
Subject: fourier transform
Name: Abdul
Who is asking: Other
Level: Secondary
Question:
 Sir, we have the Dirichlet's condition for the Fourier transform : " The function should be integral over the
real line "
But why we are we neglecting this for example when we take the Fourier transform of an impulse train?
 Suppose we want to travel from one corner of a square of side 'a' to the diagonally opposite corner. We can travel along the sides which gives a pah length of '2a'. We can also do it in steps as shown below:
_
 _PATH
 _
_____
Suppose The step size =DELTA x
Then the path length will be again '2a'. Now in the limit DELTA x >0 again we get '2a' But when we take the limit we get the straight line diagonal whose length is 'SQRT(2)X a' Where did I go wrong?
Hi Abdul,
 The Dirichlet condition is just a convenient sufficient condition. It
is possible to have a Fourier transform without satisfying that condition.
What is called the "impulse function" isn't really a function at all, but
it is a useful idea and it has a nice transform.
 You can estimate the length of a curve using the length of a polygon
ABCD... with all the points A, B, C, D,... on the curve. The length of the
polygon approaches the length of the curve as the segments AB, BC, CD, etc.
go to zero  in fact, this observation forms the basis for the definition
of "length". You cannot use a step function because (as you point out) its
length doesn't approach the length of the curve as the step size goes to
zero. (But the area between the steps and the curve does go to zero.)
This shouldn't surprize you  an easier example of this idea is to compare
the length of the unit segment PQ with a zigzag that starts at P and ends
at Q. You can keep the zigs and zags as small as you wish, while the
length of the zigzag is as large as you wish.
Chris
Go to Math Central

