Delay-And-Sum (DAS), the state-of-art in ultrasound imaging, is known to be sub-optimal, resulting in low resolution and contrast. Most proposed improvements involve ad-hoc re-weighting, or hit computational bottlenecks given real-time requirements. This paper takes a fresh perspective on the problem, leveraging a functional framework to obtain a regularized least-squares estimate of the tissue reflectivity function. An explicit solution is derived, which for specific cases - can be efficiently implemented, making it suitable for real-time imaging. In our formulation, DAS appears as a back-projection without any optimal properties. We illustrate the framework through first a one-dimensional set-up, and then a two-dimensional extension with Synthetic Aperture Focusing Technique (SAFT). The one-dimensional simulations show a 77% resolution improvement with respect to DAS, which artificially limits the available bandwidth. On a standard performance-assessment phantom, simulations show that SAFT depth resolution can be improved by 71%.