Finding basis functions for approximating transcendental function

[sauravrt]

I am working on a problem where I want to approximate a transcendental function of the form
$f(x) = x^Ne^{x}$ for $x \geq 0$ as a linear combination of functions of the form $x^v \text{where} -1 < v < 0$.
How can I find the basis functions of the desired form to represent my transcendental function as a finite linear combination?

If not, what would be approach to obtain finte approximate a transcendental function of the form above ?

 

[mathman]

Your f(x)=0 for x=0 (for N ≥ 0), while the basis functions you are suggesting are infinite at x=0. Your proposal doesn't make sense.

 

[sauravrt]

Hi mathman, I realize that error. Instead of that I would like my basis to be of the form $\frac{x^v}{1 + x^2} ~\text{where} ~ 0< v < 1$.

What is the general procedure to come up with a basis set?

 

[mathman]

I can't give you an off-hand answer. However your proposal has at least one problem. The functions go to zero for large x, while the functions you are trying to represent become infinite very rapidly.

 

[sauravrt]

Hi mathman:

I am sorry about the error in the original post. The function I am trying to approximate is $f(x) = x^Ne^{-x}$. This function goes to zero for large x.

 

[chiro]

I am working on a problem where I want to approximate a transcendental function of the form
$f(x) = x^Ne^{x}$ for $x \geq 0$ as a linear combination of functions of the form $x^v \text{where} -1 < v < 0$.
How can I find the basis functions of the desired form to represent my transcendental function as a finite linear combination?

If not, what would be approach to obtain finte approximate a transcendental function of the form above ?

You could use orthogonal polynomials and project your function onto these basis polynomials through an integral transform.

The polynomials themselves will be determined on the basis as well as the interval for projection. If you want to project onto polynomial basis, then there are some texts out there that cover this. If however you want to project onto non-polynomial basis (like discontinuous functions or transendental functions of some sort), then you will need to accommodate for that.

What you can do to derive such polynomials is to use the Gram-Schmidt process to generate the basis functions from first principles. To do this you need to define an inner product, and this is dependent on the interval that you are dealing with.

You should be aware that you will have to use a finite interval: it won't make sense to use an infinite interval as your function is not complete in the L^2 space.

So to sum up:

1) Choose a power for your highest degree for the polynomial
2) Choose the interval that you are approximating for your function
3) Using gram-schmidt and the inner product definition for an L^2 space, generate the basis polynomials
4) Project your function to the basis polynomials to get your coeffecients
5) Write out your approximate function using a linear combination of your basis coeffecients with your basis polynomials.