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

From Wikiversity
Jump to navigation Jump to search

Problem 2.6: Determination of orthogonal functions


Given[edit]

Consider the family of functions

(4.1)

on the interval [0,T], where T=

Find[edit]

A) Construct and observe its properties
B) Find
C) Is an orthogonal basis

Solution[edit]

Construct :

(4.2)

where

In order to construct the matrix we must first define

(4.3)

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

(4.4)

And therefore is a symmetric matrix

We must now evaluate the terms of the matrix

All values were checked with Wolframalpha

The Gram matrix then becomes

(4.5)

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

Finding

The determinant of a diagonal matrix is

(4.6)

Where


Based on equation 4.6

(4.7)

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.