Existence of smooth solutions to the classical moment problems

Abstract

Let s ( 0 ) , s ( 1 ) , … s(0),s(1), \ldots be a given sequence, and define s ( n ) = s ( − n ) ¯ s(n) = \overline {s( - n)} for n > 0 n > 0 . If Σ Σ ξ ¯ n ξ m s ( m − n ) ≥ 0 \Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0 holds for all finite sequences ( ξ n ) n ∈ Z {({\xi _n})_{n \in \mathbb {Z}}} , then it is known that there is a positive Borel measure μ \mu on the circle T \mathbb {T} such that s ( n ) = ∫ − π π e i n t d μ ( t ) s(n) = \smallint _{ - \pi }^\pi {{e^{int}}d\mu (t)} , and conversely. Our main theorem provides a necessary and sufficient condition on the sequence ( s ( n ) ) (s(n)) that the measure μ \mu may be chosen to be smooth. A measure μ \mu is said to be smooth if it has the same spectral type as the operator i d / d t id/dt acting on L 2 ( T ) {L^2}(\mathbb {T}) with respect to Haar measure d t dt on T \mathbb {T} : Equivalently, μ \mu is a superposition (possibly infinite) of measures of the form | w ( t ) | 2 d t |w(t){|^2}dt with w ∈ L 2 ( T ) w \in {L^2}(\mathbb {T}) such that d w / d t ∈ L 2 ( T ) dw/dt \in {L^2}(\mathbb {T}) . The condition is stated purely in terms of the initially given sequence ( s ( n ) ) (s(n)) : We show that a smooth representation exists if and only if, for some ε ∈ R + \varepsilon \in {\mathbb {R}_ + } , the a priori estimate \[ ∑ ∑ s ( m − n ) ξ ¯ n ξ m ≥ ε | ∑ n s ( n ) ξ n | 2 \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left | {\sum {ns(n){\xi _n}} } \right |}^2}} } \] is valid for all finite double sequences ( ξ n ) ({\xi _n}) . An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.