In mathematics, the term discretization stands for the transition from abstract, continuous, often infinite-dimensional objects to concrete, discrete, finite-dimensional counterparts. We define discretizations as tuples encompassing all necessary aspects involved in this transition. Let be an arbitrary set, be the set of -tuples where each component lies in . We define two mappings
The following abstract diagram visualizes a discretization:
TODO: write up in more detail
Let be the space of real-valued continuous functions on the interval , and let be ordered sampling points in .
We define the grid collocation operator as
The abstract object in this case is the input function , and the operator evaluates this function at the given points, resulting in a vector in .
This operator is implemented as
Let discrete values be given. Consider the linear interpolation of those values at a point :
where is the index such that .
Then we can define the linear interpolation operator as
where stands for the function .
Hence, this operator maps the finite array to the abstract interpolating function .
This interpolation scheme is implemented in the