%%  interpolation.tex
%%  Approximation Theory: 
%%  Exercises on Lagrange Interpolation
%%  Neal L. Carothers
%%  Bowling Green State University
%%  Bowling Green, Ohio  43403
%%  carother@math.bgsu.edu
%%  http://www.bgsu.edu/~carother/


\input 680-setup.tex

\chaptertitle = {Lagrange Interpolation}

\centerline{\hfil\tf Lagrange Interpolation\hfil}
\vskip-\baselineskip
\line{\sc Math 680 \hfil 7/12/94}

\noindent
Throughout, $x_0,x_1,\ldots,x_n$ are distinct points
in some interval $[\,a,b\,]$, and $V(x_0,x_1,\ldots,x_n)$ 
denotes the {\sl Vandermonde determinant\/}:
$$V(x_0,x_1,\ldots,x_n) \ = \ 
    \left|\,\matrix{
    \noalign{\smallskip}
        1 & x_0 & x_0^2 & \cdots & x_0^n\cr
    \noalign{\smallskip}
        1 & x_1 & x_1^2 & \cdots & x_1^n\cr
        \vdots & \vdots & \vdots &\ddots & \vdots \cr
        1 & x_n & x_n^2 & \cdots & x_n^n\cr 
    \noalign{\smallskip}
    }\,\right|.$$

\prob{\cmark\gmark}{%
Show by induction that 
$V(x_0,x_1,\ldots,x_n)=\prod\limits_{\strut 0\le j<i\le n}(x_i-x_j)$.
\vskip0pt
[Hint: 
In order to reduce to the  $n\timesÊn$  case, replace ${\bf c}_j$, 
the $j$-th
column, by  ${\bf c}_j - x_0{\bf c}_{j-1}$, starting on the
right with $j = n$.  Factor and use the induction
hypothesis.]
}

\prob{}{%
Let $y_0,y_1,\ldots,y_n\in\R$ be given.
Show that the polynomial  $p\in{\cal P}_n$
satisfying  $p(x_i) = y_i$, $i=0,1,\ldots,n$,
may be written as
$$p(x) \ = \ c  \left|\,\matrix{
    \noalign{\smallskip}
        0 & 1 & x & x^2 & \cdots & x^n\cr
    \noalign{\smallskip}
        y_0 & 1 & x_0 & x_0^2 & \cdots & x_0^n\cr
        \vdots & \vdots & \vdots &\ddots & \vdots \cr
        y_n & 1 & x_n & x_n^2 & \cdots & x_n^n\cr   
    \noalign{\smallskip}
    }\,\right|,$$ 
where  $c$  is a certain
constant.    Find  $c$ and prove the formula.
}

\prob{}{%
Given  $f \in C[\,a,b\,]$, let  $L_n(f)$ denote the 
polynomial of degree at most $n$ that agrees with  
$f$  at the  $x_i$'s.  Prove that  
$L_n$  is a linear projection onto ${\cal P}_n$.
That is, show that  
$L_n(\alpha f+\beta g) = \alpha L_n(f) + \betaÊL_n(g)$, 
%for any $f$, $g\in C[\,a,b\,]$, $\alpha$, $\beta\in\R$,
and that  $L_n(f)= f$  if and only if  $f\in{\cal P}_n$.
}

\prob{}{%
Let  $\ell_i(x)$, $i = 0, 1, \ldots , n$, denote the Lagrange
interpolating polynomials of degree at most $n$ 
associated with the nodes  $x_0,x_1,\ldots,x_n$;
that is,  $\ell_i(x_j) = \delta_{i,j}$.  Show that 
$\sum_{i=0}^n\ell_i(x)\equiv1$ and, more generally, 
that $\sum_{i=0}^nx_i^k\ell_i(x)=x^k$, for $k=0,1,\ldots, n$.
}

\prob{}{%
If $\ell_i$ and $L_n$ are as above, show that the
error in the Lagrange interpolation formula is 
$(L_n(f)-f)(x) =Ê\sum_{i=0}^n[\,f(x_i)-f(x)\,]\,\ell_i(x)$.
}

\prob{}{%
With $\ell_i$ and $L_n$ as above, show
that  $\VertÊL_n(f)\Vert\le \Lambda_n\VertÊf\Vert$, where  
$\Lambda_n=\bigl\Vert\sum_{i=0}^n|\ell_i(x)|\,\bigr\Vert$.
Show that no smaller number $\Lambda$ has this property
for all  $f \in C[\,a,b\,]$.
}



\bye


%%  end of interpolation.tex