Approximationsidentität

Approximationsidentität, often translated as "approximation identity" or "approximate identity," is a concept in mathematical analysis. It refers to a sequence of functions that "approximates" the Dirac delta function. A Dirac delta function, denoted as $\delta(x)$, is a generalized function that is zero everywhere except at zero, where it is infinitely large, and its integral is equal to one. While not a function in the traditional sense, it is a fundamental tool in the study of distributions and differential equations.

A sequence of functions, typically denoted by $\{K_\epsilon\}_{\epsilon > 0}$ or $\{K_n\}_{n \in \mathbb{N}}$, is considered an approximation identity if it satisfies certain properties. These properties usually include:

1. Non-negativity: $K_\epsilon(x) \ge 0$ for all $x$ and $\epsilon$.

2. Integral normalization: The integral of $K_\epsilon(x)$ over its domain is equal to 1.

3. Concentrated peak: For any $\delta > 0$, the integral of $K_\epsilon(x)$ over any region excluding a neighborhood of zero tends to zero as $\epsilon$ approaches zero (or as $n$ approaches infinity). This means the mass of the function becomes increasingly concentrated around zero.

When a continuous function $f(x)$ is convolved with an approximation identity $K_\epsilon(x)$, the result, $f * K_\epsilon$, approaches $f(x)$ as $\epsilon$ tends to zero. This property makes approximation identities useful for smoothing functions and for understanding the behavior of convolution operations. They appear in various areas of mathematics, including Fourier analysis, partial differential equations, and functional analysis. Different forms of approximation identities exist, such as Gaussian functions, sinc functions, or sequences of positive kernels.