Theory of Generalized Functions

The theory of generalized functions, also known as distributions, extends traditional functions to include objects like the Dirac delta, which can represent idealized point sources or impulses. This framework allows mathematicians and scientists to handle situations involving sudden changes, singularities, or discontinuities more rigorously. Essentially, it broadens the concept of what counts as a function so that complex phenomena—such as sharp spikes or instantaneous events—can be analyzed mathematically with precision, enabling advanced work in fields like physics, engineering, and signal processing.