The Local Lorentz Coefficient Kernel

Abstract

Inverse central-moment corrections give endpoint-sensitive local coefficient bounds.

The central moment is the Bernstein average of (i-mX) raised to its order. Its inverse convolution defines tau(m,n), with tau(m,0)=1. Write C(D,J,P) for the normalized Bernstein coefficient transform and R(m,p) for max(sqrt(p(1-p)/m),1/m). The polynomial Q(m,r,f) uses the prescribed jet f(n,i), the correction tau(m,n)/m^n, and the degree-m Bernstein basis, summed over n from zero to r and i from zero to m.

Theorem 1.1 (Both ends of the correction array).

$$\forall n\in \mathbb{N},\exists K\in \mathbb{R},1\le K\wedge \forall m\in \mathbb{N},0<m\Rightarrow \forall h\in \mathbb{N},h\le n\Rightarrow \Vert C\left(n,h,\frac{tau\left(m,n\right)}{m^n}\right)\Vert \le K\cdot min\left(\frac{m^h}{m^n},\frac{m^{n-h}}{m^n}\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.tau_coefficient_bound (✓ 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.

For each correction order n, one constant controls every positive grid size m and every coefficient h from zero through n. Reflection supplies the symmetric minimum of the two endpoint bounds. This is Corollary 16 of Holtz, Nazarov and Peres in the normalized Bernstein basis; the proof here obtains the coefficient estimate from complex disk bounds for the actual moments.

Theorem 1.2 (Hypergeometric averaging of the correction).

$$\forall n\in \mathbb{N},\exists K\in \mathbb{R},1\le K\wedge \forall m\in \mathbb{N},\forall D\in \mathbb{N},\forall J\in \mathbb{N},0<m\wedge 0<D\wedge 2\cdot n\le D\wedge J\le D\Rightarrow \sum _{h=0}^{n}w\left(D,J,n,h\right)\cdot \Vert C\left(n,h,\frac{tau\left(m,n\right)}{m^n}\right)\Vert \le K\cdot R{\left(m,\frac{J}{D}\right)}^n$$

Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.weighted_tau_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.

The weight w(D,J,n,h) is choose(n,h) times the falling factorials (J)h and (D-J)(n-h), divided by (D)n. When 2n is at most D, weighting the actual correction array gives the nth power of the endpoint-adapted radius, with a constant depending only on n.

Theorem 1.3 (Only the effective local jets are needed).

$$\forall r\in \mathbb{N},\exists K\in \mathbb{R},1\le K\wedge \forall m\in \mathbb{N},0<m\wedge r\le m\Rightarrow \forall J\in \mathbb{N},J\le m+r\Rightarrow \forall A\in \mathbb{R},0\le A\Rightarrow \forall f\in \mathbb{N}\to \mathbb{N}\to \mathbb{C},(\forall n\in \mathbb{N},n\le r\Rightarrow \forall i\in \mathbb{N},i\le m\wedge i\le J\wedge J\le i+r\Rightarrow \Vert f\left(n,i\right)\Vert \le \frac{A}{R{\left(m,\frac{J}{m+r}\right)}^n})\Rightarrow \Vert C\left(m+r,J,Q\left(m,r,f\right)\right)\Vert \le K\cdot A$$

Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.local_coefficient_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.

One constant depending only on r controls all output coefficients, including J=0 and J=m+r. For each derivative order, only grid entries with i at most J and J at most i+r contribute. Arbitrary complex jets satisfying the displayed local bounds are allowed. This coefficient estimate refines the function-based estimates around Lemma 24 of Holtz, Nazarov and Peres. It requires r at most m; the polynomial remainder theorem uses reproduction outside its nonzero regime.

References

• Truth anchor: D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.local_coefficient_bound
• Truth anchor: D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.tau_coefficient_bound
• Truth anchor: D5/S3/Analytic/SeriesInequalities/LorentzLocalCoefficientKernel.weighted_tau_bound
• Dependency: D5/S3/Analytic/SeriesInequalities/LorentzLowDegreeCoefficientRemainder
• Dependency: D5/S3/Observer/BlockStructure/CommonDenominatorPolynomialBasis