Bernstein Coefficient Tails of Taylor Polynomials
Abstract
Original Bernstein arrays control the coefficient tails of centered Taylor truncations.
Write B(d,c) for the polynomial with normalized Bernstein coefficients c at degree d, T(P,p,r) for its Taylor polynomial through order r at p, and C(D,J,P) for the degree-D Bernstein coefficient transform. The norm of a finite array is its maximum coefficient modulus. These conventions include both endpoints.
Theorem 1.1 (A uniform coefficient-tail estimate).
Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzLowDegreeCoefficientRemainder.taylor_coefficient_remainder_bound (✓ 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 truncation order r, two constants A and K, both at least one, work for every input degree d and output degree D satisfying A d at most D. A complex disk bound obtained from the original coefficients is combined with centered finite-population product moments. Summation of the resulting geometric tail gives exponent (r+1)/2. No derivative bound on the input is assumed.
Theorem 1.2 (An actual Bernstein representation).
Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzLowDegreeCoefficientRemainder.taylor_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.
At each chosen output index J, the Taylor center is J/D. The corresponding remainder has an actual degree-D coefficient array, whose Jth entry satisfies the uniform estimate. The entire array vanishes when d is at most r. The array can depend on J because its Taylor center does. This statement concerns ordinary Taylor truncation in the large-ratio regime; the Lorentz operator remainder is treated separately.