A moment problem for pseudo-positive definite functionals (Q2380814)

The authors study a modified moment problem for a class of signed measures, in contrast to the classical moment problem. Their results are based on the Gauss representation of a polynomial: let \(\mathcal{H}_{k}(\mathbb{R}^d)\) be the set of harmonic homogeneous complex-valued polynomials of degree \(k\) in \(d\) variables. We put \(a_{k}:=\text{dim}(\mathcal{H}_{k}(\mathbb{R}^{d}))\). Let \((Y_{k,l}:\mathbb{R}^{d}\to\mathcal{R})_{l=1\dots{}a_{k}}\) be an orthogonal basis of \(\mathcal{H}_{k}(\mathbb{R}^d)\) with respect to the scalar product \(\left\langle f,g\right\rangle_{\mathbb{S}^{d-1}}:=\int_{\mathbb{S}^{d-1}}f(\theta)\overline{g(\theta)}\,d\theta\), where \(\mathbb{S}^{d-1}:=\{x\in\mathbb{R}^{d}:|x|=1\}\) and when considering \(x\in\mathbb{R}^d\) in spherical coordinates \(x=r\theta\), \(\theta\in \mathbb{S}^{d-1}\). The set \(\{|x|^{2j}Y_{k,l}(x):j\geq 0\), \(k\geq 0\), \(l=1,\dots,a_{k}\}\) turns out to be a basis of the space of polynomials, so one can study the moment problem following this approach instead of the classical one. From this point of view, a signed measure \(\mu\) over \(\mathbb{R}^{d}\) is said to be pseudo-positive if \[ \int_{\mathbb{R}^d}h(|x|)Y_{k,l}(x)\,d\mu(x)\geq 0, \quad k\geq 0,\quad l=1,\dots,a_{k}, \] for every \(h:[0,\infty)\to[0,\infty)\) non-negative function with compact support. A functional \(T:\mathbb{C}[x_{1},\dots,x_{d}]\to\mathbb{C}\) is said to be pseudo-positive definite with respect to the orthonormal basis \(Y_{k,l}\), \(k\geq 0, l=1,\dots,a_{k}\), if \[ T_{k,l}(p^{\star}(t)p(t))\geq0,\quad T_{k,l}(tp^{\star}(t)p(t))\geq0,\quad k\geq0,\quad l=1,\dots,a_{k}, \] for every \(p(t)\in\mathbb{C}[x_{1}]\), where the functional \(T_{k,l}:\mathbb{C}[x_{1}]\to\mathbb{C}\) is given by \(T_{k,l}(p):=T(p(|x|^{2})Y_{k,l}(x))\). The authors first settle the existence of non-negative measures \(\sigma_{k,l}\) with support in \([0,\infty)\) in such a way that a pseudo-positive definite \(T\) can be written in the following manner: \[ T(f)=\sum_{k=0}^{\text{deg}(f)}\sum_{l=1}^{a_{k}}\int_{0}^{\infty}f_{k,l}(r)r^{-k}d\sigma_{k,l}(r),\quad f\in\mathbb{C}[x_{1},\dots,x_{d}], \] where \(f_{k,l}(r)=\int_{\mathbb{S}^{d-1}}f(r\theta)Y_{k,l}(\theta)\,d\theta\). The first main result states that, for a given pseudo-positive definite functional \(T:\mathbb{C}[x_{1},\dots{},x_{d}]\to \mathbb{C}\) written as above, if one has \[ \sum_{k=0}^{\infty}\sum_{l=1}^{a_{k}}\int_{0}^{\infty}r^{N}r^{-k}\,d\sigma_{k,l}<\infty, \] then there exists a pseudo-positive, signed moment measure \(\sigma\) such that \(T(f)=\int_{\mathbb{R}^{d}}fd\sigma\), \(f\in\mathbb{C}[x_{1},\dots,x_{d}]\). The point of view followed in the paper is adequate in order to define in a natural way the truncated moment problem in the class of pseudo-positive definite functionals, which is solved in Section~3. The second main result determines that, if one departs from functionals \(T_{k,l}\), defined as before, in such a way that there exists one and only one measure \(\mu_{k,l}\) in the set of non-negative moment measures on \(\mathbb{R}^{d}\) such that \[ \int_{0}^{\infty}r^{m}d\mu_{k,l}=T_{k,l}(r^m), \] for every \(m\geq0\) (it is to say, there is only one \(\mu_{k,l}\) representing \(T_{k,l}\) on \([0,\infty)\) in the sense of Stieltjes), then the pseudo-positive definite functional \(T:\mathbb{C}[x_{1},\dots,x_{d}]\to\mathbb{C}\) is unique in the class of pseudo-positive signed measures. Finally, several examples and properties of pseudo-positive definite functionals are considered.