Math 6610 • Analysis of Numerical Methods I

{Lebesgue's Lemma}

Let \(I\) be a closed, bounded interval on the real line and let
\(\|f\|_\infty:=\sup_{x\in I}|f(x)|\). Given \(N\) distinct points
\(x_1,\ldots,x_N\in I\) and a continuous function \(f\), consider the Lagrange form
of the degree-\((N-1)\) polynomial interpolant of \(f\):
\[
  I_N f := \sum_{j=1}^{N}f(x_j)\ell_j(x),
\]

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where \(\{\ell_j(\cdot)\}_{j=1}^{N}\) are cardinal Lagrange polynomials. Show
that
\[
  \sup_{\|g\|_\infty=1}\|I_N g\|_\infty
  \le \Lambda := \sum_{j=1}^{N}|\ell_j(x)|,
\]
and that
\[
  \|f-I_N f\|_\infty
  \le
  (1+\Lambda)\inf_{p\in P_{N-1}}\|f-p\|_\infty,
\]
where \(P_{N-1}=\operatorname{span}\{1,x,\ldots,x^{N-1}\}\).

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\textbf{Note:} In problems 5--8, the notations \(k=\Delta t\) and \(h=\Delta x\)
are used.