Approach function

Example of an approach function ${\displaystyle \varphi }$

Approach functions are functions that are used in mathematics to approximate a function space by a function space of lower dimension. They are used in particular in numerical mathematics , where they are used for the approximate representation of functions.

Hat function

These have the property of having the function value one at exactly one point and the function value zero at all other discrete points. They play an important role in finite element methods .

In its simplest form, such a function runs linearly between the points, and the resulting form corresponds to an isosceles triangle. The definition for the example shown above is as follows:

${\displaystyle \varphi (x)=\max(1-\left|x\right|,0).}$

Where represents the maximum function and the absolute value function . ${\displaystyle \max(\cdot ,\cdot )}$ ${\displaystyle \left|\cdot \right|}$

literature

• H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13: 1-123, 2004. PDF