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Lebesgue Constants (Lagrange Interpolation)

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Define the $n$th Lebesgue constant for the Lagrange Interpolating Polynomial by

\Lambda_n(X)\equiv \max_{-1\leq x\leq 1} \sum_{k=1}^n \left\vert{\,\prod_{j\not=k} {x-x_j\over x_k-x_j}}\right\vert.
\end{displaymath} (1)

It is true that
\Lambda_n>{4\over\pi^2}\ln n-1.
\end{displaymath} (2)

The efficiency of a Lagrange interpolation is related to the rate at which $\Lambda_n$ increases. Erdös (1961) proved that there exists a Positive constant such that
\Lambda_n>{2\over\pi}\ln n-C
\end{displaymath} (3)

for all $n$. Erdös (1961) further showed that
\Lambda_n<{2\over\pi}\ln n+4,
\end{displaymath} (4)

so (3) cannot be improved upon.


Erdös, P. ``Problems and Results on the Theory of Interpolation, II.'' Acta Math. Acad. Sci. Hungary 12, 235-244, 1961.

Finch, S. ``Favorite Mathematical Constants.''

© 1996-9 Eric W. Weisstein