Ultrasound systems are cheap, portable, and fast, which have become impressively popular over the last decades. State-of art imaging is however known to be sub-optimal. Most attempts to improve it formulate the problem on a discrete spatial grid and suffer from computational limitations as they rely on iterative optimization. For far-field applications, an alternative approach formulated the inverse problem as a sampling operator acting on an object in an infinite dimensional Hilbert space, and proposed the corresponding least-squares optimal estimate described as a continuous object. This methodology cannot be directly applied to ultrasonic imaging, whose images contain depth information. This thesis develops an appropriate mathematical extension of the functional inverse problem to acquisition of time-dependent signals. The theory allows for the signals to be interpreted as either discrete or functional data. The proposed extension formulates the inverse problem between acquisition and imaging Hilbert spaces. We prove that the current state-of-the-art can be seen as a suboptimal approximation of this right pseudo-inverse in the general Hilbert space formulation. Processing the data as a function of time, rather than as discrete-time samples makes the continuous image estimator more efficient, stable and accurate. For instance, in SAFT imaging, we demonstate that the depth resolution achieved can be 60% better when compared to the DAS image, and the contrast 30% better. In addition, the proposed framework encompasses existing techniques, mathematically captures where they fail, and brings a unifying method comparison through a analytic description of the point-spread-function. Additionally, the representation of the image through an expansion over data-independent basis functions is compact, and that can be exploited for efficient machine learning diagnosis support.