%PREAMBLE DEFINITIONS

\def\Re{{\rm Re}}
\def\reals{{{\sf l} \kern -.102em {\sf R} }}
\newcommand{\bx}{\hfill  $\Box$}
\newcommand{\nos}{\vspace*{7pt} \noindent}

\documentstyle{oulam}
%SET UP PAGE LENGTH AND WIDTH FOR ULAM ARTICLE
\setlength{\topmargin}{1.2cm}
\setlength{\textheight}{19.0cm}
\setlength{\oddsidemargin}{2.4cm}
\setlength{\evensidemargin}{2.4cm}
\setlength{\textwidth}{11.5cm}

\pagestyle{headings}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{algorithm}[theorem]{Algorithm}
\renewcommand{\thetheorem}{\arabic{theorem}}
%\renewcommand{\thetheorem}{\thesection.\arabic{theorem}}

\newcommand{\eqref}[1]{(\ref{#1})}
\newcommand{\text}[1]{\ {\rm #1} \ }
\def \dis {\displaystyle}
\def \Bbb {\bf}
\def \dotsc {\dots}

\begin{document}
%EDITOR WE WILL SET THE PAGE NUMBER HERE
\setcounter{page}{11}

%THIS IS THE FILE ulam.tex INPUT BY ulamc.tex.
%SET UP TITLE PAGE
\thispagestyle{plain}

%JK SETUP TITLE HEADER
\vspace*{-40pt}
\begin{center}
{\small Ulam Quaterly -- Volume 3, Number 2, 1996}
\end{center}
\vspace*{16pt}

\raggedbottom
\pagestyle{myheadings}
\markboth{ A $C^\infty$ Spline Construction }{ P.~Blass, J.~Dodziuk, T.~J.~Ford, and J.~Kolibal}


%SET UP TITLE AND TIDY SPACING FOR TWO LINE TITLE
\begin{center}
\begin{tabular}{c}
\LARGE \bf   A $C^\infty$ Spline Construction 
\end{tabular}





%JK\title{A $C^\infty$ Spline Construction }
%\subjclass{}
%JK\author{P.~Blass, J.~Dodziuk,
%JKT.~J.~Ford, and J.~Kolibal}


%SET UP AUTHOR LISTING
\vspace*{40pt}
\begin{tabular}{cc}
\large \bf  P.~Blass, &\large \bf  J.~Dodziuk  \\
Department of Mathematics   &  Program in Mathematics  \\
Palm Beach Atlantic College  &  Graduate Center, CUNY  \\[20pt]
\large  \bf  T.J~Ford,  & \large \bf  J.~Kolibal  \\
Department of Mathematics   &   Department of Mathematics  \\
Florida Atlantic University  &  University of Southern Mississippi
\end{tabular}

\end{center}
\vspace*{40pt}


\begin{center}
\section*{Abstract}
\end{center}

We consider the problem of determining a smooth spline through a set
of points in the plane determined using a standard partition of
unity of construction.


\vspace*{7pt}

\date{\today}
%\thanks{Preliminary report}
%JK\maketitle
%\medskip

%\begin{abstract}
%\end{abstract}
\begin{center}
\section{Introduction}
\end{center}


\begin{theorem}
\label{th1}
Let $a_0, a_1, a_2, \dotsc$ be a sequence of real numbers and 
 $d_0, d_1, d_2, \dotsc$ another sequence of real numbers defined by the
formula
\begin{equation}
\label{eq1}
\begin{array}{lcl}
d_0 &=& \text{arbitrary, and} \\
d_i &=& (a_{i+1} - a_{i-1})/2 \text{ for } i > 0 \text{ .}
\end{array}
\end{equation}
(This is the 3-point midpoint formula for approximating the derivative at
$(i,a_i)$.) Given $\epsilon > 0$, then there exists a function $f(x) \in
C^\infty[0,\infty)$ satisfying
\begin{enumerate}
\item
$f(i) = a_i$ for $i \ge 0$;
\item
$f'(i) = d_i$ for $i \ge 0$;
\item
$\dis{\sup_{x} \left| f(x) - l(x) \right| < \epsilon}$ where $l(x)$ is the
linear spline satisfying
\begin{enumerate}
\item
$l(i) = a_i$ for $i \ge 0$,
\item
$l(x) $ is linear on $[i,i+1]$ for $i \ge 0$;
\end{enumerate}
and,
\item
On each interval  $[i,i+1]$ for $i \ge 0$, $f(x)$ has at most one
inflection point. More precisely, if $m_i$ denotes the slope of $l(x)$ on
$[i,i+1]$ then, $f(x)$ has no inflection points on $[i,i+1]$ provided
$(d_i-m_i)(d_{i+1}-m_i) < 0$. 
On the other hand, if $(d_i-m_i)(d_{i+1}-m_i) \ge 0$, then $f(x)$ has a
unique inflection point in $[i,i+1]$ at the midpoint $x=i+\frac{1}{2}$.
\end{enumerate}
\end{theorem}
%\begin{pf}

\noindent
{\it Proof.}
First we review the existence of certain $C^\infty$ functions. Let

\begin{equation} \label{eq2}
r(x) = \left\{ \begin{array}{ll}
\displaystyle e^{-\frac{1}{{x}^2}-\frac{1}{({x}-1)^2}} & \text{ for } x \in (0,1) \\
0 & \text{ otherwise }
\end{array} \right. \text{ .}
\end{equation}

\nos
It is well known that $r(x)$ is infinitely differentiable at all
points $x \in
%JK \Bbb R$. Let
 \reals$. Let


\begin{equation} \label{eq3}
t(x) = \left( \int^\infty_{-\infty} r(x) \, dx \right)^{-1}
\int^{x}_{-\infty} r(u) \, du \text{ .}
\end{equation}


\nos
Then $t(x)$ is $C^{\infty}$, $t(x)=0$ for $x \le 0$, $t(x) = 1$ for $x \ge
1$ and $t'(x) = r(x) > 0$ if $0 < x< 1$. Note that derivatives of all
orders (except zeroth) of $t(x)$ vanish at $x=0$ and $x=1$. All functions
to be constructed in the proof of Theorem~\ref{th1}
will be built out of $t(x)$ by means of formulae. This will imply that all
functions constructed below will depend continuously on the parameters of
the construction.

Using affine transformations in $x$ and in $y$ ( $=t(x)$), we see easily
that for every interval $[a,b]$ and every pair of values $\alpha, \beta \in
%JK\Bbb R$ there exists a $C^ \infty$ function on $\Bbb R$, $j_ \xi(x)$, $\xi
\reals$ there exists a $C^ \infty$ function on $\reals$, $j_ \xi(x)$, $\xi
= (a,b,\alpha,\beta)$ such that

\begin{equation}
\label{eq4}
\begin{array}{lcl}
j_\xi(x)& =& \alpha \text{ for } x \le a \\[7pt]
j_\xi(x) &=& \beta \text{ for } x \ge b \text{ and} \\[7pt]
j_\xi'(x) &\not =& 0 \text{ for } x \in (a,b) \text{ .}
\end{array}
\end{equation}

\nos
Similarly, given $\zeta = (a,b,c,\alpha,\beta,\gamma)$, $a < b < c$, there
%JKexists a smooth function $k_\zeta(x)$ on $\Bbb R$ with
exists a smooth function $k_\zeta(x)$ on $\reals$ with

\begin{equation}
\label{eq5}
\begin{array}{lcl}
k_\zeta(x) &=& \alpha \text{ for } x \le a \\[7pt]
k_\zeta(x) &=& \gamma \text{ for } x \ge c \text{ and} \\[7pt]
k_\zeta(b) & =& \beta  \text{ .}
\end{array}
\end{equation}
%See Figure~1.

%JK\epsfxsize=4in
%JK\epsfclipon
%JK\centerline{\epsffile[50 275 500 486 ]{fig1.ps}}
%JK\centerline{ Figure 1: {$y = k_\zeta(x)$}}
%JK\bigskip


\begin{figure}[t] \setlength{\unitlength}{1cm}
%\begin{picture}(11.7,7.0)(5.5,3.2)
\begin{picture}(11.7,4.0)(5.0,12.8)
\put(0,0){\special{psfile="fig1.ps"}}
\end{picture}
\vspace*{0.6cm}
\caption{\sf {$y = k_\zeta(x)$.}
\label{fig:fig1}  }
\end{figure}


We can simply set


\begin{equation}
\label{eq6}
\begin{array}{lcl}
k_\zeta(x)& =& j_{(a,b,\alpha,\beta)}(x) \text{ for } x \le b \\[7pt]
k_\zeta(x) &=& j_{(b,c,\beta,\gamma)}(x) \text{ for } x \ge b  \text{ .}
\end{array}
\end{equation}

\nos
Clearly $k_\zeta(x)$ has the required properties.

Note that if $\zeta$ is as above and $a$, $b$, $c$, $\alpha$, $\gamma$ are
kept fixed, then

\begin{equation}
\label{eq7}
\lim_{\beta \to \infty} \int_a^c k_\zeta(x) \, dx = \infty
\text{ .}
\end{equation}

\nos
This observation will be used repeatedly below.

%JK
\begin{center}
\section{Construction}
\end{center}

We are now ready to start the construction. We will work on one interval
$[i,i+1]$ at a time and construct $f_i = f|[i,i+1]$ satisfying in addition
to  properties (1), (2), (3), (4) listed in the theorem, the property
\vspace*{5pt}

\begin{itemize}
\item[(5)] $f_i^{(k)}(i) = f_i^{(k)}(i+1)=0$ for all $k\ge 2$.
\end{itemize}
Condition (5) will ensure that the $f_i$'s fit together smoothly. Now,
since $l(x)$ is a known function, it is enough to construct
$\phi_i=f_i-l_i$, where $l_i=l|[i,i+1]$.
The properties that $\phi_i$ must satisfy are then
\begin{itemize}
\item[(1')] $\phi_i(i) = \phi_i(i+1)=0$
\item[(2')] $\phi'_i(i) = e_i$, $\phi'_i(i+1) = e_{i+1}$, where
$e_i=d_i-m_i$ is prescribed.
\item[(3')] $|\phi_i(x)| < \epsilon$ for every $x \in [i,i+1]$
\item[(4')] If $e_i \cdot e_{i+1}\ge 0$, then $\phi_i | (i,i+1)$ has
exactly one inflection point located at $i+\frac{1}{2}$. If 
$e_i \cdot e_{i+1} < 0$, then  $\phi_i | (i,i+1)$ has
no inflection point.
\end{itemize}
Clearly it suffices to describe the construction of $\phi = \phi_0$. The
same construction will work for every interval $[i,i+1]$.
%JK
The idea is to guess the shape of the graph of the second derivative of
$\phi$ and integrate twice.
\vspace*{5pt}


\vspace*{12pt}
\noindent
{\it Case 1.} Consider $e_0 \cdot e_1 < 0$. If necessary, multiply all functions by
$-1$, to be able to assume that $e_0 >0$, $e_1 < 0$. The functions
$y=\phi(x)$, $y=\phi'(x)=\rho(x)$ and $y=\phi''(x)=\psi(x)$ have graphs
that look like the following.

%JK\epsfxsize=6in
%JK\epsfclipon
%JK\centerline{\epsffile[125 225 475 575]{fig2.ps}}
%JK\centerline{ Figure 2: The function $\phi$ and its first 2 derivatives.}
%JK\bigskip


\begin{figure}[t] \setlength{\unitlength}{1cm}
%\begin{picture}(11.7,7.0)(5.5,3.2)
\begin{picture}(11.7,8.7)(4.6,10.8)
\put(0,0){\special{psfile="fig2.ps"}}
\end{picture}
\vspace*{1.6cm}
\caption{\sf The function $\phi$ and its first 2 derivatives.
\label{fig:fig2}  }
\end{figure}


The functions $\psi(x)$ and $\rho(x)$ must satisfy

\begin{equation}
\label{eq8}
\int_0^1 \psi(x) \, dx = e_1-e_0
\end{equation}

\begin{equation}
\label{eq9}
\int_0^1 \rho(x) \, dx = 0
\end{equation}

\begin{equation}
\label{eq10}
\left| \int_0^x \rho(u) \, du \right| \le \epsilon \text{ for all } x \in
[0,1] \text{ .}
\end{equation}


\nos
If this is the case, then

\begin{equation}
\label{eq11}
\rho(x)=e_0+\int_0^x \psi(u) \, du \text{ ,}
\end{equation}

\begin{equation}
\label{eq12}
\phi(x) = \int_0^x \rho(u) \, du 
\end{equation}

\vspace*{5pt}
\noindent
and $\phi$ satisfies the required properties.

We will take the widths of the two spikes, the values of $\psi$ at the
local minima and the value near $x=\frac{1}{2}$ as parameters and adjust
them so that \eqref{eq8}, \eqref{eq9} and \eqref{eq10} are satisfied. Thus,
let $\eta_0$, $\eta_1$, $\mu$ be three (small) positive numbers and let $h_0$ and
$h_1$ be two negative numbers.
Define


\vspace*{5pt}
\begin{equation} \label{eq13}
\psi(x) = \left\{ \begin{array}{ll}
k_{(0,\frac{\eta_0}{2},\eta_0,0,h_0,-\mu)}(x)& \text{ for } 0 \le x \le
\frac{1}{2} \\[7pt]
k_{(1-\eta_1,1-\frac{\eta_1}{2},1,-\mu,h_1,0)}(x)& \text{ for }
\frac{1}{2}\le x \le 1 \\ 
\end{array} \right. \text{ .}
\end{equation}

\vspace*{10pt}
\noindent
Thus $\eta_0$ and $\eta_1$ measure the widths of spikes, $-h_0$  and $-h_1$
are depths and $\mu$ is the depth of the horizontal portion of the graph.
Given $\eta_0$, $\eta_1$, $\mu$ (provided $\mu$ is such that $0 < \mu \le
|e_1-e_0|$)  we can choose $h_0 = h_1 = h$ so that

\vspace*{5pt}
\begin{equation}
\label{eq14}
\int_0^1 \psi(x) \, dx = e_1-e_0 <0
\end{equation}


\vspace*{5pt}
\noindent
This $h$ is actually unique for a fixed choice of $\eta_0$, $\eta_1$,
$\mu$.
Since $e_0$ and $e_1$ are fixed it is also obvious that \eqref{eq10}
holds for all choices of $\eta_0$, $\eta_1$,
$\mu$ which are sufficiently small. This is because $\eta_0$, $\eta_1$ and 
$\mu$ determine how closely the graph of $\rho(x)$ hugs the lines $x=0$,
$x=1$ and $y=0$. We still need to make sure that \eqref{eq9} holds. Suppose
this is not the  case for some choice of $\eta_0$ and $\eta_1$ and, for
example, 
\begin{equation}
\label{eq15}
\int_0^1 \rho(x) \, dx >0 \text{ .}
\end{equation}
Let $\rho_+=\frac{1}{2}(\rho + |\rho|)$, $\rho_-=\frac{1}{2}(|\rho|-\rho)$
be the positive and negative parts of $\rho$ respectively. Then


\vspace*{5pt}
\begin{equation}
\label{eq16}
\int_0^1 \rho_+(x) \, dx >
\int_0^1 \rho_-(x) \, dx 
\end{equation}



\vspace*{5pt}
\noindent
by \eqref{eq15}.
First choose $\mu$ so that 
\begin{equation}
\label{eq17}
0 < \mu \ll \int_0^1 \rho_-(x) \, dx  \text{ .}
\end{equation}

\noindent
This makes the shaded area in Figure~3 negligible.

%JK\epsfxsize=6in
%JK\epsfclipon
%JK\centerline{\epsffile[150 325 450 475]{fig3.ps}}
%JK\centerline{ Figure 3: The function $\rho$.}
%JK\bigskip

\begin{figure}[t] \setlength{\unitlength}{1cm}
%\begin{picture}(11.7,7.0)(5.5,3.2)
\begin{picture}(11.7,4.0)(4.8,12.8)
\put(0,0){\special{psfile="fig3.ps"}}
\end{picture}
\vspace*{0.6cm}
\caption{\sf The function $\rho$.
\label{fig:fig3}  }
\end{figure}



Now let $\eta_0$ approach zero keeping $\eta_1$ fixed. Clearly for $\eta_0$
sufficiently small $A_1 \ll A_2$ so that 

\begin{equation}
\label{eq18}
\int_0^1 \rho(x) \, dx <0 \text{ .}
\end{equation}

\vspace*{5pt}
\noindent
Therefore for some intermediate value of $\eta_0$

\begin{equation}
\label{eq19}
\int_0^1 \rho(x) \, dx =0 \text{ .}
\end{equation}

\vspace*{5pt}
\noindent
This finishes the construction in Case 1.

\vspace*{12pt}
\noindent
{\it Case 2.} 
Consider$e_0 \cdot e_1 \ge 0$. This is actually slightly easier. Suppose $e_0 \cdot
e_1 > 0$. The case when one of the slopes is equal to 0 requires a modified
argument and will be treated later. First consider the graphs.

%JK\epsfxsize=6in
%JK\epsfclipon
%JK\centerline{\epsffile[125 225 475 575]{fig4.ps}}
%JK\centerline{ Figure 4: The function $\phi$ and its first 2 derivatives.}
%JK\bigskip

\begin{figure}[t] \setlength{\unitlength}{1cm}
%\begin{picture}(11.7,7.0)(5.5,3.2)
\begin{picture}(11.7,9.0)(4.6,10.5)
\put(0,0){\special{psfile="fig4.ps"}}
\end{picture}
\vspace*{2.0cm}
\caption{\sf The function $\phi$ and its first 2 derivatives.
\label{fig:fig4}  }
\end{figure}



\begin{figure}[t] \setlength{\unitlength}{1cm}
%\begin{picture}(11.7,7.0)(5.5,3.2)
\begin{picture}(11.7,9.0)(4.6,10.5)
\put(0,0){\special{psfile="fig4.ps"}}
\end{picture}
\vspace*{2.0cm}
\caption{\sf The function $\phi$ and its first 2 derivatives.
\label{fig:fig5}  }
\end{figure}


In order for $\phi(x)$ to have all of the required properties, we must have
conditions \eqref{eq8}, \eqref{eq9} and \eqref{eq10} satisfied. We again
try to construct $\psi$. Let $\eta_0$, $\eta_1$ and $\eta$ be parameters
controlling the width of the spikes near $0$, $1$ and $\frac{1}{2}$
respectively. Only one parameter is needed in the middle since we require
that 


\vspace*{5pt}
\begin{equation}
\label{eq20}
\eta (\frac{1}{2}-t)= -\eta(\frac{1}{2}+t)
\end{equation}


\vspace*{5pt}
\noindent
for, say, $|t|<\frac{1}{4}$. Similarly let $\mu = |\psi(x)|$ for $x$ near
$\frac{1}{4}$ or for $x$ near $\frac{3}{4}$. We need three more parameters
that give the heights of spikes: $h_0$, $h_1$, and $h$. Observe that
because of symmetry, 


\vspace*{5pt}
\begin{equation}
\label{eq21}
\int_0^1 \psi(u) \, du 
\end{equation}
is independent of $h$. Given $\eta_0$, $\eta_1$, $\eta$, $h$ we can find
$h_0$ and $h_1$ so that
\eqref{eq8} holds. This is obvious. We can now vary $h$ to achieve 


\vspace*{5pt}
\begin{equation}
\label{eq22}
\int_0^1 \rho_+(x) \, dx =
\int_0^1 \rho_-(x) \, dx 
\end{equation}



\vspace*{5pt}
\noindent
i.e. so that \eqref{eq9} holds. Finally, if $\eta_0$, $\eta_1$, $\eta$ and
$\mu$ were chosen sufficiently small at the outset,


\vspace*{5pt}
\begin{equation}
\label{eq23}
\int_0^1 \rho_+(x) \, dx \le \epsilon
\end{equation}



\vspace*{5pt}
\noindent
which, in view of \eqref{eq22}, implies \eqref{eq10}. This finishes the
construction if $e_0 \cdot e_1 > 0$. 
%JK
Now for the final case. We will only draw the graphs for the case $e_1=0$,
$e_0 >0$. 

%JK\epsfxsize=6in
%JK\epsfclipon
%JK\centerline{\epsffile[125 225 475 575]{fig5.ps}}
%JK\centerline{ Figure 5: The function $\phi$ and its first 2 derivatives.}
%JK\bigskip

%\begin{figure}[t] \setlength{\unitlength}{1cm}
%%\begin{picture}(11.7,7.0)(5.5,3.2)
%\begin{picture}(11.7,9.0)(4.6,10.5)
%\put(0,0){\special{psfile="fig4.ps"}}
%\end{picture}
%\vspace*{2.0cm}
%\caption{\sf The function $\phi$ and its first 2 derivatives.
%\label{fig:fig5}  }
%\end{figure}


\vspace*{12pt}
\noindent
{\em Case 3}.
We can use the spike near $x=0$ to ensure that 
\begin{equation}
\label{eq24}
\int_0^1 \psi(x) \, dx = e_1-e_0 = -e_0 \text{ ,}
\end{equation}
and then adjust the height of symmetric spikes near $\frac{1}{2}$ to
achieve 
\begin{equation}
\label{eq25}
\int_0^1 \rho_+(x) \, dx =
\int_0^1 \rho_-(x) \, dx 
\end{equation}


\noindent
therebye  completing the proof of Theorem~\ref{th1}.     \bx
%\end{pf}
%JK\QED
\bigskip
\bigskip


\begin{center}
\section*{References}
\end{center}
\vspace*{-18pt}

\begin{thebibliography}{99}
\bibitem{1}
%R.~A.Adams, {\em Sobolev Spaces}, Academic Press, New York (1978).
W.~Rudin, {\em Functional Analysis}, McGraw-Hill, New York (1973).
\end{thebibliography}


\vspace*{20pt}


\clearpage

\noindent
Department of Mathematics,
Palm Beach  Atlantic College,
West Palm Beach, FL 33416
{\it E-mail address:}
PBlass@ulam.math.ufl.edu
\bigskip

\noindent
Ph.D. Program in Mathematics, Graduate Center, City University of
New York, New York, NY 10036
{\it E-mail address:}
jzdqc@cunyvm.cuny.edu
\bigskip

\noindent
Department of Mathematics, Florida Atlantic University,
Boca Raton, FL 33431
{\it E-mail address:}
{Ford@acc.fau.edu}


\bigskip
\noindent
Department of Mathematics, University of Southern Mississippi,
Hattiesburg, MS 39406
{\it E-mail address:}
kolibal@eos.st.usm.edu


\vfill
%\begin{center}
%\parbox[h]{9.0cm}{\small
\fontfamily{cm}\fontsize{8}{11pt}\selectfont
\noindent
This electronic publication and its contents are $\copyright$copyright 1996
by Ulam Quarterly.
Permission is hereby granted to give away the journal and it contents,
but no one may ``own'' it. Any and all financial interest is hereby assigned
 to the acknowledged authors of the individual texts.
This notification must accompany all distribution of Ulam Quarterly.
%}
%\end{center}

\newpage
\vspace*{4cm}
\vfill 
\thispagestyle{plain}


\end{document}

