# A Weaker Regularity Condition for the Multidimensional $\nu$-Moment Problem

## TL;DR

This paper weakens the regularity condition for solving the multidimensional $
u$-moment problem, enabling approximate solutions with smaller $
u$ values, thus broadening the applicability of spectral density reconstruction.

## Contribution

The paper demonstrates that the regularity condition for the $
u$-moment problem can be relaxed from $
u \\geq d/2+1$ to $
u \\geq d/2$, allowing for more flexible spectral density solutions.

## Key findings

- Regularity condition can be weakened to \\nu \\geq d/2.
- Broader class of spectral densities can be approximated.
- Potential for improved spectral density estimation methods.

## Abstract

We consider the problem of finding a $d$-dimensional spectral density through a moment problem which is characterized by an integer parameter $\nu$. Previous results showed that there exists an approximate solution under the regularity condition $\nu\geq d/2+1$. To realize the process corresponding to such a spectral density, one would take $\nu$ as small as possible. In this letter we show that this condition can be weaken as $\nu\geq d/2$.