First, some

**definitions**:

The least squares approximation of a function f is a function

such as:

, for every

.

Also:

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:

, where:

(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 function is also a 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.