Finding basis functions for approximating transcendental function

AI Thread Summary
The discussion focuses on approximating the transcendental function f(x) = x^Ne^x as a linear combination of basis functions. The initial proposal of using functions of the form x^v, where -1 < v < 0, was deemed inappropriate due to their behavior at x=0 and for large x. A revised suggestion involves using basis functions of the form x^v/(1 + x^2), where 0 < v < 1. The conversation highlights the importance of employing orthogonal polynomials and the Gram-Schmidt process to derive suitable basis functions, emphasizing the need for a finite interval for projection. The overall approach involves defining an inner product, generating basis polynomials, and projecting the function onto these polynomials to obtain coefficients for approximation.
sauravrt
Messages
15
Reaction score
0
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 &lt; v &lt; 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 ?
 
Mathematics news on Phys.org
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.
 
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&lt; v &lt; 1.

What is the general procedure to come up with a basis set?
 
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.
 
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.
 
sauravrt said:
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 &lt; v &lt; 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.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top