Let f:
Rn -> R a
function from a given
set of functions M.
Let K be another given set of functions. Let E: M x K ->
R be an
error function.
The
approximation problem is defined by:
Which function g out of K
minimizes E(f,.) : M ->
R ?
Sometimes it is demanded that g coincides with f on a certain finite set of points { xi} in Rn instead of an error function, but this just the case of a special error function, E(f,g) = sum(i=0,n) abs(f(xi) - g(xi)), and solutions with E(f,g)>0 are rejected.
If there exists a solution and if it's unique depends on the sets M, K and the error function.
Often not only the existance of an (unique) solution is considered but also approximations by different sets are compared. For this purpose, additional error functions are used to define the quality of an approximation.