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\begin{document}
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\title[Unbounded divergence of interpolatory product quadrature formulas]{Superdense unbounded divergence of a class \\
of interpolatory product quadrature formulas}
\author{Alexandru I. Mitrea}
\address{Technical University, Department of Mathematics\\
Str. C. Daicoviciu nr. 15,
400020 Cluj-Napoca, Romania}
\email{alexandru.ioan.mitrea@math.utcluj.ro}
%
\subjclass{41A10, 65D32}
\keywords{Superdense set, unbounded divergence, product quadrature formulas, Dini-Lipschitz convergence}

\begin{abstract}
The aim of this paper is to highlight the superdense unbounded divergence
of a class of product quadrature formulas of interpolatory type on Jacobi nodes,
associated to the Banach space of all real continuous functions
defined on $[-1,1]$, and to a Banach space of measurable and essentially
bounded functions $g:[-1,1]\to \mathbb{R}$.
Some aspects regarding the convergence of these formulas are pointed out, too.
\end{abstract}

\maketitle

\section{Introduction}
This paper deals with a class of interpolatory product quadrature formulas,
regarding their divergence and the convergence rate, as follows.
Let $C$ be the Banach space of all continuous functions
$f:[-1,1]\to \mathbb{R}$,
endowed with the supremum norm $\|\cdot \|$.
Denoting by $\mu $ the Lebesgue measure on the interval $[-1,1]$, let
$(L_p,\|\cdot \|_p)$, $1\le p\le \infty $,
be the Banach space of all measurable functions
(equivalence classes of functions, with respect to the equality $\mu $-a.e.)
$g:[-1,1]\to \mathbb{R}$, normed by
$$\|g\|_p=\left(\int_{-1}^1 |g(x)|^p dx\right)^{1/p},
\mbox{ if } 1\le p<\infty , \mbox{ and } \|g\|_\infty ={\rm esssup}|g|.$$

According to \cite{7}, \cite{8}, if $p\in [1,\infty ]$
and $\rho \in L_q$ (with $p^{-1}+q^{-1}=1$)
are given such that $\rho (x)>0$ $\mu $-a.e. on $[-1,1]$, the notation
$(L_p^{(1/\rho )},\|\cdot \|_p^{(1/\rho )})$
stands for the Banach space of all measurable functions $g$ for which
$g/\rho \in L_p$ and
$\|g\|_p^{(1/\rho )}=\|g/\rho \|_p$.

Further, let consider an arbitrary triangular node matrix
$$\mathcal{M}=\{x_{kn}:\ n\ge 1,\ 1\le k\le n\}$$
so that the $n$-th row of $\mathcal{M}$, $n\ge 1$,
contains $n$ distinct nodes of $[-1,1]$, then let us denote,
as usual, by
$\mathcal{L}_n f\in \mathcal{P}_{n-1}$ (the space of all polynomials of degree at most $n-1$)
and $\lambda _n$ the Lebesgue functions associated to the $n$-th row of $\mathcal{M}$,
respectively, i.e.,
$$(\mathcal{L}_n f)(x)=\sum_{k=1}^n f(x_{kn})l_{kn}(x),\ f\in C,\
\lambda _n(x)=\sum_{k=1}^n |l_{kn}(x)|,$$
where $l_{kn}$ are the fundamental Lagrange interpolation polynomials, \cite{2}, \cite{10}.\\
The equalities
\begin{equation}
\label{1.1}
\int_{-1}^1 g(x)f(x)dx
=\int_{-1}^1 g(x)(\mathcal{L}_nf)(x)dx+R_n(f;g),\ f\in C,\ g\in L_p^{(1/\rho )},\ n\ge 1
\end{equation}
with
\begin{equation}
\label{1.2}
R_n(P,g)=0,\ \forall \ f\in \mathcal{P}_{n-1}\mbox{ and } g\in L_p^{(1/\rho )},\ n\ge 1
\end{equation}
describe {\it product quadrature formulas of interpolatory type},
associated to the spaces $C$ and $L_p^{(1/\rho )}$.

\noindent
If $p=1$, these product quadrature formulas were intensively studied,
in their convergence aspects, for various functions
$g\in L_1$, $\rho \in L_\infty $
(including $\rho (x)=(1-x)^a(1+x)^b$, $a,b\ge 0$)
and node matrices $\mathcal{M}$,
\cite{1}, \cite{3}, \cite{4}, \cite{7}, \cite{8}.
We notice, also, the divergence result obtained by I.H. Sloan and W.E. Smith,
for arbitrary node matrices $\mathcal{M}$ and $\rho (x)=1$, $-1\le x\le 1$,
\cite[Th. 6]{8}.
A recent result, \cite{5}, refers to more general product quadrature formulas
of interpolatory type, involving polynomial projection operators
$\mathcal{L}_n:C\to \mathcal{P}_{n-1}$
(namely
$\mathcal{L}_n f\in \mathcal{P}_{n-1}$, $\forall \ f\in C$, and
$\mathcal{L}_n f=f$ if and only if $f\in \mathcal{P}_{n-1}$)
instead of Lagrange projections in (\ref{1.1}) and highlights the phenomenon
of double condensation of singularities for the corresponding product quadrature
formulas (\ref{1.1}), meaning unbounded divergence on superdense sets belonging
to the spaces $C$ and $L_1^{(1/\rho )}$,
for arbitrary node matrices $\mathcal{M}$ and $\rho \in L_\infty $, with
$\rho (x)>0$ $\mu $-a.e. on $[-1,1]$.

The aim of this paper is to point out the superdense unbounded divergence of the product
quadrature formulas described by (\ref{1.1}) and (\ref{1.2}) for $p=\infty $,
$\rho (x)=(1-x)^a(1+x)^b$, with $a,b>-1$, and
$\mathcal{M}=\mathcal{M}^{(\alpha ,\beta )}$, $\alpha >-1$, $\beta >-1$,
where $\mathcal{M}^{(\alpha ,\beta )}$ is the Jacobi node matrix
(namely, its $n$-th row contains the roots $x_n^{(\alpha ,\beta )}$, $1\le k\le n$,
of the Jacobi polynomial $P_n^{(\alpha ,\beta )}$, $n\ge 1$).
Moreover, some aspects regarding the convergence of these formulas
(for functions $f\in C$ satisfying a Dini-Lipschitz condition and arbitrary
$g\in L_\infty ^{(1/\rho )}$) will be presented in the last section.

In this paper, the notation $M_k$, $k\ge 1$, stands for some positive constants
which do not depend on $n$.
Also, we denote by $\omega (f,\cdot )$ the modulus of continuity associated
to a function $f\in C$.

\section{Unbounded divergence on superdense sets}

Suppose that
$\rho (x)=(1-x)^a (1+x)^b$, $a,b>-1$ and
$\mathcal{M}=\mathcal{M}^{(\alpha ,\beta )}$, $\alpha >-1$, $\beta >-1$.
Let $U_n$, $n\ge 1$, be the continuous linear operators defined as
\begin{equation}
\label{2.1}
\left\{\ba{lll}
U_n:C\to (L_\infty ^{(1/\rho )})^*;\ f\mapsto U_n f\medskip \\
(U_n f)(g)=\ds\int_{-1}^1 g(x)(\mathcal{L}_n f)(x)dx;\ f\in C,\ g\in L_\infty ^{(1/\rho )},
\ea\right.
\end{equation}
where $Y^*$ is the Banach space of all continuous linear functionals defined on the normed
space $Y$.

Using standard arguments and classic results of Functional Analysis, we obtain
(see also \cite{8}):
$$\|U_n\|=\sup\{\|U_n f\|:\ f\in C,\ \|f\|\le 1\}$$
and
\begin{align*}
\|U_n f\|
& =\sup\left\{\left|\int_{-1}^1 g(x)(\mathcal{L}_n f)(x)dx\right|:\
g/\rho \in L_\infty ,\ \|g/\rho \|_\infty \le 1\right\}\\
& =\sup\left\{\left|\int_{-1}^1 \rho (x)g(x)(\mathcal{L}_n f)(x)dx\right|:\ g\in L_\infty ,\
\|g\|_\infty \le 1\right\},
\end{align*}
so we get
\begin{equation}
\label{2.2}
\|U_n\|=\sup\{\|\rho \mathcal{L}_n f\|_1:\ f\in C,\ \|f\|\le 1\},\ n\ge 1.
\end{equation}

Now, we can state:

\begin{theorem}
\label{t2.1}
Suppose that $\alpha \ge 2a+3/2$ or $\beta \ge 2b+3/2$.
Then, a superdense set $X_0$ in the Banach space $L_\infty ^{(1/\rho )}$
exists such that for every $g$ in $X_0$, the set of $C$ consisting of all functions
for which the product quadrature formulas described by (\ref{1.1}) and (\ref{1.2})
are unbounded divergent, namely
$$Y_0(g)=\left\{f\in C:\ \limsup_{n\to \infty }\left|\int_{-1}^1 g(x)(\mathcal{L}_n f)(x)dx\right|
=\infty \right\},$$
is superdense in the Banach space $C$.
\end{theorem}

\noindent
{\it Proof.}
First, we show that the set
$\{\|U_n\|:\ n\ge 1\}$
is unbounded.
Similarly to \cite{9}, let consider the function
$f_n\in C$, $n\ge 1$,
defined by
$$f_n(x)=
\left\{\ba{lll}
(-1)^k, & \mbox{if} & x=x_{kn}^{(\alpha ,\beta )},\ 0\le k\le n+1\medskip \\
\mbox{linear}, & \mbox{if} & x\in [x_{kn}^{(\alpha ,\beta )},x_{k,n-1}^{(\alpha ,\beta )}],\
1\le k\le n+1,
\ea\right.$$
where $x_{0n}^{(\alpha ,\beta )}=1$ and $x_{n+1,n}^{(\alpha ,\beta )}=-1$.

It follows from (\ref{2.2}):
\begin{equation}
\label{2.3}
\|U_n\|\ge \|\rho \mathcal{L}_n f_n\|_1
=\int_{-1}^1 (1-x)^a (1+x)^b |(\mathcal{L}_n f_n)(x)|dx.
\end{equation}

Next, let us suppose that
$\alpha \ge 2a+3/2>-1/2$
and set
$q_0=1-\ds\f{4(a+1)}{2\alpha +1}$
(so, $0\le q_0<1$).
Using the estimation of \cite[formula (3.3), with $p=1$ and $q=q_0$]{9},
we get:
\begin{equation}
\label{2.4}
\left\{\ba{lll}
\|U_n\|\ge M_1\log n, & \mbox{if} & q_0=0\medskip \\
\|U_n\|\ge M_2 n^{q_0(\alpha +1/2)}, & \mbox{if} & q_0>0.
\ea\right.
\end{equation}

The relations (\ref{2.3}) and (\ref{2.4}) prove the unboundedness of the set
$\{\|U_n\|:\ n\ge 1\}$,
if $\alpha \ge 2a+3/2$;
similarly, the same assertion is true for $\beta \ge 2b+3/2>-\ds\f{1}{2}$.

Now, we apply the principle of condensation of singularities
\cite[Theorem 5.2]{3}, with
$X=L_\infty ^{(1/\rho )}$, $T=C$, $Y=\mathbb{R}$,
$J=\mathbb{N}^*$ and
$A_n(g;f)=(U_n f)(g)$.
It is easily seen that the hypotheses $1\cc$ and $2\cc$ of this principle are fulfilled.
In order to show the validity of the hypothesis $3\cc$, denote by
$\mathcal{U}=\{U_n:\ n\ge 1\}$
the family of the operators defined by (\ref{2.1}).
Using the principle of condensation of singularities,
\cite[Th. 5.4]{3}, with respect to the family $\mathcal{U}$ and taking into account
the unboundedness of the set
$\{\|U_n\|:\ n\ge 1\}$,
we infer that the set of the singularities of $\mathcal{U}$,
namely
\begin{equation}
\label{2.5}
\mathcal{S}(U)=\{f\in C:\ \sup\{\|U_n f\|:\ n\ge 1\}=\infty \},
\end{equation}
is superdense in $C$.
Now, take
$T_0=\mathcal{S}(U)$ from (\ref{2.5}) and remark that
$$\sup\{\|A_n f\|:\ n\ge 1\}=\sup\{\|U_n f\|:\ n\ge 1\}=\infty ,$$
for every $f\in T_0$, therefore the hypothesis $3\cc$ of \cite[Theorem 5.2]{3}
holds, too.
Finally, denote by $Y_0(g)$ the set of singularities of the family
$\mathcal{A}(g)=\{A_n(g,\cdot ):\ n\ge 1\}$,
which completes the proof.
\hfill $\square $

\section{Dini-Lipschitz convergence}

Let us estimate the quadrature errors $R_n(f;g)$ of (\ref{1.1}),
see also \cite{7}, \cite{8}.
Denoting by
$I:C\to (L_\infty ^{(1/\rho )})^*$,
the operator given by
$(If)(g)=\ds\int_{-1}^1 g(x)f(x)dx$
and taking into account the interpolatory condition (\ref{1.2}), we get:
\begin{equation}
\label{3.1}
|R_n(f;g)|=|(U_n-I)(f-p)(g)|\le \|U_1-I\|\cdot \|f-p\|\cdot \|g\|_\infty ^{(1/\rho )}.
\end{equation}
Further, we obtain, for every $f\in C$:
$$\|\rho \mathcal{L}_n f\|_1=\int_{-1}^1 \rho (x)|(\mathcal{L}_n f)(x)|dx
\le \left(\int_{-1}^1 \rho (x)\lambda _n(x)dx\right)\|f\|,$$
so, (\ref{2.2}) leads to:
\begin{equation}
\label{3.2}
\|U_n\|\le \|\rho \lambda _n\|_1.
\end{equation}
Similarly, we get
\begin{equation}
\label{3.3}
\|I\|\le \|\rho \|_1.
\end{equation}
Now, combining the relations (\ref{3.1}), (\ref{3.2}) and (\ref{3.3}), the estimation
\begin{equation}
\label{3.4}
\|R_n(f;g)\|\le M_3(\|\rho \|_1+\|\rho \lambda _n\|_1)\cdot \|g/\rho \|_\infty
\cdot \omega \left(f;\ds\f{1}{n}\right)
\end{equation}
holds for sufficient large $n\ge 1$.

The following step is to estimate $\|\rho \lambda _n\|_1$.
We have:
\begin{equation}
\label{3.5}
\left\{\ba{lll}
\|\rho \lambda _n\|_1=\ds\int_{-1}^1 (1-x)^a (1+x)^b \lambda _n(x)dx=I_n^{(1)}+I_n^{(2)},\mbox{ with}\medskip \\
I_n^{(1)}=\ds\int_{-1}^0 (1-x)^a (1+x)^b \lambda _n(x)dx
\mbox{ and }\medskip \\
I_n^{(2)}=\ds\int_0^1 (1-x)^a (1+x)^b \lambda _n(x)dx.
\ea\right.
\end{equation}
Using the estimation
$$\lambda _n(x)-1\sim |P_n^{(\alpha ,\beta )}|\sqrt n[1+(1-x)^{(2\alpha +1)/4}\log n],\ 0\le x\le 1,\
\mbox{\cite{6}},$$
we obtain
$$I_n^{(2)}\sim \int_0^1 (1-x)^a dx+\sqrt n\int_0^1 (1-x)^a |P_n^{(\alpha ,\beta )}(x)|dx$$
\begin{equation}
\label{3.6}
+\sqrt n(\log n)\int_0^1 (1-x)^{a+\alpha /2+1/4}|P_n^{(\alpha ,\beta )}(x)|dx.
\end{equation}
Next, the estimation \cite[formula (7.34.1)]{10}
$$\int_0^1 (1-x)^\mu |P_n^{(\alpha ,\beta )}(x)|dx\sim
\left\{\ba{lll}
n^{\alpha -2\mu -2}, & \alpha >2\mu +3/2\medskip \\
n^{-1/2}\log n, & \alpha =2\mu +3/2\medskip \\
n^{-1/2}, & \alpha <2\mu +\ds\f{3}{2}
\ea\right.;\
\alpha ,\beta ,\mu >-1,$$
gives for $\mu =a$ and $\mu =a+\alpha /2+1/4$, respectively:
\begin{equation}
\label{3.7}
\int_0^1 (1-x)^a|P_n^{(\alpha ,\beta )}(x)|dx\sim
\left\{\ba{lll}
n^{\alpha -2a-2}, & \alpha >2a+3/2\medskip \\
n^{-1/2}\log n, & \alpha =2a+3/2\medskip \\
n^{-1/2}, & \alpha <2a+3/2
\ea\right.
\end{equation}
\begin{equation}
\label{3.8}
\int_0^1 (1-x)^{a+\alpha /2+1/4}|P_n^{(\alpha ,\beta )}|dx\sim n^{-1/2}.
\end{equation}
Finally, a combination of (\ref{3.6}), (\ref{3.7}) and (\ref{3.8}) yields:
\begin{equation}
\label{3.9}
I_n^{(2)}\sim 1+\log n+
\left\{\ba{lll}
n^{\alpha -2a-3/2}, & \alpha >2a+3/2\medskip \\
\log n, & \alpha =2a+3/2\medskip \\
1, & \alpha <2a+3/2.
\ea\right.
\end{equation}
A similar estimation holds for $I_n^{(1)}$ of (\ref{3.5}),
namely:
\begin{equation}
\label{3.10}
I_n^{(1)}\sim 1+\log n+
\left\{\ba{lll}
n^{\beta -2b-3/2}, & \beta  >2b+3/2\medskip \\
\log n, & \beta  =2b+3/2\medskip \\
1, & \beta  <2b+3/2.
\ea\right.
\end{equation}

Now, we prove the following statement.

\begin{theorem}
\label{t3.1}
If
$\rho (x)=(1-x)^a (1+x)^b$, $-1<\alpha \le 2a+3/2$ and
$-1<\beta \le 2b+3/2$,
then the product quadrature formulas given by (\ref{1.1}) and (\ref{1.2}) are convergent
for each $g\in L_\infty ^{(1/\rho )}$ and for each $f\in C$ satisfying a Dini-Lipschitz
condition
$$\lim\limits_{\delta \searrow 0}\omega (f;\delta )\log \delta =0.$$
\end{theorem}

\noindent
{\it Proof.}
The relations (\ref{3.5}), (\ref{3.9}) and (\ref{3.10}) lead to the estimation
$\|\rho \lambda _n\|_1\sim \log n$.
which combined with (\ref{3.4}) gives:
$$|R_n(f;g)|\le M_n\cdot \|g/\rho \|_\infty \cdot \omega \left(f;\ds\f{1}{n}\right)\log n,$$
for sufficient large $n\ge 1$, which completes the proof.
\hfill $\square $








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\end{document}