Lebesgue Interpolation Amplification
Measure how interpolation magnifies bounded data errors through its signed basis functions.
Free theory articleMeasure how interpolation magnifies bounded data errors through its signed basis functions.
Concept & interpretation
Interpolation constructs a continuous function that exactly passes through a set of discrete data points. However, when dealing with real-world data, the values we sample are often subject to measurement errors or noise. The Lebesgue function and its associated constant provide a measure of how much an interpolation scheme amplifies these errors across the domain.
Experiment
You can explore this phenomenon interactively by adjusting the nodes and the type of perturbation in the module:
- Open the interactive module at Lebesgue Interpolation Amplification.
- Select Four equally spaced for the Nodes control.
- Set the Probe x to 0.5.
- Choose a Data error of 0.1.
- Set the Perturbation to Maximize here to observe the worst-case error at the probe point. The expected pointwise error bound should read
0.1421875. - Change the Nodes to Four Chebyshev Lobatto and note how the Lebesgue function flattens out, reducing the maximum possible amplification.