Knowlpedia

Theorem

Inverse loss when two moment exponents coalesce

A two-row moment matrix from translated positive bumps has inverse size of reciprocal order in the exponent separation.

Statement

Fix a nonnegative nonzero compactly supported smooth bump β\beta on R\mathbb R, and centers c1<c2c_1<c_2. Pair the translated bumps β(y−cj)\beta(y-c_j) with weights esiye^{s_i y}, i=1,2i=1,2, where the distinct slopes lie in a fixed compact interval. The resulting matrix BB obeys

∥B−1∥=O(∣s2−s1∣−1)as s2−s1⟶0.\|B^{-1}\|=O(|s_2-s_1|^{-1}) \quad\text{as }s_2-s_1\longrightarrow0.

This quantifies loss of invertibility of the near equal weights.

Determinant computation

Set b(s)=∫estβ(t) dt>0b(s)=\int e^{st}\beta(t)\,dt>0 and Δ=c2−c1\Delta=c_2-c_1. Dividing row ii by the bounded positive factor esic1e^{s_i c_1} gives entries (b(si),esiΔb(si))(b(s_i),e^{s_i\Delta}b(s_i)). The determinant is

b(s1)b(s2)(es2Δ−es1Δ).b(s_1)b(s_2)(e^{s_2\Delta}-e^{s_1\Delta}).

The mean value theorem makes its absolute value comparable to ∣s2−s1∣|s_2-s_1|; the numerator entries in the inverse formula remain bounded. The logarithmic substitution r=eyr=e^y converts power-weight moment problems into this exponential-weight setting, with the Jacobian included in the profile.