Approximations and expansions Approximation by linear operators Theorem

Faber's theorem on interpolation

Builds on Lebesgue constant Lagrange interpolation Continuous function Uniform convergence

Statement

Let $a < b$ . For each $n \ge 0$ , let $x_{n,0} < x_{n,1} < \cdots < x_{n,n}$ be $n + 1$ points of $[a, b]$ . For a continuous function $f$ on $[a, b]$ , let $p_n(f)$ be the Lagrange interpolation polynomial of degree at most $n$ that agrees with $f$ at $x_{n,0}, \dots, x_{n,n}$ . Then there is a continuous function $f$ on $[a, b]$ such that $p_n(f)$ does not converge uniformly to $f$ on $[a, b]$ .

Remarks

Proof idea: the norm of the interpolation map $f \mapsto p_n(f)$ on $C[a, b]$ is the Lebesgue constant $\Lambda_n$ of the $n$ -th row of nodes. For every choice of nodes, $\Lambda_n \ge c \log n$ for some constant $c > 0$ , so $\Lambda_n \to \infty$ . By the uniform boundedness principle, $\sup_n \norm{p_n(f)}_\infty = \infty$ for some continuous $f$ , and for this $f$ the sequence $p_n(f)$ cannot converge uniformly.

Example: for Chebyshev nodes $\Lambda_n$ grows only like $\frac{2}{\pi} \log n$ , which is slow but still unbounded. So even interpolation at Chebyshev nodes fails for some continuous function, although it converges for every Lipschitz function.

The theorem concerns one fixed array of nodes for all functions. For each single continuous $f$ there is an array of nodes for which $p_n(f) \to f$ uniformly, for example nodes where $f$ meets its best approximations.

Going deeper

For a student

Faber's theorem separates existence of good approximations from their construction by interpolation. It is a typical application of the uniform boundedness principle with norms that grow like a logarithm, as for divergent Fourier series of continuous functions. In practice it is mild: for Chebyshev nodes, Lebesgue's lemma gives convergence as soon as $\omega(f, 1/n) \log n \to 0$ .

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Continuous function concept
Lagrange interpolation concept
Lebesgue constant concept
Uniform convergence concept
Faber's theorem on interpolation statement

Tag	`ZTBC6G7DGD`
URI	`m2://ZTBC6G7DGD`
Kind	statement, theorem
Writing	W0 stub
MSC	41A05, 41A10
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About	Lebesgue constant
Uses	Lebesgue constant, Lagrange interpolation, Continuous function, Uniform convergence