Moment pairing matrix

Given weights $w_1,\ldots,w_m$ and profiles $\beta_1,\ldots,\beta_n$ with integrable products, their moment pairing matrix is

B_{ij}=\int w_i(r)\beta_j(r)\,dr.

For a correction $\delta u=\sum_jc_j\beta_j$ , its weighted integrals form the matrix product $Bc$ .

Solving moment equations

For a square invertible matrix, every target vector $d$ has the unique coefficient vector $c=B^{-1}d$ . If the profiles or weights depend on parameters, uniform estimates require control of the corresponding inverses. This use of “moment matrix” is a weight-profile pairing; a Hankel matrix of moments of one measure is a different special construction.

Row normalization

Multiplying each equation by a specified nonzero factor amounts to replacing $Bc=d$ by $DBc=Dd$ , where $D$ is an invertible diagonal matrix. It leaves the solutions unchanged. The same factor must be applied to the target in that row, and a parameter-dependent factor contributes to derivative estimates.