Existence of smooth solutions to the classical moment problems

Let $s(0),s(1), \ldots$ be a given sequence, and define $s(n) = \overline {s( - n)}$ for $n < 0$. If $\Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0$ holds for all finite sequences ${({\xi _n})_{n \in \mathbb{Z}}}$, then it is known that there is a positive Borel measure $\mu$ on the circle $\mathbb{T}$ such that $s(n) = \smallint_{ - \pi }^\pi {{e^{int}}d\mu (t)}$, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence $(s(n))$ that the measure $\mu$ may be chosen to be smooth. A measure $\mu$ is said to be smooth if it has the same spectral type as the operator $id/dt$ acting on ${L^2}(\mathbb{T})$ with respect to Haar measure $dt$ on $\mathbb{T}$: Equivalently, $\mu$ is a superposition (possibly infinite) of measures of the form $\vert w(t){\vert^2}dt$ with $w \in {L^2}(\mathbb{T})$ such that $dw/dt \in {L^2}(\mathbb{T})$. The condition is stated purely in terms of the initially given sequence $(s(n))$: We show that a smooth representation exists if and only if, for some $\varepsilon \in {\mathbb{R}_ + }$, the a priori estimate

$$\sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left\vert {\sum {ns(n){\xi _n}} } \right\vert}^2}} }$$

is valid for all finite double sequences $({\xi _n})$. 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.