Lebesgue Interpolation Amplification

Theory of Lebesgue Amplification

Given a set of nodes x0, x1, ..., xn-1, the Lagrange basis polynomialsli(x) are defined such that li(xj) = 1 if i = j and0 otherwise. The interpolating polynomial P(x) for data values yi is:

P(x) = Σ_i=0^n-1 y_i l_i(x)

If the data values have errors bounded by ε (i.e., |δyi| ≤ ε), the resulting error in the interpolant at any point x is:

|δP(x)| = |Σ_i=0^n-1 δy_i l_i(x)| ≤ ε Σ_i=0^n-1 |l_i(x)|

The sum of the absolute values of the Lagrange basis functions is known as the Lebesgue function, denoted as Λ(x):

Λ(x) = Σ_i=0^n-1 |l_i(x)|

The maximum value of Λ(x) over the interval is the Lebesgue constant. This module visualizes the pointwise amplification by applying specific error signs to the data points to maximize the interpolation error at a given probe point.

Worked Example

Let us consider three equally spaced nodes in the interval [-1, 1]: x0 = -1, x1 = 0, and x2 = 1. We evaluate the Lebesgue function at a probe point x = 0.5.

First, we construct the Lagrange basis polynomials for these nodes:

1. l0(x) = (x - 0)(x - 1) / ((-1 - 0)(-1 - 1)) = (x2 - x) / 2
2. l1(x) = (x - (-1))(x - 1) / ((0 - (-1))(0 - 1)) = (x + 1)(x - 1) / (-1) = 1 - x2
3. l2(x) = (x - (-1))(x - 0) / ((1 - (-1))(1 - 0)) = (x + 1)x / 2 = (x2 + x) / 2

Next, we evaluate each basis polynomial at x = 0.5:

l_0(0.5) = (0.25 - 0.5) / 2 = -0.25 / 2 = -0.125 l_1(0.5) = 1 - 0.25 = 0.75 l_2(0.5) = (0.25 + 0.5) / 2 = 0.75 / 2 = 0.375

Finally, we sum the absolute values to find the Lebesgue function value Λ(0.5):

Λ(0.5) = |-0.125| + |0.75| + |0.375| Λ(0.5) = 0.125 + 0.75 + 0.375 = 1.25

If we apply a data error of 0.1 configured to maximize the error at this probe, the output error bound is exactly0.1 * 1.25 = 0.125. This demonstrates how interpolation can amplify small input errors by a factor of 1.25 at this specific location.

Experiment

You can explore this phenomenon interactively by adjusting the nodes and the type of perturbation in the module:

1. Open the interactive module at Lebesgue Interpolation Amplification.
2. Select Four equally spaced for the Nodes control.
3. Set the Probe x to 0.5.
4. Choose a Data error of 0.1.
5. 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.
6. Change the Nodes to Four Chebyshev Lobatto and note how the Lebesgue function flattens out, reducing the maximum possible amplification.

Limitations

This module calculates the exact pointwise Lebesgue amplification at a selected probe point, rather than finding the global maximum (the true Lebesgue constant) across the entire continuous interval. The visualization operates on a fixed set of polynomial bases (equally spaced and Chebyshev Lobatto nodes) and bounds the domain to [-1, 1] for clarity. Additionally, when the baseline data error is exactly zero, the amplification ratio becomes mathematically undefined (0/0), which is correctly indicated as "Unavailable" in the interface.

References

For a deeper dive into interpolation stability, Lebesgue constants, and the advantages of Chebyshev nodes, refer to the Chebfun project's example on the Lebesgue constant:Chebfun Examples: The Lebesgue constant.