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Unit nodal data exhibit equispaced cardinal basis growth

Example

For each $m \geq 6$ , use the $2 m + 1$ equispaced nodes and the point $x_{*}$ from Equispaced interpolation has linearly growing basis amplification. Prescribe data $y_{m} = 1$ and $y_{k} = 0$ for $k \neq m$ . Then the data have maximum magnitude $1$ , while their interpolant has magnitude at least $9 m / 1280$ at $x_{*}$ . This is a degree-dependent data witness for growth of the interpolation operator. Polynomials remain uniformly dense in $C ([- 1, 1])$ by the Weierstrass theorem, but that density theorem does not claim convergence of this particular interpolation rule for any one fixed function; the data here vary with $m$ .

Facts & Assumptions

Given: An integer $m \geq 6$ , the equispaced nodes $x_{k} = - 1 + k / m$ , the query $x_{*} = - 1 + 1 / (2 m)$ , and nodal data supported at index $m$ .

[F1]

Cardinal-polynomial interpolation: $I_{2 m} y = \sum_{k = 0}^{2 m} y_{k} ℓ_{k}$ .

[F2]

Equispaced interpolation has linearly growing basis amplification: for these nodes, $∣ ℓ_{m} (x_{*}) ∣ \geq 9 m / 1280$ .

[F3]

Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval: every continuous real function on a closed interval is a uniform limit of polynomials.

Proof

1.1givenalgebraconstruct

Set $y_{m} = 1$ and $y_{k} = 0$ for $k \neq m$ . Then $\max_{0 \leq k \leq 2 m} ∣ y_{k} ∣ = 1$ .

1.2givenF1F2algebra

By [F1], $I_{2 m} y = ℓ_{m}$ . Since $m \geq 6$ , the query $x_{*} = - 1 + 1 / (2 m)$ lies in $[- 1, 1]$ ; [F2] therefore gives $∣ I_{2 m} y (x_{*}) ∣ = ∣ ℓ_{m} (x_{*}) ∣ \geq 9 m / 1280.$ The input vector depends on $m$ , so this calculation is an operator-amplification witness and does not by itself establish divergence for one fixed function.

2.1givenF3step 1.1step 1.2algebra∎

The density assertion [F3] states that for each fixed continuous function some sequence of polynomials converges uniformly to it. It makes no assertion that the polynomials selected by these changing nodal data converge for any fixed function; hence it is consistent with the finite-dimensional amplification calculation above.

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Sources

Mark Embree, Lecture Notes on Numerical Analysis, uniformly spaced nodes example (standard reference, not scraped)
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation (standard reference, not scraped)