Concrete solution to the nonsingular quartic binary moment problem

Abstract

Given real numbers β ≡ β ( 4 ) : β 00 \beta \equiv \beta ^{\left ( 4\right ) }\colon \beta _{00} , β 10 \beta _{10} , β 01 \beta _{01} , β 20 \beta _{20} , β 11 \beta _{11} , β 02 \beta _{02} , β 30 \beta _{30} , β 21 \beta _{21} , β 12 \beta _{12} , β 03 \beta _{03} , β 40 \beta _{40} , β 31 \beta _{31} , β 22 \beta _{22} , β 13 \beta _{13} , β 04 \beta _{04} , with β 00 > 0 \beta _{00} >0 , the quartic real moment problem for β \beta entails finding conditions for the existence of a positive Borel measure μ \mu , supported in R 2 \mathbb {R}^2 , such that β i j = ∫ s i t j d μ ( 0 ≤ i + j ≤ 4 ) \beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq 4) . Let M ( 2 ) \mathcal {M}(2) be the 6 × 6 6 \times 6 moment matrix for β ( 4 ) \beta ^{(4)} , given by M ( 2 ) i , j := β i + j \mathcal {M}(2)_{\mathbf {i},\,\mathbf {j}}:=\beta _{\mathbf {i}+\mathbf {j}} , where i , j ∈ Z + 2 \mathbf {i},\mathbf {j} \in \mathbb {Z}^2_+ and | i | , | j | ≤ 2 \left |\mathbf {i}\right |,\left |\mathbf {j}\right |\le 2 . In this note we find concrete representing measures for β ( 4 ) \beta ^{(4)} when M ( 2 ) \mathcal {M}(2) is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6 6 -atomic.