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

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Problem 2.6: Determination of orthogonal functions

Given[edit | edit source]

Consider the family of functions


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

Find[edit | edit source]

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

Solution[edit | edit source]

Construct :



In order to construct the matrix we must first define


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


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


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


The determinant of a diagonal matrix is



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.