Subject: fourier transform

Name: Abdul

Who is asking: Other
Level: Secondary

Question:

1. 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?
   
   
2. 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?

1. 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.
   
   
2. 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.