Lebesgue's lemma on near-best approximation

Remarks

Proof: for any $v \in V$, $P v = v$ gives $f - P f = (f - v) - P(f - v)$. So $\norm{f - P f} \le (1 + \norm{P}) \norm{f - v}$, and taking the infimum over $v$ gives the claim.

Example: let $P$ be polynomial interpolation at $n + 1$ nodes in $[a, b]$, acting on $C[a, b]$ with the uniform norm. Then $\norm{P}$ is the Lebesgue constant $\Lambda_n$, and the lemma gives $\norm{f - P f}_\infty \le (1 + \Lambda_n) E_n(f)$, where $E_n(f)$ is the error of best uniform approximation by polynomials of degree at most $n$. For Chebyshev nodes, $\Lambda_n \le \frac{2}{\pi} \log(n + 1) + 1 < 4$ for $n < 100$, so interpolation loses less than a factor $5$ compared with the best polynomial.

The same lemma applies to Fourier partial sums, with $\Lambda_n$ the Lebesgue constants of the Dirichlet kernel.

Going deeper

For a student

This is how Lebesgue constants enter convergence proofs. Combined with Jackson's inequality, it shows that interpolation at Chebyshev nodes converges uniformly when $\omega(f, 1/n) \log n \to 0$, the Dini–Lipschitz condition. Conversely, unbounded norms of projections lead through the uniform boundedness principle to divergence results such as Faber's theorem.