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**Anomander** First, some **definitions**:

The least squares approximation of a function f is a function $\displaystyle \phi\epsilon\theta_n$ such as:

$\displaystyle ||f-\phi||\le||f-\phi_n||$, for every $\displaystyle \phi_n\epsilon\theta_n$.

Also: $\displaystyle ||u||=(\int_R|u(t)|^2d\lambda t)^{1/2}$

Now the **problem**:

Let's consider $\displaystyle \theta_n$ a class of approximations with the following properties:

- all functions $\displaystyle \phi \in \theta_n$ are defined on a symmetric interval $\displaystyle [-a, a]$;

- if $\displaystyle \phi(t)\in\theta_n$, then $\displaystyle \phi(-t)\in\theta_n$;

Consider $\displaystyle d\lambda t=\omega(t)dt$, where $\displaystyle \omega$ is an even function.

Prove that if f is an even function on $\displaystyle [-a, a]$, its least squares approximation $\displaystyle \phi\epsilon\theta_n$ 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 $\displaystyle \lbrace\pi_j\rbrace_{j=1}^n$ for the given class of approximations. In this case, the least squares approximation is:

$\displaystyle \phi=\sum_{j=1}^nc_j*\pi_j$, where: $\displaystyle c_j=(\pi_j,f)/(\pi_j,\pi_j)$

(u,v) is the dot product of functions u and v. More precisely:

$\displaystyle (u,v)=\int_{-a}^a u(t)v(t)\omega(t)dt$

Using this formula and the hypothesis, it can be proven that if $\displaystyle \pi_j$ is an odd function, then $\displaystyle c_j=0$, because $\displaystyle (\pi_j,f)$ is the integral of an odd function on a symmetric interval.

This way, we eliminate from the approximation $\displaystyle \phi$ all the odd basis functions. So now $\displaystyle \phi$ 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 $\displaystyle \pi_j$ is neither even, nor odd? I tried decomposing $\displaystyle \pi_j$ in combinations of even and odd functions, but this approach does not seem to lead anywhere...

Any help would be appreciated.