Local potentials are routinely represented as matrices in finite basis sets. The inverse problem of recovering a real-space potential from its matrix is widely considered ill-posed. We show that this inverse problem admits a clean and robust solution when products of basis functions are treated as an overcomplete spanning set and their linear dependencies are properly exploited. The resulting method converges to the exact real-space potential in the basis-set limit and efficiently handles very large basis sets on standard desktop hardware. The findings provide a rigorous foundation for potential reconstruction and have direct implications for a wide range of electronic-structure applications involving local operators.