Barycentric interpolation and Lebesgue constants

For distinct real nodes $x_0,\ldots,x_n$, define weights as reciprocals of the products of all node separations from each node. Let the nodal polynomial be the product of x minus every node. Each Lagrange basis polynomial equals that nodal polynomial times its corresponding weight divided by x minus the node. Since the Lagrange basis sums to one, cancel the common nodal product to obtain the second barycentric formula. Away from nodes, its quotient represents exactly the unique interpolating polynomial in exact arithmetic.

Multiplying all weights by the same nonzero constant leaves the quotient unchanged, allowing convenient scaling. If the evaluation point is exactly a node, return its stored value instead of dividing by zero. Near a node, large numerator and denominator terms are not by themselves evidence of a singular interpolant: their quotient has a removable representation issue, not a polynomial pole. Finite-precision behavior still depends on the nodes, accurate weights, arithmetic, and evaluation region; stable implementations need specific error analysis rather than an unconditional slogan.

Write the interpolant in its Lagrange basis and perturb each supplied value by at most delta. The absolute output perturbation at x is bounded by delta times the sum of the absolute basis values there. That sum is the Lebesgue function, and its maximum over the interval is the Lebesgue constant. At a chosen point the bound is attained by assigning perturbation signs to agree with the basis signs, so it measures actual worst-case sensitivity rather than just a loose computational convenience.

For a fixed distinct node set on a compact interval, this constant is finite. Its growth with degree is the concern: good Chebyshev grids have logarithmic growth, while equally spaced polynomial grids can become highly sensitive. No exact evaluation identity can eliminate that amplification. A numerically stable algorithm aims to avoid adding much error beyond what the problem's sensitivity already permits. Relative accuracy near an interpolant's zero is a separate issue because dividing by a tiny true value changes the error metric.

Interpolation reproduces every polynomial in its degree space. For any q in that space, write f−I_nf as f−q minus the interpolation of f−q. The triangle inequality gives a factor one plus the interpolation operator norm, hence uniform interpolation error at most $(1+\Lambda_n)E_n(f)$. This links two different ingredients: how well the target can be approximated at all, and how the chosen sampling operator compares with that best possibility.

A small Lebesgue constant cannot help if the best degree-n error itself is large, and a tiny best error can be spoiled by enormous sensitivity. Real measurements add another term for noisy data, while floating-point evaluation adds an algorithmic term. Keeping those contributions separate helps decide whether to collect better nodes, increase degree, smooth noisy measurements, or change the implementation. Barycentric evaluation addresses representation and arithmetic; it does not turn noisy interpolation into a statistical denoising method.