Why is Fourier Integral Meaningless for f(x)=A*cos(ax)?

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Discussion Overview

The discussion revolves around the question of why the Fourier integral is considered meaningless for the function f(x)=A*cos(ax). Participants explore the implications of this function in the context of Fourier analysis, particularly focusing on integrability and the use of delta functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that |f(x)| is not integrable over (-∞, ∞), suggesting this as a reason the Fourier integral cannot be properly defined.
  • Another participant questions the need for analysis of a sinusoidal function, implying that the Fourier integral is more relevant for non-sinusoidal functions.
  • A participant acknowledges the concept of square integrability and mentions the use of the delta function as a resolution.
  • One participant references lectures on Fourier transforms that discuss distributions and how they relate to the Fourier transforms of sinusoidal functions.
  • Another participant states that the Fourier integral yields two delta functions, which they consider sufficient for physicists.
  • A separate query is raised regarding the forward and inverse transforms in the context of a specific equation, indicating a lack of clarity on the definitions of variables involved.
  • Another participant expresses confusion about applying Fast Fourier Transform (FFT) on non-uniform sampling and questions the validity of the results obtained.

Areas of Agreement / Disagreement

Participants express differing views on the meaning and applicability of the Fourier integral for sinusoidal functions, with some agreeing on issues of integrability while others raise questions about the necessity of the Fourier analysis in this context. The discussion remains unresolved regarding the implications of using delta functions and the specifics of Fourier transforms in various contexts.

Contextual Notes

There are limitations in the discussion regarding the assumptions about integrability and the definitions of terms used in the equations. The relationship between the Fourier integral and delta functions is also not fully explored.

caduceus
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I couldn't understand that why the Fourier integral is meaningless for f(x)=A*cos(ax) ?

Any comments will be appreciated.
 
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|f(x)| is not integrable over (-oo,oo). so the Fourier integral cannot be properly defined.
 
caduceus said:
I couldn't understand that why the Fourier integral is meaningless for f(x)=A*cos(ax) ?

Any comments will be appreciated.

Because that is a sinusoidal function. What is there to analyze?

You want to find the sinusoidal functions that make up a non sinusoidal function.
 
Oh, I guess I can see the point now. You mean square integrability. That is why I should use Delta function. Thank you.
 
Prof Brad Osgood has some excellent Fourier transform lectures on iTunesU
(The Fourier Transform and its Applications).

In particular, there's a few lectures (~lectures 10-14 I think) on distributions and Schwartz functions that help show how Fourier transforms of sines, exponentials, deltas, ... can be better justified. Suprisingly (to me after having seen and given up trying to understand Functional analysis), the basics required for application are not actually all that difficult, mostly requiring a change in approach, and an extra level of indirection.

Some lecture notes to go with the lectures can be found here:

http://www.stanford.edu/class/ee261/book/all.pdf

(chapter 4 covers distributions).
 
Last edited by a moderator:
caduceus said:
I couldn't understand that why the Fourier integral is meaningless for f(x)=A*cos(ax) ?

Any comments will be appreciated.
The Fourier integral gives two delta functions. That is good enough for physicists.
 
well i am a computer guy :) so i don't know...ok here is my question
following are two equations


$fk(k1) = \frac{1}{nj} \sum_{j=1}^{nj} c_j(j) \exp (i k1 x_j(j))$\\

$where \frac {-ms}{2} <k1 < \frac{ms-1}{2}$

a) what is k1
b) what is x_j(j)
c) what is c_j(j)

Is this a forward transform

d) What will be the inverse transform, and is inverse tranform means we are evaluating Fourier series
 
that's not really readable as is. Can you edit with [ tex ] [ / tex ] (no spaces), replacing the dollar signs.
 
I am experimenting with non uniform sampling, I applied Fast Fourier transform on the non uniform sampling in MATLAB it has given me some results. I can't understand how FFT runs on non uniform sampling. What i am getting after applying FFT on Non uniform samples is what...is it a errorfull value if yes then why
 

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