Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

In summary, to find a polynomial that is orthogonal to f(x)=x^2-1/2 using L2[0,1], one can start by looking at linear polynomials. By integrating Ax+B times x^2-1/2 from 0 to 1 and choosing appropriate values for A and B, the integral can be made to equal 0, resulting in a polynomial that is orthogonal to f(x)=x^2-1/2. This solution may seem simple, but it is effective in finding the desired polynomial.
  • #1
mandygirl22
4
0
Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1].

I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do not know anything about it and the book does not make any sense out of it. Thanks for your help!
 
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  • #2
Well, just DO it! The simplest kind of polynomial is linear. Integrate Ax+ B times \(\displaystyle x^2- 1/2\) from 0 to 1 and choose A and B so the integral is 0.
 
  • #3
*hits forehead* I knew there had to be some simple way of doing it that I was ignoring! Thanks!

(and the REALLY HARD solution ended up being f(x)=x) :smile:
 

FAQ: Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

1. What are Orthogonal Polynomials?

Orthogonal polynomials are a type of mathematical function that have special properties, including the fact that they are orthogonal to each other. This means that when two different orthogonal polynomials are multiplied together and integrated over a specific interval, the result is equal to zero.

2. What are some examples of Orthogonal Polynomials?

Some well-known examples of orthogonal polynomials include Legendre polynomials, Chebyshev polynomials, and Hermite polynomials. These polynomials have various applications in mathematics and physics, such as in approximation and solving differential equations.

3. How are Orthogonal Polynomials used in statistics?

In statistics, orthogonal polynomials are often used in regression analysis and in creating orthogonal designs for experiments. They are also used in probability distributions, such as the chi-square distribution and the Gaussian distribution.

4. What is the significance of the orthogonality property of Orthogonal Polynomials?

The orthogonality property of orthogonal polynomials allows for easier calculations and simplification of mathematical problems. It also allows for the approximation of non-orthogonal functions by using a series of orthogonal polynomials, making them a powerful tool in many areas of mathematics and science.

5. How are Orthogonal Polynomials related to Fourier series?

Orthogonal polynomials are closely related to Fourier series, as they can be used to approximate periodic functions. In fact, the Legendre polynomials are the orthogonal polynomials used in the Fourier series expansion of a function on the interval [-1, 1].

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