Generalized function — Safekipedia Adventurer

Generalized functions

In mathematics, generalized functions are special tools that extend the idea of regular functions. These tools help mathematicians and scientists work with functions that might not be smooth or continuous.

One important theory behind generalized functions is called distributions.

Generalized functions are very useful. 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. They help solve 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 began 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 back then.

Later, engineers used methods called operational calculus, 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. He worked with the idea of duality in special types of mathematical spaces called topological vector spaces. This theory is very useful in applied mathematics.

One limitation of this theory is that distributions usually cannot be multiplied together. For example, it doesn't make sense to square the Dirac delta function using this theory. This difficulty was shown by Schwartz in the 1950s and remains a challenge in the field.

Algebras of generalized functions

Mathematicians have found different ways to work with special types of functions. One way uses ideas from quantum mechanics, like something called the path integral formulation, to help these functions work better together. Another way, created by scientists Yu. M. Shirokov and E. Rosinger, separates these functions into smooth parts and parts that change suddenly.

These methods make it easier for mathematicians to study functions that have sudden jumps or sharp points. Some of these ideas also relate to theories about very large or very small numbers, much like concepts 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 began by studying the edge values of analytic functions and now use sheaf theory.

Topological groups

Bruhat created a special kind of test function called Schwartz–Bruhat functions. These are used for certain types of groups that are bigger than normal function areas. These groups are important in number theory, especially for studying adelic algebraic groups. André Weil used this idea to change Tate's thesis. This helps explain the zeta distribution on the idele group and also helps us understand L-functions.

Generalized section

A special way to expand mathematical ideas is through generalized sections of a smooth vector bundle. This method was started by Schwartz. It creates 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.