Knowlpedia

Definition

Flat logarithmic interval weight

A smooth weight on a positive interval that decays faster than every power of the logarithmic boundary margin.

For 0<a<b0<a<b and ca,cb>0c_a,c_b>0, define the flat logarithmic interval weight by

ζ(X)=exp⁡(−calog⁡2(X/a)−cblog⁡2(b/X))(a<X<b),\zeta(X)=\exp\left(-\frac{c_a}{\log^2(X/a)}-\frac{c_b}{\log^2(b/X)}\right) \quad(a<X<b),

and set it to zero outside (a,b)(a,b). This function is smooth and at both endpoints, and 0<ζ≤10<\zeta\le1 in the interval.

Boundary control

Put δ(X)=min⁡{1,log⁡(X/a),log⁡(b/X)}\delta(X)=\min\{1,\log(X/a),\log(b/X)\} on (a,b)(a,b). For every θ>0\theta>0 and finite N≥0N\ge0,

ζ(X)θδ(X)−N⟶0as X↓a or X↑b.\zeta(X)^\theta\delta(X)^{-N}\longrightarrow0 \quad\text{as }X\downarrow a\text{ or }X\uparrow b.

Near each endpoint the corresponding logarithm is comparable to distance from that endpoint, and an absorbs every polynomial loss. Differentiating ζ\zeta produces that same exponential times finite powers of reciprocal logarithms and smooth factors, proving smooth zero extension.

The minimum defining δ\delta is a pointwise weight and need not be differentiable where its branches meet. Derivative estimates involving δ\delta do not authorize differentiating it.