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The result is called the polynomial of best approximation, the Chebyshev approximation, or the [[minimax approximation]].
===On the choice of initialization===
For the initialization of the optimization problem for function ''f'' by the Lagrange interpolant ''L''<sub>n</sub>(''f''), it can be shown that this initial approximation is bounded by
<math>\lVert f - L_n(f)\rVert_\infty \le (1 + \lVert L_n\rVert_\infty) \inf_{p \in P_n} \lVert f - p\rVert</math>
with the norm or [[Lebesgue constant (interpolation)|Lebesgue constant]] of the Lagrange interpolation operator ''L''<sub>''n''</sub> of the nodes (''t''<sub>1</sub>, ..., ''t''<sub>''n'' + 1</sub>) being
<math>\lVert L_n\rVert_\infty = \left\| \Lambda_t \right\|, \quad \Lambda_t = \sum_{j = 1}^{n + 1} \left| \prod_{\stackrel{i = 1}{i \ne j}}^{n + 1} \frac{(\cdot - t_i)}{(t_j - t_i)}\right|</math>
Kilgore<ref>[http://dx.doi.org/10.1016/0021-9045(78)90013-8 T. A. Kilgore, "A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm", J. Approx. Theory 24, 273 (1978).]</ref> and de Boor & Pinkus<ref>[http://dx.doi.org/10.1016/0021-9045(78)90014-X C. de Boor and A. Pinkus, "Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation", J. Approx. Theory 24, 289 (1978).]</ref> prooved that there exists an unique ''t''<sub>''i''</sub> for each ''L''<sub>''n''</sub>, although not known explicitly for (ordinary) polynomials.
==Variants==
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