Gaussian Weighting of the Cosine-Integral Gram Product

Abstract

Gaussian weighting approximates the cosine-integral Gram integral with an explicit inverse-radius error.

Theorem 1.1 (An error bound for all positive scales and radii).

$$\forall a\in \mathbb{R},b\in \mathbb{R},beta\in \mathbb{R},R\in \mathbb{R},\left(\left(0<a\wedge 0<b\right)\wedge \left(0<beta\wedge 0<R\right)\right)\Rightarrow \left(\mathrm{Integrable}\left((z:\mathbb{R}\mapsto \mathrm{Ci}\left(a\cdot \left|z\right|\right)\cdot \mathrm{Ci}\left(b\cdot \left|z\right|\right)\cdot \exp \left(-beta\cdot {\frac{z}{R}}^2\right))\right)\wedge \left|{\int }_{\mathbb{R}}\mathrm{Ci}\left(a\cdot \left|z\right|\right)\cdot \mathrm{Ci}\left(b\cdot \left|z\right|\right)\cdot \exp \left(-beta\cdot {\frac{z}{R}}^2\right)dz-\frac{\pi }{\max \left(a,b\right)}\right|\le \frac{8\cdot beta+1}{a\cdot b\cdot R}\right)$$

Proof. Machine-checked in Lean as D5/S3/Fourier/Asymptotics/CosineGaussianGramRate.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let a, b, beta and R be arbitrary positive real numbers. The product Ci(a|z|)Ci(b|z|) multiplied by exp(-beta(z/R)^2) is Lebesgue integrable on the real line. Its integral differs from pi/max(a,b) by at most 8(beta+1)/(abR). Here Ci is the real cosine integral defined at positive x by sin(x)/x minus the integral of sin(t)/t^2 from x to infinity.D5/S3/Fourier/Asymptotics/CosineIntegralLattice.cosineIntegral

The unweighted product is integrable and has integral pi/max(a,b). Since the Gaussian factor is continuous and lies between zero and one, multiplication by it preserves integrability.D5/S3/Fourier/Asymptotics/CosineIntegralGram.result

For every x>0, the sine-tail representation gives |Ci(x)|<=2/x: the first term has absolute value at most 1/x, and the integral of t^(-2) from x to infinity is 1/x. Consequently the absolute value of the product at z different from zero is at most 4/(abz^2).

For u>=0 the Gaussian decrement satisfies 0<=1-exp(-u)<=min(u,1). On the interval |z|<=R, the absolute error integrand is therefore at most 4beta/(abR^2), except at the null singleton zero. Outside that interval it is at most 4/(abz^2). The two integrated bounds are 8beta/(abR) and 8/(abR), respectively, yielding the stated estimate.

If a and b range over compact subsets of the positive real axis and beta remains bounded above, the displayed constant is uniform. Thus the approximation error is uniformly O(1/R). This statement concerns deterministic Lebesgue integrals.

References

• Truth anchor: D5/S3/Fourier/Asymptotics/CosineGaussianGramRate.result
• Truth anchor: D5/S3/Fourier/Asymptotics/CosineIntegralGram.result
• Truth anchor: D5/S3/Fourier/Asymptotics/CosineIntegralLattice.cosineIntegral
• Dependency: D5/S3/Fourier/Asymptotics/CosineIntegralGram