The Low-Degree Lorentz Coefficient Remainder
Abstract
The actual Lorentz remainder is uniformly small in the original Bernstein coefficient norm.
Write B(d,c) for the polynomial with original normalized degree-d Bernstein array c, and Q(m,r,P) for the Lorentz operator evaluated on the actual iterated derivatives of P at i/m. Its correction polynomials are the inverse convolution of the binomial central moments. Array norms are maximum coefficient moduli, and degree elevation is included in the polynomial identity.
Theorem 1.1 (Exact polynomial reproduction).
Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzCoefficientRemainder.lorentz_reproduction (✓ std3). ∎
Citation. Olga Holtz; Fedor Nazarov; Yuval Peres (2011). New Coins from Old, Smoothly. DOI: 10.1007/s00365-010-9108-5. URL: https://arxiv.org/abs/0808.1936v3.
Commentary.
Every complex polynomial of degree at most r is reproduced for each positive grid size m. Finite Taylor expansion and the defining convolution identity cancel the nonconstant centered terms. This is the complex-scalar form of Lemma 14 of Holtz, Nazarov and Peres.
Theorem 1.2 (A uniform bound for every original array).
Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzCoefficientRemainder.low_degree_coefficient_remainder (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Olga Holtz; Fedor Nazarov; Yuval Peres (2011). New Coins from Old, Smoothly. DOI: 10.1007/s00365-010-9108-5. URL: https://arxiv.org/abs/0808.1936v3.
Commentary.
For every natural r, one constant K at least one works simultaneously for all m at least d at least one and every original complex coefficient array c. The remainder is represented at degree m+r by one array a, and the estimate holds for its full maximum norm. If d is at most r, this entire array is zero. No derivative hypothesis, selected input array, or quadratic separation of m from d is imposed.
The proof combines a local Taylor remainder with the effective Lorentz coefficient kernel. The large-ratio regime uses the centered Taylor coefficient estimate; a stability bound handles the complementary regime. Bernstein reconstruction supplies the same coefficient array for all output indices. This establishes the polynomial coefficient remainder and does not assert later smooth-function continuation or probabilistic factory constructions.