How to find a basis for the space of even functions (with some constraints)

[mnb96]

Hello,
I am considering the set of all (differentiable) even functions with the following properties:

1) $$f(x)=f(-x)$$
2) $$f(0)=a_0$$, with $a_0\in \mathbb{R}$
3) $$f(n)=0$$, where $n\in \mathbb{Z}-\{0\}$

One example of such a function is the sinc function $sin(\pi x) / \pi x$.
Is it possible to find some basis-functions that completely define this set of functions?

If so, any hint on how to find a basis?


Thanks!

 

[Tinyboss]

What kind of basis? In the strongest sense, a basis is a subset such that every element is uniquely a finite sum of basis elements. This is much stronger than the notion of "basis" used in Fourier analysis, for instance, where we allow infinite linear combinations.

In general, the first kind of basis exists even for infinite-dimensional vector spaces, but is not something you can construct or explicitly describe. If it's the second kind you're looking for, look at things like sin(kx).

 

[homology]

My first guess...if the function is even what can you say about the basis functions?

 

[mnb96]

Hi,
thanks for both your answers:

@tinyboss: I didn't know that a finite set of basis functions always existed even for infinite dimensional spaces. Is there any well-known example I am missing? Could you point out one?

@homology: well, if the functions of my space are all even, I deduce also the basis functions must be even.

Instead, what I was thinking, is that the constraint number 2) that I imposed, should be perhaps removed, because we must allow the function f(x)=0 be in our set, if we want it to be a vector space.

However, using only sin(kx) bases, would force the function to be 0 at the origin, and to be odd.
$|sin(k\pi x)|$ might do better, though it would still keep the functions passing through the origin.

 

[jambaugh]

Try this

If f_k is a basis then so is g_k= f_k + g.

So see if you can find a basis which almost satisfies all the conditions and add one function which will give a basis which does satisfy them all.

 

[mnb96]

thanks!

Perhaps I might use as a basis the sinc function itself (which satisfies all the requirements), plus the whole set of all functions $|\sin(\pi B x)|$, where B=2,3,4,...

Though I am not convinced about the orthogonality of this basis (assuming it is a basis).

EDIT:
It does not work: the functions $|\sin(\pi B x)|$ are not differentiable at 0, and they are not going to be orthogonal (as they are non-negative).

 

[HallsofIvy]

Hello,
I am considering the set of all (differentiable) even functions with the following properties:

1) $$f(x)=f(-x)$$
2) $$f(0)=a_0$$, with $a_0\in \mathbb{R}$

If $a_0$ is NOT 0, this will not be a vector space- the sum of two such functions will not be uin the space, nor will af(x) for a not equal to 1.

3) $$f(n)=0$$, where $n\in \mathbb{Z}-\{0\}$

One example of such a function is the sinc function $sin(\pi x) / \pi x$.
Is it possible to find some basis-functions that completely define this set of functions?

If so, any hint on how to find a basis?


Thanks!

 

[mnb96]

Hi HallsofIvy.

You are right. I pointed out this mistake in one of my previous posts. If it was possible I would edit the original post by removing the second "requirement", and so allowing the functions to be anything at x=0.

The requirements would be then:

1) $$f(x)=f(-x)$$
2) $$f(n)=0$$, where $n\in \mathbb{Z}-\{0\}$

 

[homology]

What's your inner product?

 

[mnb96]

the inner product of two functions f and g is given by:

$$\int_{\mathbb{R}}f(x)g(x)dx$$

 

[jambaugh]

thanks!

Perhaps I might use as a basis the sinc function itself (which satisfies all the requirements), plus the whole set of all functions $|\sin(\pi B x)|$, where B=2,3,4,...

Though I am not convinced about the orthogonality of this basis (assuming it is a basis).

EDIT:
It does not work: the functions $|\sin(\pi B x)|$ are not differentiable at 0, and they are not going to be orthogonal (as they are non-negative).

That was my thought except why not just square it instead of ||, or equivalently 1-cos(kx) for suitable k.

You didn't say anything about orthogonality. If you need an orthonormal basis you can carry out the Graham-Schmidt procedure once you find a general basis.

 

[mathwonk]

isnt this whaT FOURIeR SERIS IS ABOUT?