An arXiv preprint develops Stieltjes-polynomial interpolation and sets out conditions for unique solutions in several versions of the problem. It also states that, under its conditions, a unique best uniform approximation exists and gives an alternating-error characterization of the optimum. The work is theoretical: it concerns abstract functions and polynomial spaces rather than an empirical participant or observational sample. Its evidence comes from Stieltjes-calculus concepts, operators, mathematical identities and theorem proofs.
Uniqueness depends on the nodes
Stieltjes calculus is the paper's underlying language. The analysis introduces Stieltjes-calculus concepts and operators, then derives interpolation and approximation results through mathematical identities and theorem proofs. It works with abstract mathematical functions and polynomial spaces.
For n + 1 g-distinct nodes in R+, the Lagrange interpolation problem has a unique solution. In plain language, the theorem guarantees one interpolating polynomial for node sets that meet that condition.
For a valid sequence of n + 1 nodes in R++, the convex Lagrange interpolation problem also has a unique solution. The conclusion is tied to the sequence being valid.
The generalized interpolation operator I_n is a projection onto P_n and is bounded on R([a,b],F) when the nodes lie in [a,b]++. In mathematical terms, that says the operator lands in the selected polynomial space and remains controlled on the stated function space.
The closest fit is defined by the worst error
For any uniformly g-continuous function on [a,b], the paper states that a unique Stieltjes polynomial in P_n attains the minimum supremum-norm error. Supremum norm means the largest pointwise error on the interval, so this result concerns the worst mismatch rather than an average.
Under the theorem's compactness and cardinality conditions, best approximation is equivalent to the maximum error being attained alternately at ordered points. That makes the alternating pattern a necessary-and-sufficient characterization within those assumptions.
The paper introduces Stieltjes divided differences through Newton interpolation and obtains an explicit Peano-kernel-type error formula using a Taylor formula. This provides a formal expression for interpolation error within the Stieltjes framework.
Hermite* interpolation has a separate solvability rule: its theorem gives a unique solution exactly under the stated adjacent-interval g-cardinality condition.
The convex Hermite* theorem gives unique solvability under the corresponding adjacent-interval conditions for ordinary and convex choices of the next node.
The hypotheses define the reach of the claims
Two supporting results set upper bounds on zeros and sign changes. A real g-polynomial of order n has at most n g-distinct roots, while a real g-polynomial of degree n can change sign at most n times on R+. These are limits stated by the framework, not a claim that every polynomial reaches them.
Taken together, the results describe a proof-driven mathematical study. The analysis introduces concepts and operators and derives its conclusions through identities and theorem proofs, using abstract functions and polynomial spaces rather than an empirical sample. The results therefore apply within the framework and its stated hypotheses.
Lagrange uniqueness is stated for n + 1 g-distinct nodes in R+; convex Lagrange uniqueness for a valid sequence of n + 1 nodes in R++; and Hermite* solvability for the relevant adjacent-interval condition. The best-approximation result is for uniformly g-continuous functions on [a,b], while its alternating-error characterization adds compactness and cardinality conditions.
The supplied document is an arXiv preprint, version 1, dated 25 Aug 2026. No funding statement is reported in the supplied text or metadata.
Paper data and sources
Original title: Stieltjes polynomial interpolation
Authors: Víctor Cora, Fernando Adrián Fernández Tojo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text