SWhat changed
For every prescribed triangular array of distinct interpolation nodes in a real compact interval, some continuous function fails to be uniformly approximated by its Lagrange interpolation polynomials. Faber proves the corresponding periodic trigonometric obstruction and derives unbounded interpolation-operator norms, using a constructive continuous-function argument.
Scientific scope
For every prescribed triangular array of distinct interpolation nodes in a real compact interval, some continuous function fails to be uniformly approximated by its Lagrange interpolation polynomials. Faber proves the corresponding periodic trigonometric obstruction and derives unbounded interpolation-operator norms, using a constructive continuous-function argument. Uniform-convergence failure is the retained result: no claim of divergence everywhere or unboundedness at a single fixed point for every node array. Nodes are fixed independently of the function. This does not forbid adaptive interpolation, approximation without interpolation, or arbitrary polynomial Schauder systems with unrestricted degrees. The algebraic conclusion is original p208, and the periodic proof conclusion is p199. No assertion of divergence at every point or one fixed point for every array. Adaptive nodes, noninterpolatory approximation and unrestricted-degree polynomial systems are not ruled out. Adjacent Blaschke paper and preceding article excluded.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Uniform-convergence failure is the retained result: no claim of divergence everywhere or unboundedness at a single fixed point for every node array. Nodes are fixed independently of the function. This does not forbid adaptive interpolation, approximation without interpolation, or arbitrary polynomial Schauder systems with unrestricted degrees. The algebraic conclusion is original p208, and the periodic proof conclusion is p199. No assertion of divergence at every point or one fixed point for every array. Adaptive nodes, noninterpolatory approximation and unrestricted-degree polynomial systems are not ruled out. Adjacent Blaschke paper and preceding article excluded.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.5. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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