P. C. The degree of approximation by positive convolution operators. Mich. Math. J. 12(2) (1965), 155–160.
  * [3] Džrbašyan, M. M. On the theory of series of Fourier in terms of rational functions (Russian). Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauki 9, no. 7 (1956), 3–28.
  * [4] Dzyadyk V. K. Approximation of functions by positive linear operators and singular integrals (Russian). Matematicheskii Sbornik 112(4) (1966), 508–517.
  * [5] Frostman O. Sur les produits de Blaschke. Kungl. Fysiogr. Sâllsk. i Lund Fôrh., Bd. 12, Nr. 15 (1942), 169–182.
  * [6] Gal S. Voronovskaja’s theorem and the exact degree of approximation for the derivatives of complex Riesz–Zygmund mens. General Mathematics 16, no. 4 (2008), 61–71.
  * [7] Garnett J. Bounded Analytic Functions, Springer, New York, NY, 2007. 463 pp.
  * [8] Mashreghi J. Derivatives of inner functions. Fields Institute Monographs, 31. Springer-Verlag, New York, 2013, 170 pp.
  * [9] Pekarskii A. A. Tchebycheff rational approximation in the disk, on the circle, and on a closed interval. Math. USSR-Sb., 61:1 (1988), 87–102.
  * [10] Rovba E. A. Rational interpolation functions of Fejér–Bernstein type (Russian). Vestn. Beloruss. Gos. Univ. Im. V. I. Lenina, Ser. I, no. 2 (1991), 75–71.
  * [11] Rovba E. A. Interpolation rational operators of Fejér and de la Vallée–Poussin type. Math. Notes, 53:2 (1993), 195–200.
  * [12] Rovba E. A. On the approximation of functions of bounded variation by Fejér and Jackson rational operators (Russian). Dokl. Nats. Akad. Nauk Belarusi 42, no. 4 (1998), 13–17.
  * [13] Rusak V. N. Rational Functions as Approximation Apparatus (Russian). Beloruss. Gos. Univ., Minsk, (1979), 174 pp.
  * [14] Rusak V. N., Sharp order estimates for best rational approximations on classes of functions representable as convolutions. Math. USSR–Sb., 56:2 (1987), 491–513.
  * [15] Savchuk V. V., Chaichenko S. O., Savchuk M. V. Approximation of bounded holomorphic and harmonic functions by Fejér means. Ukr. Math. J., 71, no. 4 (2019), 589–618.
  * [16] Shapiro J. H. Composition Operators and Classical Function Theory, Springer–Verlag, New York, 1993, 223 pp.
  * [17] Stepanets A. I. Methods of Approximation Theory. VSP, 2005, 919 pp.
  * [18] Van Guhr P., Bultheel A. Bernstein equiconvergence and Fejér–type theorems for general rational Fourier series. J. Comp. Appl. Math. 133 (2001), 635–645.


# Stance Prediction for Russian: Data and Analysis

Nikita Lozhnikov 1Innopolis University, Russian Federation
{n.lozhnikov<EMAIL_ADDRESS>1ITU Copenhagen, Denmark
<EMAIL_ADDRESS>Leon Derczynski and Manuel Mazzara ITU Copenhagen, Denmark
<EMAIL_ADDRESS>
###### Abstract

Stance detection is a critical component of rumour and fake news
identification. It involves the extraction of the stance a particular author
takes related to a given claim, both expressed in text. This paper
investigates stance classification for Russian. It introduces a new dataset,
RuStance, of Russian tweets and news comments from multiple sources, covering
multiple stories, as well as text classification approaches to stance
detection as benchmarks over this data in this language. As well as presenting
this openly-available dataset, the first of its kind for Russian, the paper
presents a baseline for stance prediction in the language.

##  1 Introduction

The web is rife with half-truths, deception, and lies. The rapid spread of
such information, facilitated and accelerated by social media can have
immediate and serious effects. Indeed, such false information affects
perception of events which can lead to behavioral manipulation [1]. The
ability to identify this information is important, especially in the modern
context of services and analyses that derive from claims on the web [2].

However, detecting these rumours is difficult for humans, let alone machines.
Evaluating the veracity of a claim in for example social media conversations
requires context – e.g. prior knowledge – and strong analytical skills [3].
One proxy is “stance". Stance is the kind of reaction that an author has to a
claim. Measuring the stance of the crowd as they react to a claim on social
media or other discussion fora acts as a reasonable proxy of claim veracity.

The problem of stance detection has only been addressed for a limited range of
languages: English, Spanish and Catalan [4, 5, 6]. With these, there are
several datasets and shared tasks. While adopting now more mature standards
for describing the task and structuring data, RuStance enables stance
prediction in a new context.

Debate about media control and veracity has a strong tradition in Russia; the
populace can be vocal, for example overthrowing an unsatisfactory ruling
empire in 1918. Veracity can also be questionable, for example in the case of
radio transmitters in key areas being used to send targeted messages before
the internet became the medium of choice [7]. Indeed, news on events and
attitudes is Russia is often the focus of content in US and European media,
with no transparent oversight or fact checking. The context is therefore one
that may benefit greatly from, or least be highly engaged with, veracity and
stance technology.

This paper relates the construction of a stance dataset for Russian, RuStance.
The dataset is available openly for download, and accompanied by baselines for
stance prediction on the data and analysis of the results.

##  2 Data and Resources

Before collecting data, we set the scope of the task and criteria for data
likely to be “interesting". Collection centred around a hierarchical model of
conversation, with stories at the top having source rumours/claims, which are
referenced as “source tweets" in prior Twitter-centric work [8], that are the
root of discussion threads, with responses potentially expressing a stance
toward that source claim. The dataset is available on GitHub.
111https://github.com/npenzin/rustance

###  2.1 Requirements

The data was collected during manual observation in November 2017. Tweets that
started a useful volume of conversationl activity on a topic we had been
observing were considered “interesting”, and were added to the list of the
potentially valuable sources.

For individual messages, we needed to determine the objective support towards
a rumour, an entire statement, rather than individual target concepts.
Moreover, we were to determine additional response types to the rumourous
tweet that are relevant to the discourse, such as a request for more
information (questioning) and making a comment (C), where the latter doesn’t
directly address support or denial towards the rumour, but provides an
indication of the conversational context surrounding rumours. For example,
certain patterns of comments and questions can be indicative of false rumours
and others indicative of rumours that turn out to be true.

Following prior work [9, 10], we define Stance Classification as predicting a
class (label) given a text (features) and topic (the rumour). Classes are
Support, Deny, Query, Comment.

  * •

Support: the author of the response supports the veracity of the rumour,
especially with facts, mentions, links, pictures, etc. For instance, “Yes,
that’s what BBC said.”

  * •

Deny: the author of the response denies the veracity of the rumour, the
opposite case of the Support class. For instance, “Under the bed???? I’ve been
there and there were not any monsters!”

  * •

Query: the author of the response asks for additional evidence in relation to
the veracity of the rumour. This one is usually is said in a qustionable
manner. For instance, “Could you provide any proof on that?”

  * •

Comment: the author of the response makes their own comment without a clear
contribution to assessing the veracity of the rumour. The most common class,
but not the worst. The examples of the class usually contains a lot of
emotions and personal opinions, for example, “Hate it. This is awesome!” -
tells us nothing about the veracity.

Claim: The Ministry of Defense published irrefutable evidence of US help for
ISIS. • Reply 1: Come’on. This is a screenshot from ARMA. [deny]. • Reply 2:
Good job! [comment] • Reply 3: Is that for real? [query] • Reply 4: That’s it!
RT also say so [support]

Figure 1: A synthetic example of a claim and related reactions (English).

Claim: #СИРИЯ Минобороны России публикует неоспоримое подтверждение
обеспечения Соединенными Штатами прикрытия боеспособны…
https://t.co/auTz1EBYX0 • Reply 1: Это не фейк, просто перепутали фото. И,
кстати, заметили и исправили. Это фото из других ново… https://t.co/YtBxWebenL
[deny]. • Reply 2: Министерство фейков, пропаганды и отрицания. [comment] •
Reply 3: Что за картинки, на которых вообще не понятно - кто,где и когда?
[query] • Reply 4: Эту новость даже провластная Лента запостила, настолько она
ржачная) [support]

Figure 2: A real example of a claim and related reactions (Russian).

###  2.2 Sources

In order to create variety, the data is drawn from multiple sources. This
increases the variety in the dataset, thus in turn aiding classifiers in
generalizing well. For RuStance, the sources chosen were: Twitter, Meduza
[11], Russia Today [RT] [12], and selected posts that had an active
discussion.

  * •

Twitter: is one of the most popular sources of claims and reactions. We paid
attention to a well-known ambiguous and false claims, for example the one [13]
mentioned – Russian Ministry Of Defense.

  * •

Meduza: is an online and independent media located in Latvia and focused on
Russian speaking audience. Meduza is known to be in opposition to Kremlin and
Russian politicians. The media has ability for registered users to leave
comments to a particular events. We collected comments on some popular
political events that were discussed more than the others [14, 15].

  * •

Russia Today: is an international media that targets world wide auditory. It
is supposed that the editors policy of Russia Today is to support Russian
goverment. Its main topics are politics and international relation which means
there are always debates. We gathered some political and provocative
publications with a lot of comments [16].

To capture claims and reactions in Twitter we used software developed as part
of the PHEME project [17] which allows to download all of the threads and
replies of a claim-tweet. For other sources we downloaded it by hand or
copied.

The dataset sources are Twitter and Meduza with 700 and 200 entities
respectively.

Firstly, Twitter is presented with over 700 interconnected replies, i.e.
replies both to the claim and to other replies. The latter might be a cause of
a large number of arguments and aggression (i.e. have high emotional tension)
and as a result, the replies are poorly structured from the grammatical
perspective, contain many non-vocabulary words in comparison with national or
web corpora. Tweets that were labeled as "support" and "deny" tend to have
links to related sources or mentions. Users of Twitter also use more
multimedia, which brings in auditory content not included in this text corpus.

Secondly, Meduza comments are discovered to be more grammatically correct and
less aggressive but still non-neutral and sarcastic. Meduza users are mostly
deanonymized, but unfortunately, this is our empirical observation and not
mentioned in the data. Comments on the articles vary in amounts of aggression,
however still less aggressive than tweets. We hypothesize that this is caused
by the fact that news articles provide more context and have teams of editors
behind it. Users of Meduza tend to provide fewer links and other kinds of
media, which may be due to the user interface of the site, or a factor of the
different nature of social interactions on this platform.

Finally, Russia Today content is very noisy, and difficult to parse. This
provided the smallest contribution to the dataset, and had the least structure
and coherence in its commentary. Overall, the dataset contains both structured
and grammatically correct comments and unstructured messy documents. This is
indicative of a good sample; one hopes that a dataset for training tools that
operate over social media will contain a lot of the noise characteristic of
that text type, enabling models equipped to handle the noise in the wild.

Typical headlines for collection included (translated):

  * •

The Ministry of Defense accused “a civil co-worker" in the publication of the
screenshot from the game instead of the photo of the terrorists. And presented
a new “indisputable evidence";

  * •

The Ministry of Defense posted an “indisputable" evidence of US cooperation
with ISIS: a screenshot from a mobile game;

  * •

The Bell has published a wiretapping of Sechin and Ulyukaev phone calls;

  * •

Navalny seized from “Life" 50 thousand rubles. He didn’t receive this money as
a “citizen journalist" for shooting himself;

  * •

“If the commander in chief will call into the last fight, Uncle Vova, we are
with you." Deputy of the Duma with the Cadets sang a song about Putin;

  * •

“We are very proud of breaking this law." Representatives of VPN-services told
"Medusa" why they are not going to cooperate with Russian authorities;

  * •

Muslims of Russia suggested to teach “The basics of religious cultures" from 4
til 11 grade;

  * •

“Auchan" (supermarket) will stop providing free plastic bags.

Table 1: Dataset class distribution Support | Deny | Query | Comment
---|---|---|---
58 (6%) | 46 (5%) | 192 (20%) | 662 (69%)

The most valuable classes – Support and Deny – are outnumbered by more
general-purpose classes. This is similar to the class distribution that other
stance classification datasets usually have [4, 5, 6]. Indeed, FNC-I, the Fake
News Challenge dataset [18], also has a quite similar class distribution: 70%
- comments, 20% - queries, 10% - support & deny.

In the interests of describing the origins and potential biases in the
dataset, a brief data statement [19] follows.

  * •

Curation rationale Text was drawn from sources likely to hold debate and
discussion, incorporating many different viewpoints. The dataset should be
useful to those building systems to be applied to commentary on Russian news.

  * •

Language variety The BCP-47 descriptors relevant are ru-Cyrl-RU and ru-Cyrl-
LV.

  * •

Speaker demographic Most speakers are L1 with other information hidden. It is
assumed that all have internet access and most are adept internet users.

  * •

Annotator demographic Annotators are males between 25-40 of European descent,
with high degrees of education.

  * •

Speech situation Utterances are from November 2017, written, without editing,
and spontaneous, in a public internet conversation context.

##  3 Implementation

As a baseline, and to provide a platform for analysis of the data, we built a
pipeline of corpora preprocessing, feature extaction and classifier training.

###  3.1 Preprocessing

Dealing with natural languages is often a complicated task with many caveats;
this is no better with social media. Phenomena prevalent in social media text
include typos, acronyms, slang and another examples of non-dictionary and
unexpected words. This can mean that finding representations for words can be
very noisy, because the embedding models are usually trained using normal and
grammatically correct corpora. In the case of RuStance, this is exacerbated by
the paucity of large Russian datasets or embedding collections.

In order to be able to process in a form of vectors in a meaningful
n-dimensional space, the texts of the dataset have to be converted from a
human friendly representation to Word Embeddings [20]. First, a strict
filtering step removes all of the social-media-specific entities like -like
mentions, hashtags, URLs. We decided to proceed only with words and standard
punctuation.

To convert a token to the expected format we had to pick a threshold that
would cut out the outliers. Also, by having a fixed length input (array of
tokens) it is possible to fit smaller texts into the input by substituting the
missing tokens with zero-tokens.

[*]KutuzovKuzmenko2017 provide a pretrained model of word embeddings for
Russian called RusVectores. Since our dataset covers mostly web and social
conversations, we used a model that was trained on a web corpora and
Wikipedia. This hopefully increases the probability that words from our
dataset will be in the vocabulary.

Figure 3:  Dataset symbols/words length distribution.

To maintain a uniform feature representation, we set the input length to 25
words. We set the length of the input to be 25 words because over 95% of the
records would pass the threshold (Figure 3) and the outliers will be
abandoned. For the records that have less than 25 words we used pad sequence
from Keras [36] which appends empty word embeddings in order to equalize
lengths. Unknown tokens (the tokens that are not in the vocabulary) are
substituted with zero-vectors.

So far an entry of the dataset for the classifier has 25 features = word
embeddings and a class label. Then features can be used as an input for both
Bayesian and Deep Learning models. At this step we have prepared dataset that
later will be split for delayed cross-validation, model training and
evaluation.

After the tokenization Keras Tokenizer is next to be trained on the texts to
substitute words with Term Frequency - Inverse Document Frequency [TFIDF]
indices. Using that indices and Gensim package [62] we create Embedding layer
of the Deep Learning model [54].

It is known from the documentation that the models are initialized with the
most common defaults, thus we expect our models to perform at a non-zero
level.

##  4 Evaluation and Discussion

The dataset is split into train and test partitions, and then perform cross-
validation.

We evaluate model performance using accuracy and f1-measure. The dataset
consists of short messages, so accuracy will be more representative in the
context of overall analysis, whereas f1-measure will partially compensate the
imbalance of classes. In this case, accuracy tends to be more effective, since
we want to exclude as many false positives as possible in order to not to call
arbitrary media to be fake.

Table 2: Evaluation metrics after Cross-validation, k = 5 | Bagging | AdaBoost | Boosting | SGD | Logistic Regression
---|---|---|---|---|---
F1 | 0.832 | 0.530 | 0.865 | 0.266 | 0.259
Accuracy | 0.925 | 0.766 | 0.925 | 0.582 | 0.678

With trained split stratified validation and precision as a metric we built
Confusion Matrices (fig. 4) for top-scoring classifiers. The fit-predict part
was performed with GridSearch cross-validation with K equals 5 and train/test
split coefficient equals 0.1.

###  4.1 Analysis

To analyse misclassification errors, we generated confusion matrixes. These
enable visual comparison of accurate and error predictions for each class. The
ideal outcome of such a visualization would be a unit matrix since the main
diagonal would contain ones which means that every class was predicted 100%
correct. In practice, the stronger (close to 1) the main diagonal – the better
the classifier performs.

Figure 4:  Confusion matrices for SQDC classification of stance on RuStance.

According to the confusion matrices one can infer that models tend to
overfitting by predicting Comment-class all of the time, which would result in
around 70% of true positives. However, the models that use Bagging and
Boosting are more reliable and predict all of the classes more or less
equally.

##  5 Related Work

Evaluation of the reliability of the information is a difficult and time
consuming task, even for trained professionals [58]. Fortunately, the process
can be divided into steps or stages, some of which can later be automated.

The first step towards classifying claims as either fake or trustworthy is to
find out how others react to said claims. This process of Stance Detection
plays significant role in fact-checking pipelines [67, 60, 58].

Since initial work on determining veracity of social media [9], powerful
systems and annotation schemes have been developed to support the analysis of
rumours and misinformation in text. In this paper, the authors introduced
methods to enhance the performance of classifiers trained on relatively long-
term rumours in tweets. The idea was later extended by [47] who, in addition,
tracked the presence of positive or negative markers in a tweet. Later [4]
provides a novel hand crafted dataset of rumoured claims. Our work is to
provide a similar dataset for Russian tweets.

[10] exploited the temporal sequence of tweets, although the conversational
structure was ignored and each tweet was treated as a separate unit. In other
domains where debates or conversations are involved, the sequence of responses
has been exploited to make the most of the evolving discourse and perform an
improved classification of each individual post after learning the structure
and dynamics of the conversation as a whole.

Classification of stance towards a claim on Twitter has been mentioned in
SemEval-2016 task 6 [67]. Subtask B tested stance detection towards an
unlabelled target, which required a weakly supervised or unsupervised
approach. The dataset of this competition was not related to rumours or
breaking news, it only considered a 3-way classification and did not provide
any relations between tweets, which were treated as individual instances.

Recent state-of-the-art work on stance classification includes the top-scoring
system in the RumourEval exercise [66], which decomposed conversational
branches into lines for use as context input to an LSTM [72], and another
approach based on proximity to English words representative of certain stance
classes [69]. Stance detection has also been used to develop datasets of
diverse stance, enabling construction of balanced summaries and examination of
arugmentation and counter-argumentation [70]. Finally, while RumourEval, the
Fake News Challenge, FEVER [71] and others have provided datasets, and some
have developed creative uses of this stance data, all of these resources are
in English; RuStance is the first such dataset for Russian, a language hosting
an active, nuanced and passionate political debate.

##  6 Conclusion

RuStance [21] is the first dataset for stance classification in Russian. We
think it is a highly area to study, collecting data for future researchers
with tools and data, and opening up the arena of fake news in Russia to global
researchers. It comprises a dataset from multiple sources, including many
conversation threads, and a mixture of social interaction.

A baseline is included. As we assumed, a thousand examples is definitely not
enough to fit the LSTM layer. Nevertheless, we achieved accuracy over 90%
using classifiers without any tuning whatsoever. Confusion matrices suggest
that even if a single precision rate is high the class imbalance is still a
huge issue and the bottleneck that stops us from fitting a really accurate
classifiers.

New metrics are also to be considered and developed. Russian tweets seem to be
as reliable as English in terms of consistency and class distribution. This
dataset and baseline provide first steps into analyzing fake news spread and
generation among Russian speakers, and we hope, with further work,
multilingually.

## References

  * [1] K. Rapoza, “These Two Russian ’Fake News’ Outfits Get Billions Of Hits On Facebook.” https://www.forbes.com/sites/kenrapoza/2017/09/22/these-two-russian-fake-news-outfits-get-billions-of-hits-on-facebook, 2017\.
  * [2] A. Zubiaga, A. Aker, K. Bontcheva, M. Liakata, and R. Procter, “Detection and Resolution of Rumours in Social Media: A Survey,” ArXiv e-prints, Apr. 2017.
  * [3] D. Mrowca, E. Wang, and A. Kosson, “Stance detection for fake news identification,” 2017.
  * [4] W. Ferreira and A. Vlachos, “Emergent: a novel data-set for stance classification,” in Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, ACL, 2016.
  * [5] A. F. Anta, L. N. Chiroque, P. Morere, and A. Santos, “Sentiment analysis and topic detection of spanish tweets: A comparative study of of nlp techniques,” Procesamiento del lenguaje natural, vol. 50, pp. 45–52, 2013\.
  * [6] M. Taulé, M. A. Martí, F. M. Rangel, P. Rosso, C. Bosco, V. Patti, et al., “Overview of the task on stance and gender detection in tweets on catalan independence at ibereval 2017,” in 2nd Workshop on Evaluation of Human Language Technologies for Iberian Languages, IberEval 2017, vol. 1881, pp. 157–177, CEUR-WS, 2017.
  * [7] R. Enikolopov, M. Petrova, and E. Zhuravskaya, “Media and political persuasion: Evidence from Russia,” American Economic Review, vol. 101, no. 7, pp. 3253–85, 2011.
  * [8] A. Zubiaga, E. Kochkina, M. Liakata, R. Procter, and M. Lukasik, “Stance classification in rumours as a sequential task exploiting the tree structure of social media conversations,” arXiv preprint arXiv:1609.09028, 2016.
  * [9] V. Qazvinian, E. Rosengren, D. R. Radev, and Q. Mei, “Rumor has it: Identifying misinformation in microblogs,” in Proceedings of the Conference on Empirical Methods in Natural Language Processing, pp. 1589–1599, Association for Computational Linguistics, 2011.
  * [10] M. Lukasik, P. Srijith, D. Vu, K. Bontcheva, A. Zubiaga, and T. Cohn, “Hawkes processes for continuous time sequence classification: an application to rumour stance classification in twitter,” in Proceedings of 54th Annual Meeting of the Association for Computational Linguistics, pp. 393–398, Association for Computational Linguistics, 2016.
  * [11] Meduza, “Meduza.io.” http://meduza.io, 2018.
  * [12] Channel RT TV, “Russia Today.” https://rt.com, 2018.
  * [13] The Guardian, “Russia’s ’irrefutable evidence’ of us help for isis appears to be video game still.” https://www.theguardian.com/world/2017/nov/14/russia-us-isis-syria-video-game-still, 2017\.
  * [14] Meduza, “Meduza.io: On Fake Evidence.” https://meduza.io/shapito/2017/11/14/minoborony-vylozhilo-neosporimoe-dokazatelstvo-sotrudnichestva-ssha-i-ig-skrinshot-iz-mobilnoy-igry, 2017\.
  * [15] Meduza, “Meduza.io: On Ministry of Defense.” https://meduza.io/shapito/2017/11/14/minoborony-vylozhilo-neosporimoe-dokazatelstvo-sotrudnichestva-ssha-i-ig-skrinshot-iz-mobilnoy-igry, 2017\.
  * [16] Channel RT TV, “Russia Today: On Russian President Candidates 2018.” https://russian.rt.com/inotv/2017-11-14/Rukovoditel-internet-kampanii-Sobchak-eyo-uchastie, 2017\.
  * [17] L. Derczynski and K. Bontcheva, “Pheme: Veracity in digital social networks.,” in UMAP Workshops, 2014.
  * [18] D. Pomerleau and D. Rao, “Fake news challenge.” http://www.fakenewschallenge.org, 2017.
  * [19] Anonymous, “Data Statements for NLP: Toward Mitigating System Bias and Enabling Better Science.” OpenReview.net, 2018.
  * [20] T. Mikolov, I. Sutskever, K. Chen, G. S. Corrado, and J. Dean, “Distributed representations of words and phrases and their compositionality,” in Advances in neural information processing systems, pp. 3111–3119, 2013.
  * [21] “Rustance.” https://figshare.com/articles/dataset_csv/7151906/2.
  * [22] B. Pang and L. Lee, Opinion mining and sentiment analysis.  Now Publishers Inc., Foundations trends in information retrieval, available at http://portal. acm. org/citation. cfm, 2008.
  * [23] B. Liu, “Sentiment analysis and subjectivity.,” Handbook of natural language processing, vol. 2, pp. 627–666, 2010.
  * [24] A. Go, R. Bhayani, and L. Huang, “Twitter sentiment classification using distant supervision,” CS224N Project Report, Stanford, vol. 1, no. 12, 2009\.
  * [25] G. Laboreiro, L. Sarmento, J. Teixeira, and E. Oliveira, “Tokenizing micro-blogging messages using a text classification approach,” in Proceedings of the fourth workshop on Analytics for noisy unstructured text data, pp. 81–88, ACM, 2010.
  * [26] A. Pak and P. Paroubek, “Twitter as a corpus for sentiment analysis and opinion mining.,” in LREC, 2010.
  * [27] K. Kukich, “Techniques for automatically correcting words in text,” Acm Computing Surveys (CSUR), vol. 24, no. 4, pp. 377–439, 1992.
  * [28] A. Bermingham and A. F. Smeaton, “Classifying sentiment in microblogs: is brevity an advantage?,” in Proceedings of the 19th ACM international conference on Information and knowledge management, pp. 1833–1836, ACM, 2010\.
  * [29] C. Engstrom, “Topic dependence in sentiment classification,” Master’s thesis, University of Cambridge, 2004.
  * [30] J. Read, “Using emoticons to reduce dependency in machine learning techniques for sentiment classification,” in Proceedings of the ACL student research workshop, pp. 43–48, Association for Computational Linguistics, 2005\.
  * [31] L. Padró, S. Reese, E. Agirre, and A. Soroa, “Semantic services in freeling 2.1: Wordnet and ukb,” in 5th Global WordNet Conference, pp. 99–105, 2010.
  * [32] C.-C. Chang and C.-J. Lin, “Libsvm: a library for support vector machines,” ACM transactions on intelligent systems and technology (TIST), vol. 2, no. 3, p. 27, 2011.
  * [33] M. Mathioudakis and N. Koudas, “Twittermonitor: trend detection over the twitter stream,” in Proceedings of the 2010 ACM SIGMOD International Conference on Management of data, pp. 1155–1158, ACM, 2010.
  * [34] A. Vakali, M. Giatsoglou, and S. Antaris, “Social networking trends and dynamics detection via a cloud-based framework design,” in Proceedings of the 21st International Conference on World Wide Web, pp. 1213–1220, ACM, 2012\.
  * [35] L. Buitinck, G. Louppe, M. Blondel, F. Pedregosa, A. Mueller, O. Grisel, V. Niculae, P. Prettenhofer, A. Gramfort, J. Grobler, R. Layton, J. VanderPlas, A. Joly, B. Holt, and G. Varoquaux, “API design for machine learning software: experiences from the scikit-learn project,” in ECML PKDD Workshop: Languages for Data Mining and Machine Learning, pp. 108–122, 2013\.
  * [36] F. Chollet et al., “Keras.” https://github.com/keras-team/keras, 2015\.
  * [37] K. M. Hermann, T. Kocisky, E. Grefenstette, L. Espeholt, W. Kay, M. Suleyman, and P. Blunsom, “Teaching machines to read and comprehend,” in Advances in Neural Information Processing Systems, pp. 1693–1701, 2015.
  * [38] Z. Wang, W. Hamza, and R. Florian, “Bilateral multi-perspective matching for natural language sentences,” arXiv preprint arXiv:1702.03814, 2017.
  * [39] I. Augenstein, T. Rocktäschel, A. Vlachos, and K. Bontcheva, “Stance detection with bidirectional conditional encoding,” arXiv preprint arXiv:1606.05464, 2016.
  * [40] W. Wang, S. Yaman, K. Precoda, C. Richey, and G. Raymond, “Detection of agreement and disagreement in broadcast conversations,” in Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies: short papers-Volume 2, pp. 374–378, Association for Computational Linguistics, 2011.
  * [41] A. Zubiaga, M. Liakata, R. Procter, K. Bontcheva, and P. Tolmie, “Towards detecting rumours in social media,” in AAAI Workshop: AI for Cities, 2015\.
  * [42] R. Abbott, M. Walker, P. Anand, J. E. Fox Tree, R. Bowmani, and J. King, “How can you say such things?!?: Recognizing disagreement in informal political argument,” in Proceedings of the Workshop on Languages in Social Media, pp. 2–11, Association for Computational Linguistics, 2011.
  * [43] N. FitzGerald, G. Carenini, G. Murray, and S. Joty, “Exploiting conversational features to detect high-quality blog comments,” Advances in Artificial Intelligence, pp. 122–127, 2011.
  * [44] P. Nakov, L. Màrquez, W. Magdy, A. Moschitti, J. Glass, and B. Randeree, “Semeval-2015 task 3: Answer selection in community question answering.,” SemEval@ NAACL-HLT, vol. 2015, 2015.
  * [45] Z. Qu and Y. Liu, “Interactive group suggesting for twitter,” in Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies: short papers-Volume 2, pp. 519–523, Association for Computational Linguistics, 2011.
  * [46] Z. Zhao, P. Resnick, and Q. Mei, “Enquiring minds: Early detection of rumors in social media from enquiry posts,” in Proceedings of the 24th International Conference on World Wide Web, pp. 1395–1405, International World Wide Web Conferences Steering Committee, 2015.
  * [47] X. Liu, A. Nourbakhsh, Q. Li, R. Fang, and S. Shah, “Real-time rumor debunking on twitter,” in Proceedings of the 24th ACM International on Conference on Information and Knowledge Management, pp. 1867–1870, ACM, 2015.
  * [48] O. Enayet and S. R. El-Beltagy, “Niletmrg at semeval-2017 task 8: Determining rumour and veracity support for rumours on twitter.,” in Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017), pp. 470–474, 2017.
  * [49] H. Bahuleyan and O. Vechtomova, “Uwaterloo at semeval-2017 task 8: Detecting stance towards rumours with topic independent features,” in Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017), pp. 461–464, 2017.
  * [50] A. Zubiaga, M. Liakata, R. Procter, G. W. S. Hoi, and P. Tolmie, “Analysing how people orient to and spread rumours in social media by looking at conversational threads,” PloS one, vol. 11, no. 3, p. e0150989, 2016.
  * [51] C. Shao, G. L. Ciampaglia, A. Flammini, and F. Menczer, “Hoaxy: A platform for tracking online misinformation,” in Proceedings of the 25th International Conference Companion on World Wide Web, pp. 745–750, International World Wide Web Conferences Steering Committee, 2016.
  * [52] Q. Zhang, S. Zhang, J. Dong, J. Xiong, and X. Cheng, “Automatic detection of rumor on social network,” in Natural Language Processing and Chinese Computing, pp. 113–122, Springer, 2015.
  * [53] K. K. Kumar and G. Geethakumari, “Detecting misinformation in online social networks using cognitive psychology,” Human-centric Computing and Information Sciences, vol. 4, no. 1, p. 14, 2014.
  * [54] A. Kutuzov and E. Kuzmenko, WebVectors: A Toolkit for Building Web Interfaces for Vector Semantic Models, pp. 155–161.  Cham: Springer International Publishing, 2017.
  * [55] Y. Goldberg and O. Levy, “word2vec explained: deriving mikolov et al.’s negative-sampling word-embedding method,” arXiv preprint arXiv:1402.3722, 2014.
  * [56] S. Tavernise, “As Fake News Spreads Lies, More Readers Shrug at the Truth.” https://www.nytimes.com/2016/12/06/us/fake-news-partisan-republican-democrat.html, 2016\.
  * [57] J. H. Michael Barthel, Amy Mitchell, “Many americans believe fake news is sowing confusion.” http://www.journalism.org/2016/12/15/many-americans-believe-fake-news-is-sowing-confusion, 2016\.
  * [58] D. Ghulati, “Introducing factmata — artificial intelligence for political fact-checking.” https://medium.com/factmata/introducing-factmata-artificial-intelligence-for-political-fact-checking-db8acdbf4cf1, 2016\.
  * [59] S. Ruder, “On word embeddings.” http://ruder.io/word-embeddings-1, 2016\.
  * [60] Y. P. Sean Baird, Doug Sibley, “Talos targets disinformation with fake news challenge victory.” http://blog.talosintelligence.com/2017/06/talos-fake-news-challenge.html, 2017\.
  * [61] A. Hanselowski, “Team athene on the fake news challenge.” https://medium.com/@andre134679/team-athene-on-the-fake-news-challenge-28a5cf5e017b, 2017\.
  * [62] R. Řehůřek and P. Sojka, “Software Framework for Topic Modelling with Large Corpora,” in Proceedings of the LREC 2010 Workshop on New Challenges for NLP Frameworks, (Valletta, Malta), pp. 45–50, ELRA, May 2010.  http://is.muni.cz/publication/884893/en.
  * [63] B. Riedel, I. Augenstein, G. P. Spithourakis, and S. Riedel, “A simple but tough-to-beat baseline for the fake news challenge stance detection task,” arXiv preprint arXiv:1707.03264, 2017.
  * [64] N. Rakholia and S. Bhargava, ““is it true?”–deep learning for stance detection in,”
  * [65] P. Krejzl, B. Hourová, and J. Steinberger, “Stance detection in online discussions,” arXiv preprint arXiv:1701.00504, 2017.
  * [66] L. Derczynski, K. Bontcheva, M. Liakata, R. Procter, G. W. S. Hoi, and A. Zubiaga, “Semeval-2017 task 8: Rumoureval: Determining rumour veracity and support for rumours,” arXiv preprint arXiv:1704.05972, 2017.
  * [67] S. M. Mohammad, P. Sobhani, and S. Kiritchenko, “Stance and sentiment in tweets,” ACM Transactions on Internet Technology (TOIT), vol. 17, no. 3, p. 26, 2017.
  * [68] S. Hamidian and M. T. Diab, “Rumor identification and belief investigation on twitter.,” in Proceedings of WASSA at NAACL-HLT, pp. 3–8, 2016.
  * [69] A. Aker, L. Derczynski, and K. Bontcheva, “Simple open stance classification for rumour analysis,” in Proc. RANLP, 2017.
  * [70] S. Ruder, J. Glover, A. Mehrabani, and P. Ghaffari, “360∘ stance detection,” in Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations, pp. 31–35, Association for Computational Linguistics, 2018.
  * [71] J. Thorne, A. Vlachos, C. Christodoulopoulos, and A. Mittal, “Fever: a large-scale dataset for fact extraction and verification,” in Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pp. 809–819, Association for Computational Linguistics, 2018.
  * [72] E. Kochkina, M. Liakata, and I. Augenstein, “Turing at SemEval-2017 Task 8: Sequential Approach to Rumour Stance Classification with Branch-LSTM,” in Proc. SemEval, 2017.