This thesis investigates the practicality and utility of mathematical models to represent continuous and occasionally unavailable data stream attributes, and processing relational-style queries in a stream processing engine directly on these models. We present Pulse, a framework for processing continuous queries over stream attributes modeled as piecewise polynomial functions. We use piece-wise polynomials to provide a compact, approximate representation of the input dataset and provide query language extensions for users to specify precision bounds to control this approximation. Pulse represents queries as simultaneous equation systems for a variety of relational operators including filters, joins and standard aggregates. In the stream context, we continually solve these equation systems as new data arrives into the system. We have implemented Pulse on top of the Borealis stream processing engine and evaluated it on two real-world datasets from financial and moving object applications. Pulse is able to achieve significant performance improvements by processing queries directly on the mathematical representation of these polynomials, in comparison to standard tuple-based stream processing, thereby demonstrating the viability of our system in the face of having to meet precision requirements.