mollifikátor

The term "mollifikátor" is a Hungarian word that translates to "mollifier" in English. In mathematics, particularly in the field of analysis, a mollifier is a smooth, compactly supported function that is used to smooth out other functions. Mollifiers are essential tools for proving theorems about the existence and regularity of solutions to differential equations and for constructing approximations of functions.

A mollifier, denoted by the Greek letter eta (η), is typically a non-negative, infinitely differentiable function with compact support. This means it is zero outside a certain bounded region. The most common form of a mollifier is constructed using the function e^(-1/(1-|x|^2)) for |x| < 1 and 0 for |x| >= 1, and then normalizing it appropriately.

The process of smoothing a function using a mollifier is called mollification or convolution with a mollifier. When a function f is convolved with a mollifier η, the resulting function f*η is a smoother version of f. This means that f*η will have more continuous derivatives than f, and if f itself is sufficiently regular, f*η will be infinitely differentiable.

Mollifiers are used in various contexts, including the theory of distributions, where they are used to define derivatives of distributions and to show that smooth functions are dense in certain function spaces. They are also crucial in the study of partial differential equations, particularly in establishing the existence of weak solutions and proving their regularity properties. The ability of mollifiers to approximate functions and smooth them makes them a fundamental concept in modern analysis.