# Hessianizability of surface metrics

Robert L. Bryant  Duke University Mathematics Department
P.O. Box 90320
Durham, NC 27708-0320<EMAIL_ADDRESS>http://www.math.duke.edu/~bryant
Dedicated to the memory of Joseph A. Wolf, whose generous advice, inspiring
works, and seemingly inexhaustible knowledge of geometry has been a boon to me
and so many others.

(Date: January 30, 2024)

###### Abstract.

A symmetric quadratic form $g$ on a surface $M$ is said to be _locally
Hessianizable_ if each $p\in M$ has an open neighborhood $U$ on which there
exists a local coordinate chart $(x^{1},x^{2}):U\to\mathbb{R}^{2}$ and a
function $f:U\to\mathbb{R}$ such that, on $U$, we have

$g=\frac{\partial^{2}f}{\partial x^{i}\partial
x^{j}}\,\mathrm{d}x^{i}\circ\mathrm{d}x^{j}.$

In this article, I show that, when $g$ is nondegenerate and smooth, it is
always smoothly locally Hessianizable.

###### Key words and phrases:

Hessian metrics, Cartan-Kähler theory

###### 1991 Mathematics Subject Classification:

53B05, 58A15

Thanks to Duke University for its support via a research grant, to the NSF for
its support via DMS-0103884 and DMS-1359583 (originally DMS-1105868), and to
Columbia University for its support via a Samuel Eilenberg Visiting
Professorship during Spring 2004, during which this work was begun (and mostly
completed).

###### Contents

  1. 1 Introduction
  2. 2 The structure equations
  3. 3 An exterior differential system for Hessianizability
    1. 3.1 A prolongation
  4. 4 Involutivity
  5. 5 Hyperbolicity and smooth Hessianizability
    1. 5.1 The characteristic variety
    2. 5.2 Hyperbolicity

##  1\. Introduction

A symmetric quadratic differential form $g$ on an $n$-manifold $M^{n}$ is said
to be of _Hessian type_ if there exist $(n{+}1)$ functions
$x^{1},\ldots,x^{n},f$ on $M$ such that
$\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{n}\not=0$ and such that111Note
the use of the Einstein summation convention, which will be employed
throughout this article. Also, note the standard use of $\alpha\circ\beta$ for
$\tfrac{1}{2}(\alpha\otimes\beta+\beta\otimes\alpha)$, i.e., the symmetrized
tensor product.

$g=\frac{\partial^{2}f}{\partial x^{i}\partial
x^{j}}\,\mathrm{d}x^{i}\circ\mathrm{d}x^{j}\,.$

Note that the independence of the differentials $\mathrm{d}x^{i}$ is needed in
order to define the ‘partial derivatives’ in the formula. One says that $g$ is
_locally of Hessian type_ if each point of $M$ has an open neighborhood
$U\subset M$ on which there exists a coordinate chart $x:U\to\mathbb{R}^{n}$
and a function $f\in C^{\infty}(U)$ such that the above formula holds on $U$.
The notion of Hessian type has been considered in many places in the
literature (cf. [4, 5, 6] and the papers cited therein) and is connected with
affine and Kähler geometry and many other areas.

Since symmetric quadratic differential forms $g$ in dimension $n$ depend on
$\tfrac{1}{2}n(n{+}1)$ functions of $n$ variables and the data of a Hessian
representation depends only on $(n{+}1)$ functions of $n$ variables, it is
clear that, when $n>2$, not every $g$ is locally of Hessian type. As $n$
increases, the condition of being locally of Hessian type becomes more and
more restrictive. When $g$ is nondegenerate, so that $(M^{2},g)$ is a pseudo-
Riemannian manifold, being of Hessian type implies algebraic conditions on the
Riemann curvature tensor once $n$ is sufficiently large [6, 1].

Meanwhile, when $n=2$, finding Hessianizing data $(x^{1},x^{2},f)$ for a given
quadratic form $g$ is a determined PDE problem, being three equations for
three unknowns, so one might expect formal solvability. However, it is not
necessarily true that the PDE system involved is nondegenerate, so even formal
solvability is not guaranteed.

For example, the smooth (in fact, real-analytic) quadratic form
$g=c_{ijk}y^{i}\,\mathrm{d}y^{j}\circ\mathrm{d}y^{k}$, where $c_{ijk}=c_{ikj}$
are constants, is not Hessianizable on a neighborhood of $y=0$ unless
$c_{ijk}=c_{jik}$.

However, when $g\in S^{2}(T^{*}M)$ is real-analytic _and nondegenerate_ at
$p\in M^{2}$, it is straightforward to show, using the Cartan-Kähler Theorem,
that $g$ is Hessianizable on an open neighborhood of $p$. See Theorem 1 below,
which is the result to which I alluded in [3]. (The reader might also want to
compare the argument in [1], which was arrived at independently.)

It is natural to ask whether this Hessianizability result continues to hold in
the smooth setting. It turns out that the PDE involved is never elliptic, so
one never gets existence and/or regularity by applying results from elliptic
theory. In fact, the complex characteristic variety of the PDE along regular
initial conditions consists of $3$ points, so at least one of the
characteristics has to be real.

Now, it turns out that, when $g$ is nondegenerate, one can always find regular
initial conditions for the PDE with $3$ real, distinct characteristics, i.e.,
initial conditions for which the PDE problem is hyperbolic. Thus, one finds
that one can invoke results of Yang [7] on involutive hyperbolic systems to
prove smooth local Hessianizability of nondegenerate $g$ that are merely
smooth. See Theorem 2.

For details on the notation and definitions in the use of the Cartan-Kähler
Theorem, which will be used to prove Theorem 1, the reader is referred to [2].

##  2\. The structure equations

Let $M^{2}$ be a surface, and let $g$ be a metric on $M$, i.e., a
nondegenerate quadratic form on $M$. Thus, in a local coordinate chart
$y=(y^{i}):U\to{\mathbb{R}}^{2}$, one has
$g=g_{ij}(y)\,\mathrm{d}y^{i}\,\mathrm{d}y^{j}$, where
$g_{11}g_{22}-g_{12}^{2}$ is nonvanishing. (One does not need to assume that
$g$ be definite.)

Let $\pi:F\to M$ denote the coframe bundle of $M$, whose typical element $u\in
F$ is an isomorphism $u:T_{\pi(u)}M\to{\mathbb{R}}^{2}$, with its right action
by
$A\in\operatorname{GL}(2,{\mathbb{R}})=\operatorname{Aut}({\mathbb{R}}^{2})$
defined to be $R_{A}(u)=u\cdot A=A^{-1}\circ u$. There is an
${\mathbb{R}}^{2}$-valued $1$-form $\omega:TF\to{\mathbb{R}}^{2}$ defined by
$\omega_{u}(v)=u\bigl{(}\pi^{\prime}(v)\bigr{)}$ for $v\in T_{u}F$. Its
components $\omega=(\omega^{i})$ are the canonical (‘soldering’) forms
$\omega^{i}$ (where $1\leq i\leq 2$). Let $G_{ij}=G_{ji}$ be the functions on
$F$ such that $\pi^{*}g=G_{ij}\,\omega^{i}{\circ}\omega^{j}$ and, as usual,
let $G^{ij}=G^{ji}$ satisfy $G^{ik}G_{kj}=\delta^{i}_{j}$.

Let $\omega^{i}_{j}$ be the Levi-Civita connection forms of $g$ on $F$. These
forms, together with the $\omega^{i}$, define a coframing of $F$, and they are
uniquely specified by the requirements that they satisfy _compatibility_ with
$g$,

(2.1) $\mathrm{d}G_{ij}=G_{ik}\,\omega^{k}_{j}+G_{kj}\,\omega^{k}_{i}\,,$

and the _first structure equation_ ,

(2.2)
$\mathrm{d}\omega^{i}=-\omega^{i}_{j}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}.$

Furthermore, they satisfy the _second structure equation_ ,

(2.3)
$\mathrm{d}\omega^{i}_{j}=-\omega^{i}_{k}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{k}_{j}+G_{jl}K\,\omega^{i}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{l},$

where the function $K$ is the _Gauss curvature_ of $g$ pulled up to $F$ via
the projection $\pi:F\to M$.

It will be convenient to define $\omega_{ij}=G_{ik}\,\omega^{k}_{j}$, so that
the equations above become $\mathrm{d}G_{ij}=\omega_{ij}+\omega_{ji}$, and to
define $\omega_{i}=G_{ij}\,\omega^{j}$, so that
$\pi^{*}g=\omega_{i}\circ\omega^{i}$. Note that one has
$\omega_{i}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{i}=0$,
since $G_{ij}=G_{ji}$.

##  3\. An exterior differential system for Hessianizability

If $U\subset M$ is an open subset on which there exist coordinates
$x=(x^{i}):U\to{\mathbb{R}}^{2}$ and a function $f$ such that, on $U$, one has

$g=\frac{\partial^{2}f}{\partial x^{i}\partial
x^{j}}\,\mathrm{d}x^{i}\circ\mathrm{d}x^{j}=\mathrm{d}\left(\frac{\partial
f}{\partial x^{1}}\right)\circ\mathrm{d}x^{1}+\mathrm{d}\left(\frac{\partial
f}{\partial x^{2}}\right)\circ\mathrm{d}x^{2},$

then the coordinate coframing $\xi=(\mathrm{d}x^{i})$ defines a section
$\xi:U\to F$ that satisfies $\xi^{*}(\omega^{i})=\mathrm{d}x^{i}$ and
$\xi^{*}(G_{ij})=\frac{\partial^{2}f}{\partial x^{i}\partial x^{j}}$.
Consequently, $\xi^{*}(\omega_{i})=\mathrm{d}\left(\frac{\partial f}{\partial
x^{i}}\right)$. In particular, the section $\xi:U\to F$ satisfies

$\xi^{*}\bigl{(}\mathrm{d}\omega^{i}\bigr{)}=\mathrm{d}\bigl{(}\mathrm{d}x^{i})=0\quad\text{and}\quad\xi^{*}(\mathrm{d}\omega_{i})=\mathrm{d}\left(\mathrm{d}\left(\frac{\partial
f}{\partial x^{i}}\right)\right)=0$

while
$\xi^{*}(\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})=\mathrm{d}x^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}x^{2}\not=0$.

Conversely 222Here and throughout this article, when $S\subset M$ is a
submanifold and $\tau$ is a contravariant tensor field on $M$ (e.g., a
differential form or a metric on $M$), the notation $S^{*}(\tau)$ denotes the
_pullback_ of $\tau$ to $S$, which the reader should take care not to confuse
with $\tau_{|S}$, i.e., the _restriction_ of $\tau$ to $S$. , if $S\subset F$
is a surface such that
$S^{*}(\mathrm{d}\omega^{i})=S^{*}(\mathrm{d}\omega_{i})=0$ while
$S^{*}(\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\not=0$,
then each point of $S$ has an open neighborhood $V\subset S$ that is the image
of a local section $\xi:U\to F$ for some open set $U\subset M$. When $U$ is
simply connected, one has $\xi^{*}\omega^{i}=\mathrm{d}x^{i}$ for some
independent functions $x^{i}$ on $U$, while
$\xi^{*}\omega_{i}=\mathrm{d}p_{i}$ for some functions $p_{i}$ on $U$. Since
$\omega_{i}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{i}=0$,
it follows that
$\mathrm{d}(p_{i}\,\mathrm{d}x^{i})=\mathrm{d}p_{i}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\mathrm{d}x^{i}=0$,
so that there exists a function $f$ on $U$ such that
$\mathrm{d}f=p_{i}\,\mathrm{d}x^{i}$, i.e., $p_{i}=\frac{\partial f}{\partial
x^{i}}$. Thus,

$g=\xi^{*}(\omega_{i}\circ\omega^{i})=\mathrm{d}p_{i}\circ\mathrm{d}x^{i}=\frac{\partial^{2}f}{\partial
x^{i}\partial x^{j}}\,\mathrm{d}x^{i}\circ\mathrm{d}x^{j},$

as desired.

Letting ${\mathcal{I}}$ be the differential ideal on $F$ generated by the four
$2$-forms $\mathrm{d}\omega^{i}$ and $\mathrm{d}\omega_{i}$ for $i=1,2$, it
then follows that the problem of locally writing $g$ in Hessian form is
equivalent to finding the integral surfaces of ${\mathcal{I}}$ on which the
$2$-form
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}$
is nonvanishing.

###  3.1. A prolongation

The ideal with independence condition
$\bigl{(}{\mathcal{I}},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
is not quite the right ideal to consider, for it does not contain all of the
forms on $F$ that vanish on Hessianizing sections. To see this, note that, by
definition,
$G_{ij}\mathrm{d}\omega^{j}=-\omega_{ij}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}$
while, since $\mathrm{d}G_{ij}=\omega_{ij}+\omega_{ji}$, one has that

$\mathrm{d}\omega_{i}=\mathrm{d}G_{ij}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}+G_{ij}\mathrm{d}\omega^{j}=\omega_{ji}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}.$

Thus, ${\mathcal{I}}$ is generated by the four $2$-forms
$\omega_{ij}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}$
and
$\omega_{ji}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}$
for $i=1,2$. In particular, ${\mathcal{I}}$ contains the pair of $2$-forms
$(\omega_{12}-\omega_{21}){\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}$
for $j=1,2$, and, consequently, any integral surface of ${\mathcal{I}}$ on
which
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\not=0$
must also be an integral surface of the $1$-form
$\theta=\tfrac{1}{2}(\omega_{12}-\omega_{21})$. Thus, one can add $\theta$ and
its exterior derivative $\mathrm{d}\theta$ to ${\mathcal{I}}$ to form a larger
ideal ${\mathcal{I}}_{+}$ with the property that the integral surfaces of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
are the same as the integral surfaces of
$\bigl{(}{\mathcal{I}},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$.
Henceforth, I will work with this larger ideal, which is a partial
prolongation of
$\bigl{(}{\mathcal{I}},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$.

Write $\sigma_{ij}=\tfrac{1}{2}(\omega_{ij}+\omega_{ji})$
($=\tfrac{1}{2}\mathrm{d}G_{ij}$), so that
$\omega_{ij}\equiv\sigma_{ij}\mod\theta$ and

$-G_{ij}\mathrm{d}\omega^{j}\equiv\mathrm{d}\omega_{i}\equiv\sigma_{ij}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}\mod\theta.$

Note that the forms $\omega^{1}$, $\omega^{2}$, $\theta$, $\sigma_{11}$,
$\sigma_{12}$, and $\sigma_{22}$ are everywhere linearly independent and hence
define a coframing of $F$. Using the second structure equations, one finds

$\mathrm{d}\theta\equiv
G^{kl}\sigma_{k1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma_{l2}+K\,\omega_{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega_{2}\mod\theta.$

Thus, the differential ideal ${\mathcal{I}}_{+}$ is generated algebraically by
$\theta$ and three $2$-forms:

(3.1) $\displaystyle\Omega_{1}$
$\displaystyle=\sigma_{1j}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}\,,$
$\displaystyle\Omega_{2}$
$\displaystyle=\sigma_{2j}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{j}\,,$
$\displaystyle\Theta$
$\displaystyle=G^{kl}\sigma_{k1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\sigma_{l2}+K\,\omega_{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega_{2}\,.$

It is easy to see that the right action of $\mathrm{GL}(2,\mathbb{R})$ on $F$
preserves the ideal with independence condition
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$.
In particular, this right action prolongs to an action of
$\mathrm{GL}(2,\mathbb{R})$ on the space
${\mathcal{V}}_{2}\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}\subset\operatorname{Gr}_{2}(TF)$,
which consists of the $2$-dimensional integral elements of ${\mathcal{I}}_{+}$
on which
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}$
is nonvanishing.

##  4\. Involutivity

The pieces are now in place to prove the main result.

######  Theorem 1.

The differential ideal with independence condition
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
is involutive, with Cartan characters $s_{0}=1$, $s_{1}=3$, and $s_{2}=0$.

If $g$ is real analytic, then the integral surfaces of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
depend on three arbitrary functions of one variable. In particular, each point
of $M$ has an open neighborhood on which $g$ can be written in Hessian form.

###### Proof.

Because ${\mathcal{I}}_{+}$ is generated in degree $1$ by the nonvanishing
$1$-form $\theta$, all of the $0$-dimensional integral elements are both
ordinary and regular, and all of its $1$-dimensional integral elements are
ordinary.

Thus, to prove involutivity, it suffices to show that, at each $u\in F$, there
exists a regular $1$-dimensional integral element $E_{1}\subset T_{u}F$ for
which
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}$
is nonvanishing on the polar space $H(E_{1})$.

To this end, let $v\in T_{u}F$ be the basis of a $1$-dimensional integral
element, i.e., $\theta(v)=0$ while $v$ is nonzero. Then the polar space of
$E_{1}={\mathbb{R}}v$ is

$H(E_{1})=\\{\ w\in T_{u}F\ \vrule\
\theta(w)=\Omega_{1}(v,w)=\Omega_{2}(v,w)=\Theta(v,w)=0\ \\},$

and hence has dimension at least $2$. Set $a^{i}=\omega^{i}(v)$ and
$s_{ij}=\sigma_{ij}(v)$. Then

(4.1) $\displaystyle v\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\Omega_{1}$
$\displaystyle=-a^{1}\,\sigma_{11}-a^{2}\,\sigma_{12}+s_{11}\,\omega^{1}+s_{12}\,\omega^{2},$
$\displaystyle v\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\Omega_{2}$
$\displaystyle=-a^{1}\,\sigma_{12}-a^{2}\,\sigma_{22}+s_{12}\,\omega^{1}+s_{22}\,\omega^{2},$
$\displaystyle v\mathbin{\hbox{\vrule
height=1.4pt,width=4.0pt,depth=-1.0pt\vrule
height=4.0pt,width=0.4pt,depth=-1.0pt}}\Theta$
$\displaystyle=G^{kl}(s_{k1}\sigma_{l2}-s_{l2}\sigma_{k1})+K(G_{11}G_{22}-{G_{12}}^{2})\,(a^{1}\,\omega^{2}-a^{2}\,\omega^{1}).$

Thus, $H(E_{1})$ has dimension $2$ and
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}$
is nonvanishing on $H(E_{1})$ exactly when

(4.2)
$(a^{1})^{2}\,s^{2}_{1}-a^{1}a^{2}\,(s^{1}_{1}-s^{2}_{2})-(a^{2})^{2}\,s^{1}_{2}\not=0,$

where, for brevity, I have written $s^{i}_{j}=G^{ik}(u)s_{kj}$. In particular,
except when $s_{ij}=\lambda G_{ij}(u)$ for some $\lambda$, it will always be
possible to choose $a^{1}$ and $a^{2}$ so that the inequality (4.2) holds. For
a $v\in T_{u}F$ so chosen, the integral element $E_{1}={\mathbb{R}}v$ is
regular and, hence, its $2$-dimensional polar space $H(E_{1})=E_{2}$ is an
ordinary integral element of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$.
Moreover, the characters of the ordinary flag $(0)\subset E_{1}\subset E_{2}$
are visibly $s_{0}=1$, $s_{1}=3$, and $s_{2}=0$ and
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
is involutive.

Moreover, when $g$ is real analytic, the Cartan-Kähler Theorem applies,
yielding that any analytic curve $\Gamma\subset F$ that satisfies
$\Gamma^{*}\theta=0$ and $\Gamma^{*}C\not=0$, where $C$ is the characteristic
cubic form333For use below, I note that $R_{A}^{*}(C)=\det(A)\,C$ for
$A\in\operatorname{GL}(2,{\mathbb{R}})$.

(4.3)
$C=(\omega^{1})^{2}\circ(G^{2j}\sigma_{j1})-\omega^{1}\circ\omega^{2}\circ(G^{1j}\sigma_{j1}-G^{2j}\sigma_{j2})-(\omega^{2})^{2}\circ(G^{1j}\sigma_{j2}),$

lies in a unique (ordinary) analytic integral surface of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$.

When $g$ (and hence ${\mathcal{I}}_{+}$) is real analytic, it is easy to
construct such analytic curves $\Gamma\subset F$: Let $A\subset M$ be any
connected, noncompact, analytic, embedded curve, on which there is a parameter
$t:A\to(a,b)\subset{\mathbb{R}}$ and let $F_{A}=\pi^{-1}(A)\subset F$. Then
$F_{A}$ is a $5$-dimensional embedded, analytic submanifold on which
$\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}$
vanishes, although $\omega^{1}$ and $\omega^{2}$ do not simultaneously vanish.
Thus, there exist two functions $a^{i}:F_{A}\to{\mathbb{R}}$ for $i=1,2$ that
do not simultaneously vanish such that
$\omega^{i}=a^{i}\,\pi^{*}(\mathrm{d}t)$ on $F_{A}$. Then let $\Gamma\subset
F_{A}$ be any real analytic curve in $F_{A}$ that projects diffeomorphically
onto $A$ and satisfies $\Gamma^{*}\theta=0$ while $\Gamma^{*}\psi\not=0$ where
$\psi$ is the $1$-form on $F_{A}$ defined by

$\psi=(a^{1})^{2}\,G^{2k}\sigma_{k1}-a^{1}a^{2}\,(G^{1k}\sigma_{k1}-G^{2k}\sigma_{k2})-(a^{2})^{2}\,G^{1k}\sigma_{k2}\,.$

Such a $\Gamma$ will then be a regular integral curve of ${\mathcal{I}}_{+}$
lying in a unique integral surface $S\subset F$ of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$.
Moreover, an open neighborhood of $\Gamma$ in $S$ will project
diffeomorphically onto an open neighborhood $U$ of $A$ in $M$, and the inverse
map $\xi:U\to S$ will be a Hessianizing section for $g$ on $U$.

In particular, for any given point $u\in F$, an integral surface of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
exists that passes through $u$.

Since the Cartan characters of the system are $s_{0}=1$, $s_{1}=3$, and
$s_{2}=0$, it follows that the ordinary integral surfaces depend on three
functions of one variable [2].

As explained above, this implies that the metric $g$ is locally Hessianizable
if it is real analytic. ∎

##  5\. Hyperbolicity and smooth Hessianizability

In order to prove Hessianizability in the smooth category, it will be
necessary to take a closer look at the structure of the ideal with
independence condition
$({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$.

###  5.1. The characteristic variety

Let
${\mathcal{V}}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\subset\operatorname{Gr}_{2}(TF)$
denote the locus of $2$-dimensional integral elements of
$({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$.
A $2$-plane $E\subset T_{u}F$ belongs to
${\mathcal{V}}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
if and only if it is defined by $4$ Pfaffian equations of the form

(5.1) $\theta=\sigma_{ij}-p_{ijk}(E)\,\omega^{k}=0,$

where the numbers $p_{ijk}(E)=p_{jik}(E)$ satisfy the linear relations
$p_{ijk}(E)=p_{ikj}(E)$ (which are equivalent to the vanishing of $\Omega_{i}$
on $E$) and the quadratic relation

(5.2) $\displaystyle 0$
$\displaystyle=G^{kl}(u)\bigl{(}p_{k11}(E)p_{l22}(E){-}p_{k12}(E)p_{l12}(E)\bigr{)}$
$\displaystyle\qquad\qquad+K(u)\bigl{(}G_{11}(u)G_{22}(u){-}G_{12}(u)^{2}\bigr{)}$

(which is equivalent to the vanishing of $\Theta$ on $E$). The quadratic form
in the $4$ independent quantities $p_{111}(E)$, $p_{112}(E)$, $p_{122}(E)$,
and $p_{222}(E)$ that makes up the first term of this relation is easily seen
to be nondegenerate and indefinite. Hence, this exhibits
${\mathcal{V}}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
as a hypersurface in $F\times{\mathbb{R}}^{4}$ whose fiber over $u\in F$ is a
(nonempty) $3$-dimensional hyperquadric that is smooth away from the $2$-plane
defined by $p_{ijk}(E)=K(u)=0$.

Meanwhile, the integral element $E$ will be ordinary (i.e., it will contain a
regular $1$-dimensional integral element) if and only if $E^{*}(C)\not=0$,
since $\Xi_{E}\subset{\mathbb{P}}E^{*}$, i.e., the characteristic variety of
$E$, consists of the points represented by the linear factors of the cubic
form

(5.3)
$E^{*}(C)=\epsilon_{jk}G^{kl}(u)p_{lmn}(E)\,\,\omega^{j}\circ\omega^{m}\circ\omega^{n}.$

(Here, $\epsilon_{ij}=-\epsilon_{ji}$ and $\epsilon_{12}=1$.) It follows that
$E$ is ordinary unless $p_{ijk}(E)=0$ for all $(i,j,k)$ which, by the above
quadratic relation, can only happen if $K(u)=0$.

######  Remark 1 (Non-ordinary integral surfaces).

An integral surface of the ideal
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
on which $C$ vanishes identically is a so-called ‘non-ordinary’ integral
surface, i.e., it is an integral surface of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$
that contains no regular integral curves. Since $E^{*}(C)$ vanishes if and
only if $p_{ijk}(E)=0$, it follows from (5.2) that non-ordinary integral
surfaces can only exist over open sets in $M^{2}$ on which the curvature $K$
vanishes, and, in this case, they are the integral surfaces of the Frobenius
system generated by $\theta$ and the $\sigma_{ij}$. Thus, when $K\equiv 0$,
the coframe bundle $F$ is foliated by the $4$-parameter family of leaves of
this system.

Conversely, if $g$ is a nondegenerate quadratic form on $M$ with vanishing
curvature, then each point $p\in M$ has a neighborhood on which there exist
$p$-centered local coordinates $(x^{1},x^{2})$ in which
$g=c_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}$, where the $c_{ij}=c_{ji}$ are
constants, so that $g$ is in Hessian form with
$f=\tfrac{1}{2}c_{ij}\,x^{i}x^{j}$. Such constant coefficient Hessianizations
describe the non-ordinary integral surfaces.

###  5.2. Hyperbolicity

The (complex) characteristic variety of the ideal ${\mathcal{I}}_{+}$ meets
each cotangent space of each ordinary integral surface in the three linear
factors of the cubic form $C$ (as defined in (4.3)). Since a nonvanishing
cubic in $2$ variables always has at least one real linear factor, it follows
that the real characteristic variety is never empty, so the system is never
elliptic. Therefore, the best behavior one could hope for is that the
characteristic variety at each point would consist of three real distinct
factors, which would be the hyperbolic case.

Recall that, for a cubic form $c\in S^{3}(E^{*})$, where $E$ is a
$2$-dimensional vector space over $\mathbb{R}$, its discriminant $\Delta(c)$
is an element of $S^{6}\bigl{(}\Lambda^{2}(E^{*})\bigr{)}$. When
$(e^{1},e^{2})$ is a basis of $E^{*}$ and $c$ is expressed in the form

(5.4)
$c=c_{0}\,(e^{1})^{3}+3c_{1}\,(e^{1})^{2}e^{2}+3c_{2}\,e^{1}(e^{2})^{2}+c_{3}\,(e^{2})^{3},$

then

(5.5)
$\Delta(c)=\bigl{(}6{c_{0}}{c_{1}}{c_{2}}{c_{3}}+3{c_{1}}^{2}{c_{2}}^{2}-4{c_{0}}{c_{2}}^{3}-4{c_{1}}^{3}{c_{3}}-{c_{0}}^{2}{c_{3}}^{2}\bigr{)}\,(e^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}e^{2})^{\otimes
6}.$

Then $\Delta(c)=0$ if and only if $c$ has a multiple linear factor. Because
the scalar field is $\mathbb{R}$, the inequality $\Delta(c)>0$ makes sense,
and this inequality holds if and only if $c$ has three real, distinct, linear
factors.

Let
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\subset{\mathcal{V}}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
denote the open subset that consists of integral elements $E$ for which the
characteristic cubic $E^{*}(C)$ is nonzero and has three real, distinct,
linear factors. By definition, the elements of
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
are ordinary. I will refer to
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
as the _hyperbolic locus_.

######  Lemma 1.

The basepoint projection
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\to
M$ is a surjective submersion.

###### Proof.

It suffices to prove that the basepoint projection
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\to
M$ is surjective, since every element of
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
is a Kähler-regular integral element and the ideal ${\mathcal{I}}_{+}$ is
generated in positive degree.

Also, because the right action by $\mathrm{GL}(2,\mathbb{R})$ on $F$ is
transitive on the fibers of $F\to M$, it suffices to consider an element $u\in
F$ at which all of $G_{11}$, $G_{12}$ and $G_{22}$ are nonzero, so I will
assume this hencforth.

Consider the locus
$L_{u}\subset{\mathcal{V}}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
in the fiber over $u$ that is defined by the two equations

$G_{22}p_{122}-G_{12}p_{222}=0\quad\text{and}\quad
G_{12}p_{111}-G_{11}p_{112}=0.$

Because $G_{11}G_{22}-(G_{12})^{2}\not=0$, there will exist unique functions
$s_{1}$ and $s_{2}$ on $L$ such that

$p_{111}=G_{11}s_{1}\,,\ p_{112}=G_{12}s_{1}\,,\ p_{122}=G_{12}s_{2}\,,\
p_{222}=G_{22}s_{2}\,.$

These values for $p_{ijk}$ will define an integral element if and only if they
satisfy (5.2), which turns out to be

(5.6)
$(G_{11}G_{22}{-}{G_{12}}^{2})^{2}\,K-G_{12}(G_{11}s_{2}{-}G_{12}s_{1})(G_{21}s_{2}{-}G_{22}s_{1})=0.$

Because $G_{12}$ and $G_{11}G_{22}-{G_{12}}^{2}$ are nonvanishing, this
equation defines either a hyperbola or a pair of transverse lines in the
$s_{1}s_{2}$-plane, depending on whether $K$ is zero or not.

Meanwhile, the characteristic cubic at points of $L_{u}$ is (up to a nonzero
multiple) given by

$\displaystyle C$
$\displaystyle=-\bigl{(}(G_{11}G_{22}+{G_{12}}^{2})s_{1}-2G_{11}G_{12}s_{2}\bigr{)}\,(\omega^{1})^{2}\omega^{2}$
$\displaystyle\qquad+\bigl{(}(G_{11}G_{22}+{G_{12}}^{2})s_{2}-2G_{22}G_{12}s_{1}\bigr{)}\,\omega^{1}(\omega^{2})^{2}.$

Obviously $C$ will have three real distinct factors as long as both quantities
$(G_{11}G_{22}+{G_{12}}^{2})s_{1}-2G_{11}G_{12}s_{2}$ and
$(G_{11}G_{22}+{G_{12}}^{2})s_{2}-2G_{22}G_{12}s_{1}$ are nonzero. The
vanishing of these quantities define two lines $\ell_{1}$ and $\ell_{2}$ in
the $s_{1}s_{2}$-plane that intersect transversely at $s_{1}=s_{2}=0$.

When $K$ is nonzero, the hyperbola (5.6) in the $s_{1}s_{2}$-plane is not
contained in the union of the two lines $\ell_{1}$ and $\ell_{2}$. Hence there
is a choice of $(s_{1},s_{2})$ that satisfies (5.6) for which $C$ has three
real distinct factors.

When $K=0$, one checks that the two (distinct) lines in the $s_{1}s_{2}$-plane
defined by $G_{11}s_{2}{-}G_{12}s_{1}=0$ and $G_{21}s_{2}{-}G_{22}s_{1}=0$
only meet the lines $\ell_{1}$ and $\ell_{2}$ in the origin $s_{1}=s_{2}=0$.
Hence, again there is a choice of $(s_{1},s_{2})$ that satisfies (5.6) for
which $C$ has three real distinct factors.

Such a choice of $(s_{1},s_{2})$ defines an element of $L_{u}$ that lies in
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$,
proving the claimed surjectivity. ∎

######  Remark 2 (Universal hyperbolicity).

It is not always true that each $u\in F$ has a $2$-dimensional integral
element $E\subset T_{u}F$ for which the characteristic cubic $E^{*}(C)$ is the
product of a linear factor and an irreducible quadratic factor (the other open
$\operatorname{GL}(2,{\mathbb{R}})$-orbit). For example, if $g>0$, then such
integral elements exist only when $K(u)<0$. In fact, when $g>0$ and $K>0$, one
finds that _all_ of the Kähler-ordinary integral elements are hyperbolic.

######  Theorem 2.

Let ${\mathcal{I}}^{(1)}$ denote the contact ideal on
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})\subset\operatorname{Gr}_{2}(TF)$,
endowed with its canonical independence condition $\omega$. Then
$({\mathcal{I}}^{(1)},\omega)$ is involutive hyperbolic in the sense of Yang
[7].

Consequently, any smooth nondegenerate quadratic form on $M^{2}$ is locally
Hessianizable.

###### Proof.

Since $({\mathcal{I}}^{(1)},\omega)$ on
${\mathcal{V}}^{h}_{2}({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$
is an open subset of the ordinary prolongation of the involutive system
$({\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2})$,
it is involutive, with Cartan characters $s_{0}=4$, $s_{1}=3$, $s_{2}=0$.
Moreover, because ordinary prolongation preserves the characteristic variety,
it follows that the characteristic variety of each integral element of
$({\mathcal{I}}^{(1)},\omega)$ consists of three real, distinct points. Thus,
$({\mathcal{I}}^{(1)},\omega)$ is involutive hyperbolic in the sense of Yang
[7].

In particular, it follows from Theorem 4.11 in [7] that, when $g$ is a smooth
nondegenerate quadratic form on $M^{2}$, each $u\in F$ lies in a smooth,
ordinary, _hyperbolic_ integral surface of
$\bigl{(}{\mathcal{I}}_{+},\omega^{1}{\mathchoice{\,{\scriptstyle\wedge}\,}{{\scriptstyle\wedge}}{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}\omega^{2}\bigr{)}$.
Consequently, any smooth nondegenerate quadratic form on a surface can locally
be written in Hessian form. ∎

## References

  * [1] S. Amari and J. Armstrong, _Curvature of Hessian manifolds_ , Differential Geometry and Its Applications, 33 (2014), 1–12.
  * [2] R. Bryant, S.-s. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, _Exterior differential systems_ , Springer-Verlag, New York, 1991.
  * [3] R. Bryant, _Answer to mathoverflow question “When a Riemannian manifold is of Hessian Type”_ , http://mathoverflow.net/questions/122308/.
  * [4] J. Duistermaat, _On Hessian Riemannian structures_ , Asian J. Math. 5 (2001), 79–91.
  * [5] H. Shima, _The geometry of Hessian structures_ , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
  * [6] B. Totaro, _The curvature of a Hessian metric_ , Int. J. Math. 15 (2004), 369–391.
  * [7] D. Yang, _Involutive hyperbolic differential systems_. Mem. Amer. Math. Soc. 68 (1987), no. 370.


# Uniform convergence for sequences of best $L^{p}$ approximation

Donatella Bongiorno  Lucian Coroianu  Dipartimento di Ingegneria, Università
degli Studi di Palermo,
Viale delle Scienze, 90100 Palermo, Italy  Department of Mathematics and
Computer Science, University of Oradea,
Universitatii 1, 410610, Oradea, Romania

###### Abstract

Let $f$ be a continuous monotone real function defined on a compact interval
$[a,b]$ of the real line. Given a sequence of partitions of $[a,b]$,
$\Delta_{n}$, $\left\|{\Delta}_{n}\right\|\rightarrow 0$, and given $l\geq
0,m\geq 1$, let $\mathbf{S}_{m}^{l}(\Delta_{n})$ be the space of all functions
with the same monotonicity of $f$ that are $\Delta_{n}$-piecewise polynomial
of order $m$ and that belong to the smoothness class $C^{l}[a,b]$.
In this paper we show that, for any $m\geq 2l+1$,
$\bullet$ sequences of best $L^{p}$-approximation in
$\mathbf{S}_{m}^{l}(\Delta_{n})$ converge uniformly to $f$ on any compact
subinterval of $(a,b)$;
$\bullet$ sequences of best $L^{p}$-approximation in
$\mathbf{S}_{m}^{0}(\Delta_{n})$ converge uniformly to $f$ on the whole
interval $[a,b]$.

MSC 41A10; 41A29; 41A30; 41A52

###### keywords:

Piecewise polynomial function, $L^{p}$ approximation, Best approximation,
Uniform convergence.

##  1 Introduction

It is well known that the $L^{p}$-convergence of a sequence $(f_{n})$ to a
continuous function $f\colon[a,b]\rightarrow{\mathbb{R}}$ doesn’t imply the
uniform convergence. However J.T. Lewis and O. Shisha, in [8], proved that in
the case of increasing sequences of functions $(f_{n})$, the
$L^{p}$-convergence of $(f_{n})$ to a continuous function $f$ implies the
uniform convergence on any compact interval strictly included in $(a,b)$. The
problem of the uniform convergence of $(f_{n})$ on the whole interval $[a,b]$
was proved by E. Kimchi and N. Richter-Dyn in [5] by considering sequences
that have first-order divided differences uniformly bounded from above and
from below.

In this paper we study sequences $(f_{n})$ of piecewise polynomial functions
obtained from the $L^{p}$ minimization under monotonicity and smoothness
constraints. More precisely, we will investigate the local and the global
uniform convergence of $(f_{n})$.

Our result comes also from an idea of E. Passow [11].

Given a nondecreasing and continuous function
$f\colon[a,b]\rightarrow{\mathbb{R}}$, a real $p\in[1,\infty)$, a partition
$\Delta$ of $[a,b]$ and integers $m,l$ with $m\geq 1$ and $0\leq l\leq m-1$,
let us consider the projection, with respect to the $L^{p}$ norm, of $f$ onto
the class of all nondecreasing $\Delta$-piecewise polynomial functions with
degree less than or equal to $m$ that belongs to the smoothness class
$C^{l}[a,b]$. Such a projection, called the best $L^{p}$ approximation of $f$
with respect to the partition $\Delta$, there exists and it is unique (Theorem
1). Then, given a sequence $(\Delta_{n})_{n\geq 1}$ of partitions of $[a,b]$
whose norm tends to zero, by [11, Theorem 1] it is possible to generate a
sequence of $(\Delta_{n})$-piecewise polynomial functions that converges to
$f$ in the $L^{p}$ norm whenever $m\geq 2l+1$. Combining this results with the
one of Lewis and Shisha mentioned earlier, we will prove that the sequence of
best $L^{p}$ approximations of $f$ along the partitions’ sequence
$(\Delta_{n})_{n\geq 1}$ converges uniformly to $f$ on any compact interval
included in $\left(a,b\right)$, whenever $m\geq 2l+1$. Finally, in the special
case $l=0$ and $m$ arbitrary, we extend this results to the uniform
convergence of such a sequence on the whole interval $\left[a,b\right]$ (Main
Theorem). It is obvious that the most stringent question is whether such
result also holds for other values of $l$ if not in general (that is, higher
order of smoothness), as well as for the cases when the condition $m\geq 2l+1$
could be dropped or replaced by a weaker one. Clearly, this is an important
open question that worth further investigation. The proof of the Main Theorem
which is long and uses many auxiliary results (among them, the main results in
[8], [11] and [12]), however, it seems to work only for the case $l=0$.

This paper is organized as follows: in Section 2 we present some basic notions
about piecewise polynomial functions and about $L^{p}$-type approximations for
some given real $p\in[1,\infty)$. In the same section, we present, also, some
properties of the projection operator onto the set of monotone piecewise
polynomial functions from the class $C^{l}[a,b]$ of degree less than or equal
to $m\geq 1$ on any compact subinterval of a given partition. In Section 3 we
give some useful results on the uniform convergence of the sequence of best
$L^{p}$ approximation of $f$ and we enunciate the Main Theorem. In Section 4
we prove the Main Theorem. Finally, in Section 5 some open questions, with the
purpose to complete and generalize the result of this paper, are presented.

In conclusion, remark that, since piecewise polynomial functions are used as
tools in approximation or interpolation problems, data analysis and many
others (see [3], [4], [9]), $\cdots$, for example), this note could be
interesting also from the point of view of approximation theory even if the
sequences of piecewise polynomial functions, used in the paper, cannot be
expressed analytically.

##  2 Piecewise polynomial approximation of $L^{p}$\- integrable functions

For $p\in[1,+\infty)$ and for a compact interval $\left[a,b\right]$, let
$L^{p}[a,b]$ be the space of all $L^{p}$-integrable functions on $[a,b]$, with
the norm $\left\|\cdot\right\|_{p}:L^{p}[a,b]\rightarrow[0,\infty),$

$\left\|f\right\|_{p}=\left(\int\limits_{a}^{b}\left|f(x)\right|^{p}dx\right)^{1/p}.$

It is well known that $\left(L^{p}[a,b],\left\|\cdot\right\|_{p}\right)$ is a
uniformly convex Banach space for any $p\in(1,+\infty)$.

It is also well known that, for each $p\in[1,+\infty)$, $L^{p}[a,b]$ includes
the class of all continuous functions on $[a,b]$ and the class of all monotone
functions on $[a,b]$. Moreover it is also well known that, for some
$j\in\mathbb{N}$, the class of all functions that are continuously
differentiable of order $j$ on $[a,b]$ is denoted by $C^{j}[a,b]$. In
particular, $C[a,b]$ denotes the class of all continuous functions defined on
$\left[a,b\right]$, with the norm
$\left\|\cdot\right\|:C[a,b]\rightarrow[0,\infty),$

$\left\|f\right\|=\max_{x\in[a,b]}\left|f(x)\right|\text{.}$

Let $\mathcal{D}[a,b]$ be the set of all partitions of $[a,b]$. If
$a=\alpha_{1}<\alpha_{2}<...<\alpha_{n}=b$ is a partition of $[a,b],$ then we
identify it with the vector
${\Delta}_{n}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$. As usual, we
denote by $\left\|{\Delta}_{n}\right\|$ the norm of this partition, where
$\left\|{\Delta}_{n}\right\|=\max\\{\alpha_{i+1}-\alpha_{i}:1\leq i\leq
n-1\\}$. A function $f:[a,b]\rightarrow\mathbb{R}$ is called
${\Delta}_{n}$-piecewise polynomial of order $m$, with $m\geq 1$, if the
restriction of $f$ on any interval $[\alpha_{i},\alpha_{i+1}]$,
$i=\overline{1,n-1}$ is a polynomial function of degree less than or equal to
$m$. Let $\mathbf{P}_{m,{\Delta}_{n}}$ be the space of all functions that are
${\Delta}_{n}$-piecewise polynomial of order $m$ and let
$\mathbf{S}_{m}^{l}(\Delta_{n})=\mathbf{P}_{m,{\Delta}_{n}}\cap C^{l}[a,b]$ be
the class of all ${\Delta}_{n}$-piecewise polynomials of order $m$ which are
continuously differentiable up to order $l$, for some nonnegative integer
$l\leq m-1$. Moreover let $\mathbf{SI}_{m}^{l}(\Delta_{n})$ and
$\mathbf{SD}_{m}^{l}(\Delta_{n})$ be the set of all nondecreasing and
nonincreasing functions in $\mathbf{S}_{m}^{l}(\Delta_{n})$, respectively.
Clearly, $\mathbf{SD}_{m}^{l}(\Delta_{n})=-\mathbf{SI}_{m}^{l}(\Delta_{n})$.
In all that follows in this paper we will consider only the case of best
$L^{p}$ approximation in $\mathbf{SI}_{m}^{l}(\Delta_{n})$ and in the main
results we will consider only nondecreasing functions to be approximated.
Obviously the case of approximation of nondecreasing functions in
$\mathbf{SD}_{m}^{l}(\Delta_{n})$ has the same treatment and therefore we omit
the details that the reader can easily deduce from the other case.

It is easy to observe that $\mathbf{S}_{m}^{l}(\Delta_{n})$ is a finite
dimensional linear subspace of $L^{p}[a,b]$ and that
$\mathbf{SI}_{m}^{l}(\Delta_{n})$ and $\mathbf{SD}_{m}^{l}(\Delta_{n})$ are
nonempty closed cones in $\mathbf{S}_{m}^{l}(\Delta_{n})$. Therefore, by the
existence and uniqueness of the projection onto a closed convex subset of a
Banach uniformly convex space, it follows the existence and uniqueness of the
best approximation with respect to $\left\|\cdot\right\|_{p}$, with
$p\in(1,+\infty)$, in the spaces $\mathbf{SI}_{m}^{l}(\Delta_{n})$ and
$\mathbf{SD}_{m}^{l}(\Delta_{n}),$ respectively. In additions, even if
$L^{1}[a,b]$ is no longer a Banach uniformly convex space (nor even
reflexive), the best approximation with respect to $\left\|\cdot\right\|_{1}$
always exists and in the particular case $f\in C[a,b]$, the projection of $f$
onto $\mathbf{SI}_{m}^{l}(\Delta_{n})$ and $\mathbf{SD}_{m}^{l}(\Delta_{n})$
respectively, not only there exists but also it is unique (see Theorem 2.3 in
[12] and see also [10]). Note that in [12] the results is obtained in a more
general setting for piecewise polynomial functions satisfying generalized
convexity constraints. Consequently, we can state the following result:

###### Theorem 1

If $p\in[1,\infty)$, $m\geq 1$, $0\leq l\leq m-1$, $f\in C[a,b]$ and
${\Delta}_{n}\in\mathcal{D}[a,b]$, then there exists uniquely
$\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\in\mathbf{SI}_{m}^{l}(\Delta_{n})$ so
that

$\left\|f-\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right\|_{p}=\min_{g\in\mathbf{SI}_{m}^{l}(\Delta_{n})}\left\|f-g\right\|_{p}\text{.}$

Taking into account the facts from above, for some given real
$p\in[1,+\infty)$, we consider the approximation operator
$\mathbf{U}_{m,\Delta_{n}}^{l}:C[a,b]\rightarrow\mathbf{SI}_{m}^{l}(\Delta_{n})$
defined by
$\mathbf{U}_{m,\Delta_{n}}^{l}(f)=\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)$.

The following properties are immediate and therefore we omit their proofs.

###### Proposition 2

Let $f\in C[a,b]$. For some $p\in[1,+\infty)$, $m\geq 1$, $0\leq l\leq m-1$
and ${\Delta}_{n}\in\mathcal{D}[a,b]$ with
${\Delta}_{n}=(\alpha_{1},\alpha_{2},...,\alpha_{n})$, we have that:

(i)
$\mathbf{SI}_{m}^{l}(\Delta_{n},f-c,p)=\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)-c$,
for all $c\in\mathbb{R};$

(ii)
$\mathbf{SI}_{m}^{l}(\Delta_{n},cf,p)=c\cdot\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)$
and
$\mathbf{SD}_{m}^{l}(\Delta_{n},cf,p)=c\cdot\mathbf{SD}_{m}^{l}(\Delta_{n},f,p)$,
for all $c\in\left[0,\infty\right];$

In all that follows, we no longer impose that
${\Delta}_{n}\in\mathcal{D}[a,b]$ has exactly $n$ knots. Indeed, later on we
construct sequences of partitions of $[a,b]$ such that any partition of the
sequence has an arbitrary number of knots. Therefore, in general, we will
consider a vector of the form
${\Delta}_{n}=(\alpha_{1},\alpha_{2},...,\alpha_{k_{n}})$.

In the next result we will provide sufficient conditions which will imply that
the sequence $\left(\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right)_{n\geq 1}$
converges to $f$ in the $L^{p}$ norm $\left\|\cdot\right\|_{p}$ for some given
$p\in[1,\infty)$. Combining this result with several auxiliary results, in the
next section we will be able to prove a local uniform convergence for this
sequence and even a global uniform convergence in the special case $l=0$. At
the moment, the next result will not suffice to obtain these uniform
convergence properties since we will rely on a rate of convergence for an
estimation in the $L^{p}$ norm (see relation (1) in the proof) and it is well
known that in general such estimations will not necessarily imply convergence
in the uniform norm based on the simple fact that these two norms are not
equivalent.

###### Proposition 3

Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is a continuous function and
consider in $\mathcal{D}[a,b]$ a sequence $\left({\ \Delta}_{n}\right)_{n\geq
1}$ with $\left\|{\Delta}_{n}\right\|\rightarrow 0$. Moreover, let $l$ be an
arbitrary nonnegative integer. Then if $f$ is nondecreasing, for any
$p\in[1,\infty)$ and $m\geq 2l+1$ we have:

$\left\|f-\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right\|_{p}\leq(b-a)^{1/p}\omega_{f}(\left\|{\Delta}_{n}\right\|)\text{,
for all }n\geq 1\text{,}$

where

$\omega_{f}(\delta)=\sup\\{\left|f(x)-f(y)\right|:x,y\in[a,b]\text{,}\left|x-y\right|\leq\delta\\}\text{}$

denotes the modulus of uniform continuity of $f.$

So, in particular, we have:

$\lim_{n\rightarrow\infty}\left\|f-\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right\|_{p}=0.$

Proof. Let us choose arbitrary $n\geq 1$ and suppose that
${\Delta}_{n}=(\alpha_{1},\alpha_{2},...,\alpha_{k_{n}})$. In [11] (see
Theorem 1 there) it is proved that if $\left(x_{i},y_{i}\right)_{i=1,s}$ are
in $\mathbb{R}^{2}$ with $x_{1}<x_{2}<\cdot\cdot\cdot<x_{s}$ then there exists
$P\in\mathbf{S}_{2l+1}^{l}(\Delta_{s})$, where $\Delta_{s}=(x_{1},...,x_{s})$,
such that $P(x_{i})=y_{i}$, $i=\overline{1,s}$ and $P$ is monotone on any
interval $\left[x_{i},x_{i+1}\right]$, $i=\overline{1,s-1}$. Then, applying
this result to $(\alpha_{i},f(\alpha_{i}))_{i=1,k_{n}}$ we get
$P\in\mathbf{SI}_{2l+1}^{l}(\Delta_{n})$ that interpolates $f$ on the knots of
${\Delta}_{n}$, and since $m\geq 2l+1$, it follows that
$P\in\mathbf{SI}_{m}^{l}(\Delta_{n})$.

So, since $f$ and $P$ are both nondecreasing and since
$f(\alpha_{i})=P(\alpha_{i})$, $i=\overline{1,k_{n}}$, it easily results that
for any $i\in\\{1,...,k_{n}-1\\}$ and $x\in[\alpha_{i},\alpha_{i+1}]$ we have
$\left|f(x)-P(x)\right|\leq f(\alpha_{i+1})-f(\alpha_{i}).$ Moreover, since

$f(\alpha_{i+1})-f(\alpha_{i})\leq\omega_{f}(\alpha_{i+1}-\alpha_{i})\leq\omega_{f}(\left\|{\Delta}_{n}\right\|)$

for all $i\in\\{1,...,k_{n}-1\\}$, we have

$\left|f(x)-P(x)\right|\leq\omega_{f}(\left\|{\Delta}_{n}\right\|)\text{, for
all }x\in[a,b]\text{.}$

This easily implies

$\left\|f-P\right\|_{p}\leq(b-a)^{1/p}\omega_{f}(\left\|{\Delta}_{n}\right\|).$

Then, by the definition of $\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)$ and since
$P\in\mathbf{SI}_{m}^{l}(\Delta_{n})$, it results that

$\left\|f-\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right\|_{p}\leq\left\|f-P\right\|_{p}\leq(b-a)^{1/p}\omega_{f}(\left\|{\Delta}_{n}\right\|).$
(1)

The continuity of $f$ and the fact that
$\left\|{\Delta}_{n}\right\|\rightarrow 0$ implies
$\omega_{f}(\left\|{\Delta}_{n}\right\|)\rightarrow 0$, hence by (1) we have
$\lim_{n\rightarrow\infty}\left\|f-\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right\|_{p}=0$
and the proof is complete.

##  3 Uniform convergence of best piecewise polynomial approximations

This section is dedicated to the study of the uniform convergence of the
sequence of all piecewise polynomial functions $(f_{n})_{n\geq 1}$ to a
continuous nondecreasing function $f$ of the sequence
$(\mathbf{SI}_{m}^{l}(\Delta_{n},f,p))_{n\geq 1}$ whenever
$\left\|{\Delta}_{n}\right\|\rightarrow 0$.

It is well known that if $f$ and $(f_{n})_{n\geq 1}$ are continuous functions
on $[a,b]$ such that $(f_{n})_{n\geq 1}$ converges uniformly to $f$ on $[a,b]$
then $(f_{n})_{n\geq 1}$ converges towards $f$ in the $L^{p}$ norm for any
$p\in[1,\infty)$. But it is also well known that the converse property does
not hold in general. However, if $(f_{n})_{n\geq 1}$ is a sequence of
nondecreasing functions then the following theorem provides us a local
convergence property in the uniform norm.

###### Theorem 4

(see [8, Theorem 1]) Let $f$ be a real, continuous function on the finite
interval $(a,b)$, and let $(f_{n})_{n\geq 1}$ be a sequence of nondecreasing
functions on $(a,b)$ such that
$\lim\limits_{n\rightarrow\infty}\left\|f_{n}-f\right\|_{p}=0$, where $1\leq
p<\infty$. Then for any $c$ and $d$ with $a<c<d<b$, the sequence
$(f_{n})_{n\geq 1}$ converges uniformly to $f$ on $[c,d]$.

Combining Proposition 3 with Theorem 4, we obtain the following result.

###### Corollary 5

Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is a continuous nondecreasing
function and consider in $\mathcal{D}[a,b]$ a sequence
$\left({\Delta}_{n}\right)_{n\geq 1}$, such that
$\left\|{\Delta}_{n}\right\|\rightarrow 0$. Then for any nonnegative integer
$l$ and for any $m\geq 2l+1$, the sequence
$\left(\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)\right)_{n\geq 1}$ is uniformly
convergent to $f$ on any interval $\left[c,d\right]$, with $a<c<d<b$.

Proof. By Proposition 3, we have
$\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)-f\right\|_{p}=0$.
Hence, we obtain the conclusion taking
$f_{n}=\mathbf{SI}_{m}^{l}(\Delta_{n},f,p)$ for any $n$ in Theorem 4.

Now let us consider the function $f\equiv 0$ and the sequence $(f_{n})_{n\geq
1}$ defined on $[0,1]$ by $f_{n}(x)=x^{n},n=1,2,\cdots$. It is trivial to
observe that $\|f_{n}\|_{p}\rightarrow 0$, for any $p\in[1,\infty)$ and that
$(f_{n})_{n\geq 1}$ is uniformly convergent to zero on any $[c,d]$, with
$0<c<d<1$. However, $(f_{n})_{n\geq 1}$ is not uniformly convergent to zero on
$[0,1]$.

Let us note that in the previous corollary as well as in Proposition 3 which
implies this corollary, we need to assume that $m\geq 2l+1$ because otherwise
we cannot use Theorem 1 in [11] used to prove Proposition 3\. Actually, in the
case $m\leq 2l$ the interpolant used to prove Proposition 3 may not exist (see
Remark 2 in [11]). Of course, it is an open question whether these results
would hold also for $m\leq 2l$ but then another proof is needed which does not
use Theorem 1 in [11].

In this paper we prove that:

Main Theorem If $f:[a,b]\rightarrow\mathbb{R}$ is a continuous nondecreasing
function and if $\left({\Delta}_{n}\right)_{n\geq 1}\in\mathcal{D}[a,b]$ such
that $\left\|{\Delta}_{n}\right\|\rightarrow 0$, then, for any real
$p\in[1,\infty)$ and $m\geq 1$, the sequence
$\left(\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)\right)_{n\geq 1}$ is uniformly
convergent to $f$ on $[a,b]$; that is
$\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},f)-f\right\|=0$.

To prove this Theorem we need some auxiliary results.

###### Lemma 6

(see [1, Lemma 4.3]) Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is a
continuous monotone function and suppose that $(f_{n})_{n\geq 1}$ is a
sequence of monotone functions, all with the same monotonicity as $f$, such
that for some $p\geq 1$ we have
$\lim\limits_{n\rightarrow\infty}\left\|f_{n}-f\right\|_{p}=0$. If
$\lim\limits_{n\rightarrow\infty}f_{n}(a)=f(a)$ and
$\lim\limits_{n\rightarrow\infty}f_{n}(b)=f(b)$, then
$\lim\limits_{n\rightarrow\infty}\left\|f_{n}-f\right\|=0$.

First of all we give a simple sufficient condition to obtain the pointwise
convergence needed in the previous lemma.

###### Lemma 7

Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is a nondecreasing continuous
function and suppose that $(f_{n})_{n\geq 1}$ is a sequence of nondecreasing
functions, such that for some $p\geq 1$ we have
$\lim\limits_{n\rightarrow\infty}\left\|f_{n}-f\right\|_{p}=0$. If
$\left(x_{n}\right)_{n\geq 1}$ is a sequence in $[a,b]$ such that
$x_{n}\rightarrow a$ and $f_{n}(x_{n})\geq f(x_{n})$ for all $n\geq 1$, then
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})=\lim\limits_{n\rightarrow\infty}f(x_{n})=f(a)$.
In particular, if $f_{n}(a)\geq f(a)$ for $all$ $n\geq 1$, then
$\lim\limits_{n\rightarrow\infty}f_{n}(a)=f(a)$.

Proof. To avoid unnecessary reasonings with subsequences, we may assume that
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})$ exists, including the case when
it is unbounded. By way of contradiction, suppose that
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})=f(a)$ does not hold. Then, since
$f(x_{n})\rightarrow f(a)$, there exists $\varepsilon>0$ such that for
sufficiently large $n$ we have $f_{n}(x_{n})-f(x_{n})>\varepsilon$. Letting
$n\rightarrow\infty$ we get
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})-f(a)>\varepsilon/2$. From the
continuity of $f$ it results the existence of $y_{0}\in[a,b]$ such that
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})-f(x)>\varepsilon/2$ for all
$x\in[a,y_{0}]$. In particular we get
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})-f(y_{0})>\varepsilon/2$. This
implies the existence of $n_{1}\in\mathbb{N}$ such that

$f_{n}(x_{n})-f(y_{0})>\varepsilon/2,\text{ for all }n\geq n_{1}.$ (2)

Let us choose arbitrary $y_{1}\in(a,y_{0})$. Since $x_{n}\rightarrow a$ there
is $n_{2}\in\mathbb{N}$ (which may depends only on $y_{1}$) such that
$x_{n}\leq y_{1}$ for all $n\geq n_{2}$. By the monotonicity of $f$ and
$f_{n}$, for each $n$, and by (2), it results

$f(x)\leq f(y_{0})<f_{n}(x_{n})\leq f_{n}(x)\text{, for all
}n\geq\max\\{n_{1},n_{2}\\}\text{ and }x\in[y_{1},y_{0}]\text{.}$

This implies

$\displaystyle\int\limits_{a}^{b}\left|f_{n}(x)-f(x)\right|^{p}dx$
$\displaystyle\geq$
$\displaystyle\int\limits_{y_{1}}^{y_{0}}\left|f_{n}(x)-f(x)\right|^{p}dx\geq(y_{0}-y_{1})\left|f_{n}(x_{n})-f(y_{0})\right|^{p}>(y_{0}-y_{1})\cdot\left(\frac{\varepsilon}{2}\right)^{p}$

for all $n\geq\max\\{n_{1},n_{2}\\}$.
Clearly this implies that we cannot have
$\lim\limits_{n\rightarrow\infty}\left\|f_{n}-f\right\|_{p}=0$ and this is a
contradiction. In conclusion, we have
$\lim\limits_{n\rightarrow\infty}f_{n}(x_{n})=\lim\limits_{n\rightarrow\infty}f(x_{n})=f(a)$.

The following lemma is inspired by the inequality of Markov for real
polynomials.

###### Lemma 8

Let $P:[a,b]\rightarrow\mathbb{R}$ be a nondecreasing polynomial function of
degree at most $m\geq 1$. Then,

$P\left(a+\frac{b-a}{2m^{2}+1}\right)\leq
P(a)+\frac{2m^{2}}{2m^{2}+1}\cdot(P(b)-P(a))\text{.}$

Proof. We start with the special case $a=0$, $b=1$, $P(0)=0$ and $P(1)=1$. Let
us define on $[-1,1]$ the polynomial $Q(x)=P((x+1)/2)$. From the Markov’s
inequality we have

$\max_{-1\leq x\leq 1}\left|Q^{\prime}(x)\right|\leq m^{2}\cdot\max_{-1\leq
x\leq 1}\left|Q(x)\right|$

. Since $Q^{\prime}(x)=\frac{1}{2}P^{\prime}\left(\frac{x+1}{2}\right)$, this
implies

$\max_{-1\leq x\leq 1}\left|P^{\prime}\left(\frac{x+1}{2}\right)\right|\leq
2m^{2}\cdot\max_{-1\leq x\leq 1}\left|P\left(\frac{x+1}{2},\right)\right|$

hence

$\max_{0\leq x\leq 1}\left|P^{\prime}(x)\right|\leq 2m^{2}\cdot\max_{0\leq
x\leq 1}\left|P(x)\right|=2m^{2}\text{.}$

Then, by the mean value theorem, we have

$P\left(\frac{1}{2m^{2}+1}\right)=\int\limits_{0}^{1/(2m^{2}+1)}P^{\prime}(x)dx=\frac{1}{2m^{2}+1}P^{\prime}(x_{m})\leq\frac{2m^{2}}{2m^{2}+1}.$

Now, let us consider the general case. It is immediate that
$P_{1}:[0,1]\rightarrow\mathbb{R}$, defined as
$P_{1}(x)=(P(a+(b-a)x)-P(a))/(P(b)-P(a))$, is a nondecreasing polynomial of
degree at most $m$, such that $P_{1}(0)=0$ and $P_{1}(1)=1$. It means that
$P_{1}\left(\frac{1}{2m^{2}+1}\right)\leq\frac{2m^{2}}{2m^{2}+1}$, from which
the desired conclusion easily follows.

##  4 Proof of the Main Theorem

If $f$ is a constant, then the conclusion is immediate since
$\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)=f$ for any $n$. Therefore, in what it
follows, we assume that $f$ is not a constant. By Corollary 5, it results that
$\left(\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)\right)_{n\geq 1}$ converges
uniformly to $f$ on any interval $[c,d]$ with $a<c<d<b$. In order to prove the
desired uniform convergence on $\left[a,b\right]$, it is enough to prove that
$\lim\limits_{n\rightarrow\infty}\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)(a)=f(a)$
and
$\lim\limits_{n\rightarrow\infty}\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)(b)=f(b)$,
respectively (see, e. g. Lemma 4.3 in [1]). Without loss of generality we may
assume that $f(a)=0$. In fact, supposed that the thesis holds for such
functions, given a continuous nondecreasing generic function $f$ we set
$g(x)=f(x)-f(a).$ Then $g$ is continuous nondecreasing and $g(a)=0$. So
$\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},g,p)-g\right\|=0$.
Now, by Proposition 2(i), we have $\mathbf{SI}_{m}^{0}(\Delta_{n},g,p)$
$=\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)-f(a)$, for all $n\geq 1$. Therefore

$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)-f\right\|$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)-f(a)+f(a)-f\right\|$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},g,p)-g\right\|$
$\displaystyle=$ $\displaystyle 0.$

From now on, until the end of the proof, we will set
$f_{n}=\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)$ for all $n\geq 1$. To avoid the
use of subsequences we may suppose that there are only two cases: either
$f_{n}(a)<f(a)=0$ for all $n\geq 1$, or $f_{n}(a)\geq f(a)=0$ for all $n\geq
1$. But in this latter one, by Lemma 7, it results that
$\lim\limits_{n\rightarrow\infty}f_{n}(a)=f(a)$ and therefore in this case
there is nothing to be proved. So, in all that follows we will suppose that
$f_{n}(a)<f(a)=0$ for all $n\geq 1$.

The idea of the proof is to show that any subsequence of
$\left(f_{n}(a)\right)_{n\geq 1}$ contains a subsequence that converges to
$0$. Clearly, this will imply that
$\lim\limits_{n\rightarrow\infty}f_{n}(a)=0$. For this reason, without loosing
generality we may suppose that there exists the limit
$\lim\limits_{n\rightarrow\infty}f_{n}(a)$ (including the case when this limit
is $-\infty$). For any $n\geq 1$ let
${\Delta}_{n}=(\alpha_{1}(n),\alpha_{2}(n),...,\alpha_{k_{n}}(n))$. We must
observe that for each $n$ the function $f_{n}$ cannot be a constant function.
Indeed, if $f_{n}$ would be constant (with strictly negative constant value
according to our assumption) then, taking $h(x)=0$, $x\in[a,b]$, we obviously
have $h\in\mathbf{SI}_{m}^{0}(\Delta_{n})$ and in addition one can easily
prove that $\left\|h-f\right\|_{p}<\left\|f_{n}-f\right\|_{p}$, which
contradicts the fact that $f_{n}$ is the best approximation of $f$ in
$\mathbf{SI}_{m}^{0}(\Delta_{n})$ with respect to $\left\|\cdot\right\|_{p}$.
Therefore, since $f_{n}$ is not constant, it results that there exists
$j\in\\{1,...,k_{n}-1\\}$ such that
$f_{n}(\alpha_{j}(n))=f_{n}(a)<f_{n}(\alpha_{j+1}(n))$. In addition, we notice
that necessarily $f_{n}$ is strictly increasing on
$\left[\alpha_{j}(n),\alpha_{j+1}(n)\right]$ (otherwise $f_{n}$ is not
polynomial on this interval). Since $f$ is not a constant function let
$a_{1}\in[a,b]$ such that $f(a_{1})=f(a)=0<f(x)$, for all $x\in(a_{1},b]$.
Without any loss of generality, suppose that
$\left(\alpha_{j}(n)\right)_{n\geq 1}$ is convergent and denote its limit with
$u_{0}$. If $u_{0}>a_{1}$ then for some fixed value $x_{0}\in(a_{1},u_{0})$
and sufficiently large $n$ we have $f_{n}(x_{0})\leq
f_{n}(\alpha_{j}(n))<0<f(x_{0})$. It means that $f_{n}(x_{0})$ does not
converge to $f(x_{0}).$ This clearly contradicts Corollary 5. Therefore we
have $u_{0}\leq a_{1}$. In particular, we have that $f(u_{0})=f(a)=0$. Now,
let us prove that $f_{n}(\alpha_{j+1}(n))>f(a)=0$ for all $n\geq 1$. By way of
contradiction, let us consider, for some $n\geq 1,$ the function
$h_{n}:[a,b]\rightarrow\mathbb{R}$, $h_{n}(x)=f_{n}(\alpha_{j+1}(n))$ if
$x\in[a,a_{j+1}(n)]$ and $h_{n}(x)=f_{n}(x)$ elsewhere. Then
$h_{n}\in\mathbf{SI}_{m}^{0}(\Delta_{n})$ and $f_{n}(x)<h_{n}(x)\leq f(x)$,
for all $x\in[a,\alpha_{j+1}(n))$ (here it is important that $f_{n}$ is
strictly increasing on $\left[\alpha_{j}(n),\alpha_{j+1}(n)\right]$, that
$f_{n}(\alpha_{j+1}(n))\leq 0$ as well as the monotonicity of $f$).
Consequently, this implies that

$\int\limits_{a}^{\alpha_{j+1}(n)}\left|f(x)-h_{n}(x)\right|^{p}dx<\int\limits_{a}^{\alpha_{j+1}(n)}\left|f(x)-f_{n}(x)\right|^{p}dx$

that easily implies

$\int\limits_{a}^{b}\left|f(x)-h_{n}(x)\right|^{p}dx<\int\limits_{a}^{b}\left|f(x)-f_{n}(x)\right|^{p}dx.$

Thus, $\left\|h_{n}-f\right\|_{p}<\left\|f_{n}-f\right\|_{p}$ which, since
$h_{n}\in\mathbf{SI}_{m}^{0}(\Delta_{n})$, contradicts the fact that $f_{n}$
is the best approximation of $f$ in $\mathbf{SI}_{m}^{0}(\Delta_{n})$ with
respect to $\left\|\cdot\right\|_{p}$. Therefore, we have
$f_{n}(\alpha_{j+1}(n))>f(a)=0$ for all $n\geq 1$.

Now let us show that
$\lim\limits_{n\rightarrow\infty}f_{n}(\alpha_{j+1}(n))=0$. By way of
contradiction we suppose that this is not true. In this case we may assume
that there exists $\gamma_{1}>0$ such that $f_{n}(\alpha_{j+1}(n))>\gamma_{1}$
for sufficiently large $n$. Since $f(u_{0})=0$ and by the continuity of $f$,
there exists $\delta>u_{0}$ such that $f(x)<\gamma_{1}/2$ for all
$x\in[u_{0},\delta]$. Since
$\lim\limits_{n\rightarrow\infty}\alpha_{j+1}(n)=u_{0}$, for sufficiently
large $n$, we have $\alpha_{j+1}(n)<\left(u_{0}+\delta\right)/2$. Therefore,
by the monotonicity of $f_{n}$, for sufficiently large $n$ we get

$f(x)<\gamma_{1}/2<\gamma_{1}<f_{n}(\alpha_{j+1}(n))\leq f_{n}(x)\text{, for
all }x\in[\left(u_{0}+\delta\right)/2,\delta].$

This implies that, for some $x_{0}\in[\left(u_{0}+\delta\right)/2,\delta]$,
$f_{n}(x_{0})$ does not converge to $f(x_{0})$; a contradiction. Therefore, we
have $\lim\limits_{n\rightarrow\infty}f_{n}(\alpha_{j+1}(n))=0$.

Now, suppose that $n\geq 1$ is fixed and, to simplify the notations, let us
denote $v_{n}=f_{n}(\alpha_{j+1}(n))$. Let
$g_{n}:\left[a,b\right]\rightarrow\mathbb{R}$, defined by

$g_{n}(x)=\left\\{\begin{array}[]{ccc}0,&if&x\in[a,\alpha_{j}(n)];\\\
\frac{\left(x-\alpha_{j}(n)\right)v_{n}}{\alpha_{j+1}(n)-\alpha_{j}(n)},&if&x\in\left[\alpha_{j}(n),\alpha_{j+1}(n)\right];\\\
f_{n}(x),&if&x\in\left[\alpha_{j+1}(n),b\right].\end{array}\right.$

It is trivial to observe that $g_{n}\in\mathbf{SI}_{m}^{0}(\Delta_{n})$, so

$\left\|f_{n}-f\right\|_{p}\leq\left\|g_{n}-f\right\|_{p}\text{.}$

We also easily notice that

$\int\limits_{a}^{\alpha_{j}(n)}\left|f(x)-f_{n}(x)\right|^{p}dx>\int\limits_{a}^{\alpha_{j}(n)}\left|f(x)-g_{n}(x)\right|^{p}dx$

and

$\int\limits_{\alpha_{j+1}(n)}^{b}\left|f(x)-f_{n}(x)\right|^{p}dx=\int\limits_{\alpha_{j+1}(n)}^{b}\left|f(x)-g_{n}(x)\right|^{p}dx\text{.}$

Hence

$\int\limits_{\alpha_{j}(n)}^{\alpha_{j+1}(n)}\left|f(x)-f_{n}(x)\right|^{p}dx<\int\limits_{\alpha_{j}(n)}^{\alpha_{j+1}(n)}\left|f(x)-g_{n}(x)\right|^{p}dx\text{.}$

This further implies that

$\int\limits_{\alpha_{j}(n)}^{u_{n}}\left|f(x)-f_{n}(x)\right|^{p}dx<\int\limits_{\alpha_{j}(n)}^{\alpha_{j+1}(n)}\left|f(x)-g_{n}(x)\right|^{p}dx,$
(3)

where
$u_{n}=\alpha_{j}(n)+\frac{1}{2m^{2}+1}\left(\alpha_{j+1}(n)-\alpha_{j}(n)\right)$.

Applying the mean value theorem in both integrals in (3), there exist
$c_{n}\in\left(\alpha_{j}(n),u_{n}\right)$ and
$d_{n}\in\left(\alpha_{j}(n),\alpha_{j+1}(n)\right)$, such that

$\frac{\alpha_{j+1}(n)-\alpha_{j}(n)}{2m^{2}+1}\cdot\left|f(c_{n})-f_{n}(c_{n})\right|^{p}<\left(\alpha_{j+1}(n)-\alpha_{j}(n)\right)\left|f(d_{n})-g_{n}(d_{n})\right|^{p}\text{,}$

which by simple calculations gives

$\left|f(c_{n})-f_{n}(c_{n})\right|<\left(2m^{2}+1\right)^{1/p}\left|f(d_{n})-g_{n}(d_{n})\right|\text{.}$
(4)

As $f_{n}$ is nondecreasing, by Lemma 8, we get that

$\displaystyle f_{n}\left(\alpha_{j}(n)\right)$ $\displaystyle\leq$
$\displaystyle f_{n}(c_{n})\leq f_{n}\left(u_{n}\right)$ $\displaystyle\leq$
$\displaystyle
f_{n}\left(\alpha_{j}(n)\right)+\frac{2m^{2}}{2m^{2}+1}\cdot(f_{n}\left(\alpha_{j+1}(n)\right)-f_{n}\left(\alpha_{j}(n)\right))\text{.}$

This implies that there exists
$t_{n}\in\left[0,\frac{2m^{2}}{2m^{2}+1}\right]$, such that

$\displaystyle f_{n}(c_{n})$ $\displaystyle=$ $\displaystyle
f_{n}\left(\alpha_{j}(n)\right)+t_{n}\cdot(f_{n}\left(\alpha_{j+1}(n)\right)-f_{n}\left(\alpha_{j}(n)\right))$
$\displaystyle=$ $\displaystyle(1-t_{n})\cdot
f_{n}\left(\alpha_{j}(n)\right)+t_{n}\cdot f_{n}\left(\alpha_{j+1}(n)\right)$
$\displaystyle=$ $\displaystyle(1-t_{n})\cdot f_{n}\left(a\right)+t_{n}\cdot
v_{n}\text{.}$

Returning to inequality (4), we get

$\left|(1-t_{n})\cdot f_{n}\left(a\right)+t_{n}\cdot
v_{n}-f(c_{n})\right|<\left(2m^{2}+1\right)^{1/p}\left|f(d_{n})-g_{n}(d_{n})\right|\text{.}$
(5)

As $c_{n},d_{n}\in\left(\alpha_{j}(n),\alpha_{j+1}(n)\right)$, and
$\lim\limits_{n\rightarrow\infty}\alpha_{j+1}(n)\leq a_{1}$ and
$f(a_{1})=f(a)=0$, by the monotonicity of $f$, it easily results
$f(c_{n})\rightarrow 0$ and $f(d_{n})\rightarrow 0$. Then, by the construction
of $g_{n}$, we have $0\leq g_{n}(x)\leq f_{n}(\alpha_{j+1}(n))$, for all
$x\in\left[\alpha_{j}(n),\alpha_{j+1}(n)\right]$. In particular we have $0\leq
g_{n}(d_{n})\leq f_{n}(\alpha_{j+1}(n))$. Hence, since
$\lim\limits_{n\rightarrow\infty}f_{n}(\alpha_{j+1}(n))=0$, it follows that
$\lim\limits_{n\rightarrow\infty}$ $g_{n}(d_{n})=0$. This means that both
expressions in the inequality (5) converge to $0$.

Next, since $0\leq t_{n}<1$, it follows that $t_{n}\cdot
f_{n}\left(\alpha_{j+1}(n)\right)\rightarrow 0$. On the other hand, as
$1-t_{n}\geq\frac{1}{2m^{2}+1}$, we have that $f_{n}(a)\rightarrow 0.$ In
fact, if $(1-t_{n})\cdot f_{n}\left(a\right)+t_{n}\cdot
f_{n}\left(\alpha_{j+1}(n)\right)-f(c_{n})$ would not converge to $0$ we have
a contradiction, since the absolute value of the expression in (5) is bounded
by an expression converging to $0$. In conclusion, we just proved that any
subsequence of $\left(f_{n}\left(a\right)\right)_{n\geq 1}$ contains a
subsequence that converges to $0$ (please note again that all subsequences
were denoted the same to avoid too complicated notations). Clearly this
implies that $\lim\limits_{n\rightarrow\infty}f_{n}(a)=0$.

It remains to prove that $\lim\limits_{n\rightarrow\infty}f_{n}(b)=f(b)$.
First, let us notice that that $\lim\limits_{n\rightarrow\infty}f_{n}(a)=f(a)$
if $f$ is continuous and nonincreasing with
$f_{n}=\mathbf{SD}_{m}^{0}(\Delta_{n},f,p)$, for all $n\geq 1$. Indeed, this
is immediate from the first part of the proof taking into account that
obviously we have
$\mathbf{SI}_{m}^{0}(\Delta_{n},-f,p)=-\mathbf{SD}_{m}^{0}(\Delta_{n},f,p)$,
for all $n\geq 1$. Now, suppose again that $f$ is nondecreasing. Considering
the function $g(x)=f(a+b-x)$, it is easily seen that
$\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)(b)=\mathbf{SD}_{m}^{0}({\Delta}_{n}^{\prime},g,p)(a)$,
where
${\Delta}_{n}^{\prime}=\left(\beta_{1}(n),\beta_{2}(n),...,\beta_{k_{n}}(n)\right)$
and $\beta_{i}=a+b-\alpha_{k_{n}-i+1}$, $i=\overline{1,k_{n}}$. Since $g$ is
continuous and $\left\|{\Delta}_{n}^{\prime}\right\|\rightarrow 0$, the
previous result implies
$\lim\limits_{n\rightarrow\infty}\mathbf{SD}_{m}^{0}({\Delta}_{n}^{\prime},g,p)(a)=g(a)$
which implies that
$\lim\limits_{n\rightarrow\infty}\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)(b)=f(b)$.
It is clear now that we can apply Lemma 6, which means that
$\lim\limits_{n\rightarrow\infty}\left\|\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)-f\right\|=0$,
and the proof is complete. $\blacksquare$

##  5 Some open questions

It remains, of course, to extend the results obtained here for $l=0$, to an
arbitrary value of $l$. At this end, let us briefly discuss the case $l\geq
1$. To do this, let us notice that the crucial point in the proof of the Main
Theorem is that the best $L^{p}$ approximation is constantly negative until a
given knot $\alpha_{j}(n)$ and positive on the following knot
$\alpha_{j+1}(n)$. In the case $l\geq 1$ this property may not hold, or at
least a different type of reasoning is needed to prove it. Actually, for
important classes of partitions (see the equidistant partitions or even the
partitions based on the Chebyshev knots of the first kind, for instance), it
would be suffice to prove that if the function is constantly negative until a
given knot $\alpha_{j}(n)$ then it is positive on the knot $\alpha_{j+k}(n)$
where $k$ is a constant that does not depend on $n$.

Another interesting problem is to consider instead of the $L^{p}$ norms the
more general approach with monotone norms.

Moreover, it would interesting to find an estimation for the rate of the
uniform convergence of the sequences
$\left(\mathbf{SI}_{m}^{0}(\Delta_{n},f,p)\right)_{n\geq 1}$ and of the
sequence $\left(\mathbf{SD}_{m}^{0}(\Delta_{n},f,p)\right)_{n\geq 1}$,
respectively.

Then of course, as we already mentioned this problem earlier, it would be
interesting to study whether the assumption $m\geq 2l+1$ can be relaxed in the
the statements of Proposition 3 and Corollary 5 by only assuming that $m\geq
l+1$. Of course, in this case another approach is needed that does not use
Theorem 1 in [11] which necessarily implies the limitation $m\geq 2l+1$.

Finally, it remains the question whether the main theorem remains true if we
drop the assumptions on the monotonicity of $f$ and of the best piecewise
polynomial approximations. But again, these limitations are necessarily
implied by Theorem 1 in [8] and Theorem 1 in [11]. Therefore, in order to
generalize the main theorem for arbitrary not necessarily monotone functions,
a new approach is needed that does not use the aforementioned theorems.

Acknowledgment

The contribution of Lucian Coroianu was possible with the support of Ministry
of Research and Innovation, CNCS-UEFISCDI, project number PN-
III-P1-1.1-PD-2016-1416, within PNCDI III.

## References

  * [1] Bogachev L. V. and Zarbaliev S. M. , Universality of the limit shape on convex lattice polygonal lines, The Annals of Probability, 39 (2011) 2271-2317.
  * [2] Caravantes J., Gomez-Molleda M.A. and Gonzales-Vega L., A canonical form for the continuous piecewise polynomial functions, Journal of Computational and Applied Mathematics, 283 (2015) 17-27.
  * [3] De Boor C. and Swartz B., Piecewise monotone interpolation, Journal of Approximation Theory, 21 (1977) 411-416.
  * [4] Greiner H. , A survey on univariate data interpolation and approximation by splines of given shape, Mathematical and Computer Modelling, 15 (1991) 97-106.
  * [5] Kimchi E. and Richter-Dyn N. , Convergence Properties of Sequences of Functions with Application to Restricted Derivative Approximation, Journal of Approximation Theory, 22 (1978) 289-303.
  * [6] Kopotun K. A., Leviatan D. and Shevchuk I. A., Interpolatory pointwise estimates for monotone polynomial approximation, Journal of Mathematical Analysis and Applications, 459 (2018) 1260-1295.
  * [7] Kopotun K. A., Leviatan D. and Shevchuk I. A., Interpolatory estimates for convex piecewise polynomial approximation, Journal of Mathematical Analysis and Applications, 474 (2019) 467-479.
  * [8] Lewis J. T. and Shisha O., $L_{p}$ convergence of Monotone Functions and their Uniform Convergence, Journal of Approximation Theory, 14 (1975) 281-284.
  * [9] Li J., Wang R. , Xu M. and Fang Q., Piecewise linear approximation methods with stochastic sampling sites, Journal of Computational and Applied Mathematics, 329 (2018) 173-178.
  * [10] Lorentz R. A. , Uniqueness of best approximation by monotone polynomials, Journal of Approximation Theory, 4 (1971) 401-418.
  * [11] Passow E., Piecewise monotone spline interpolation, Journal of Approximation Theory, 12 (1974), 240-241.
  * [12] Pence D. D., Best mean approximation by splines satisfying generalized convexity constraints, Journal of Approximation Theory, 28 (1980) 333-348.
  * [13] Richards F. B. , A Gibbs phenomenon for spline functions, Journal of Approximation Theory, 66 (1991) 344-351.
  * [14] Saff E. B. and Tachev S., Gibbs phenomenon for best $L^{p}$ approximation by polygonal lines, East Journal of Approximation, 5 (1999) 235-251.