A Note on the Two‐Dimensional Moment Problem

Abstract

Let 9, denote the *-algebra of all polynomials in n real variables with complex coefficients (p* = p ) . A linear functional f on 9, is called positive if f(p*p) 2 0 for all p E 9,. A classical result of the theory of moment problem is that a linear functional f on 9, is represented by a positive BomL-measure p on Rn, in the sense that f ( p ) = / p ( z , , x2, .. ., z,) dp for all p E 9,, if and only if f ( p ) 2 0 for all non-negative p E P,,. It is well-known that for n = 1 each positive linear functional is represented m(i, j ) = (i + j ) (i + j + 1)/2 + j + 1 for i + j 2 4 and m(0,O) = 1; m(1, 2) = 2; m(2, 1) = 3; m(1, 1) = 4 ; m(1, 0 ) = 5 ; m(0, 1) = 6; m(2,O) = 7 ; m(0, 2) = 8 ; m(3,O) = 9; m(0, 3) = 10; and a sequence (gJnZ1 by g1 = g2 = g, = 1, ga = 4 and gn = n!(n+l)' for n 2 5. Proposition. The functional f on 9,, given by ) = 0 and f (x?"ly'j+l) = f ( z 2 1 + l y 2 1 ) = f ( z " y 2 i + l f ( 3 y z j ) = Y,!,(,,~) for all nm-negative integers i, j , i s positive and f(q) = -1.Proof. We first define the matrix (ars)r,sll by

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