Hamburger moment problem: Parseval equality and \(A\)-singularity
To solve the Nehari problem, one has to find all \(L^\infty\) functions on the unit circle, bounded in modulus by 1, whose Fourier coefficients with negative index are prescribed. All the solutions are of the form \(w=(a/\overline{a})(\omega-\overline{b})/(1-\omega b)\) with \(\omega\) an arbitrary Schur function and \((a,b)\) a pair of functions defined by the data. The pair \((a,b)\) is called a Nehari pair. A Nehari pair has the properties: (1) \(b\) is a Schur function and \(b(0)=0\), (2) \(\log(1-| b| ^2)\in L^1\), \(| a| ^2=1-| b| ^2\), and \(a\) is an outer function of the Schur class, (3) \(b\) has the (ac) property, that means that the Riesz-Herglotz measure associated with the Schur function \(b\omega\) is absolutely continuous for all Schur functions \(\omega\). In this paper it is proved that properties (1)--(3) are necessary but not sufficient for \((a,b)\) to be a Nehari pair. More precisely, it is proved that if \((a,b)\) parametrizes the solutions of an infinite indeterminate Hamburger moment problem, then it satisfies (1)-(3), but it is not a Nehari pair because it is Arov singular. To prove this, the machinary of the abstract interpolation problem is used, which lifts many classical complex interpolation and moment problems to an abstract Hilbert space setting. The Parseval identity from the title refers to a general Parseval identity in this setting which is used to prove the (ac) property of the pair \((a,b)\) when related to the Hamburger moment problem.
scientific article; zbMATH DE number 950735