First, some definitions:
The least squares approximation of a function f is a function such as:
, for every .
Now the problem:
Let's consider a class of approximations with the following properties:
- all functions are defined on a symmetric interval ;
- if , then ;
Consider , where is an even function.
Prove that if f is an even function on , its least squares approximation is even.
My attempt of a solution started by trying to prove that the approximation is a linear combination of even functions. So we consider an orthogonal basis for the given class of approximations. In this case, the least squares approximation is:
(u,v) is the dot product of functions u and v. More precisely:
Using this formula and the hypothesis, it can be proven that if is an odd function, then , because is the integral of an odd function on a symmetric interval.
This way, we eliminate from the approximation all the odd basis functions. So now is a linear combinations of even functions and functions that are neither even, nor odd. The even functions are good, because a linear combination of even functions is also an even function. But what happens when the basis function is neither even, nor odd? I tried decomposing in combinations of even and odd functions, but this approach does not seem to lead anywhere...
Any help would be appreciated.