University of Florida/Eml5526/s11.team2.reiss.HW

Given
Consider the family of functions on the interval [0,T], where T=$$2\Pi/\omega$$

Find
A) Construct $$\Gamma (\Im )$$ and observe its properties B) Find $$\det [\Gamma (\Im )]$$ C) Is $$\Im$$ an orthogonal basis

Solution
Construct $$\Gamma (\Im )$$:

where

In order to construct the matrix we must first define $$ < {b_i}, {b_j} > $$

Because multiplication of continuous functions is communicative it can be shown from equation 4.3 that

And therefore $$\Gamma (\Im )$$ is a symmetric matrix

We must now evaluate the terms of the matrix All values were checked with Wolframalpha The Gram matrix then becomes

As we can see the Gram matrix based constructed from this set of functions is a diagonal matrix

Finding $$\det [\Gamma (\Im )]$$

The determinant of a diagonal matrix is Where Based on equation 4.6

For the set to be an orthogonal basis the Gram matrix must be a diagonal matrix with a non-zero determinant. As we can see from equations 4.5 and 4.7 both of these criteria are satisfied. Thus the set of functions is an orthogonal basis.