Generalized function — Safekipedia Discoverer

In mathematics, generalized functions are special tools that extend the idea of regular functions used with real or complex numbers. These tools help mathematicians and scientists work with functions that might not be smooth or continuous in the usual sense. One important theory behind generalized functions is called distributions.

Generalized functions are very useful because they let experts treat discontinuous functions — functions that jump or break — more like smooth functions that change gradually. They are also great for describing things in the real world that appear as single points, like point charges in physics.

These ideas are used a lot in physics and engineering, especially when solving complex problems involving partial differential equations and understanding patterns in nature through group representations. Many of these tools grew from early work on operational calculus and continue to develop through the work of mathematicians like Mikio Sato in the area of algebraic analysis.

Some early history

In the 1800s, ideas about generalized functions started to appear in different areas of math, like in the Green's function and Riemann's work on trigonometric series. These ideas were not all connected at the time.

Later, engineers used methods called operational calculus, which were based on the Laplace transform. These methods were useful but sometimes used ideas that were not fully proven in math. A famous book on this topic was written by Oliver Heaviside in 1899.

When the Lebesgue integral was developed, it gave a new way to think about functions. This helped lead to the idea of generalized functions. In the late 1920s and 1930s, scientists like Paul Dirac and Sergei Sobolev made big steps in this area. They worked on treating certain math objects, like measures, more like regular functions.

Schwartz distributions

The theory of distributions was developed by Laurent Schwartz, who worked with the idea of duality in special types of mathematical spaces called topological vector spaces. This theory is very useful in applied mathematics, though it has a limitation: distributions usually cannot be multiplied together. For instance, it doesn't make sense to square the Dirac delta function in this theory. This difficulty was shown by Schwartz in the 1950s and is a fundamental challenge in the field.

Algebras of generalized functions

Mathematicians have proposed different ways to solve problems with multiplying generalized functions. One method uses ideas from quantum mechanics, specifically a concept called the path integral formulation, to ensure that products of these functions behave correctly. Another method, suggested by researchers like Yu. M. Shirokov and E. Rosinger, involves creating rules for multiplying these functions by separating them into smooth and singular parts.

These approaches help mathematicians work with functions that have sudden jumps or sharp points, making them easier to study and apply in various problems. Some of these methods also connect to theories that allow numbers to be "infinitely large" or "infinitesimally small," similar to ideas used in advanced calculus.

F G = F s m o o t h G s m o o t h + F s m o o t h G s i n g u l a r + F s i n g u l a r G s m o o t h . {\displaystyle FG~=~F_{\rm {smooth}}~G_{\rm {smooth}}~+~F_{\rm {smooth}}~G_{\rm {singular}}~+F_{\rm {singular}}~G_{\rm {smooth}}.}

Other theories

Other ideas about generalized functions include the convolution quotient theory by Jan Mikusinski. This theory uses the field of fractions of convolution algebras that are integral domains. There are also theories of hyperfunctions. These started by looking at the edge values of analytic functions and now use sheaf theory.

Topological groups

Bruhat introduced a special kind of test functions, called Schwartz–Bruhat functions, for use on certain types of groups that are larger than usual function domains. These groups are important mainly in number theory, especially for studying adelic algebraic groups. André Weil used this idea to rewrite Tate's thesis, which helps describe the zeta distribution on the idele group and also applies to understanding L-functions.

Generalized section

A special way to expand mathematical ideas is through generalized sections of a smooth vector bundle. This follows a method started by Schwartz, creating objects that are opposites to test objects—smooth sections of a bundle that have compact support. The most complete theory is about De Rham currents, which are opposites to differential forms. These are related to De Rham cohomology and help in forming a very general version of Stokes' theorem.