Consider the space of smooth functions of the type f defined on $[0,R]$:
R is given
$0<=f(0)$ on $[0,R]$
$f(R)=0$
$f'(0)=0$
$df/dx<=0$ on $[0,R]$
Linear combinations of functions of the type f are again of type f.
Is there some sequence of polynomials p_n of type f such that linear combinations of p_n upto n=i can approximate any function of the type f arbtitrarily well by increasing i?
I am not looking for that sequence, but rather the first few polynomials of that sequence, upto i=3-7 perhaps
I am not sure if there is a definite first few terms, but I need some simple polynomials of type f that approximate most f to some extent
Polynomials can approximate some functions
In our study of mathematics, we’ve found that some functions are easier to work with than others. For instance, if you are doing calculus, typically polynomials are “easy” to work with because they are easy to differentiate and integrate. Other functions, like f(x)={sin(x)x1if x≠0if x=0 are more difficult to work with. However, there are polynomials that mimic the behavior of f near zero.