Exploring Fourier Series and Gibbs Phenomenon

In summary: Choose a function f(x) such thatf(x) is a continuous function on the closed interval from a to bf(x) possesses a smooth derivative on the closed interval from a to bf(x) is integrable over the closed interval from a to bThe derivative of f(x) is a monotonic functionFor the function f(x) you can use the following:The LaplacetransformThe FourierTransformThe FourierInverseTransformThe first two are convolution and summation respectively.The second two are multiplication and division respectively.
  • #1
likearollings
27
0

Homework Statement



http://img684.imageshack.us/img684/4496/fourier.png [Broken]

Homework Equations



http://en.wikipedia.org/wiki/Small-angle_approximation

The Attempt at a Solution



This is part of a larger question, I have underlined in Red the areas I am struggling with and have cut out bits I have done.

Any ideas?

I have done some research and found out about 'Gibbs Phenonemon', http://www.sosmath.com/fourier/fourier3/gibbs.html

can't really get it all together to answer the question.

any help is appreciated, thanks! :)
 
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  • #2
I am not sure which small angle formula to use, Taylor series or sine, or how to get the RHS of that equation...
 
  • #3
Thanks for all the replies...

I have managed to do the first problem. Now I just need help with the wordy question at the end.

Any help will be massively appreciated
 
  • #4
You have

[tex]f_N(\frac{\pi}{2N}) = \frac 2 \pi\int_0^{\frac \pi {2N}}\frac{\sin(2Nt)}{\sin(t)}\, dt[/tex]

Letting u = 2Nt, du = 2Ndt gives

[tex]\frac 2 \pi\int_0^\pi \frac {\sin u}{\sin(\frac 1 {2N}u)}\frac 1 {2N}\, du[/tex]

Now if N is large, u/(2N) is small and you can use the fact that for small angles, sin(θ) ≈ θ
in the denominator. That will get you your equation.
 
  • #5
likearollings said:
Thanks for all the replies...

All what replies? I don't see any except for the one I just posted.
 
  • #6
LCKurtz said:
All what replies? I don't see any except for the one I just posted.

Hey sorry, I posted that right before you replied, was getting really frustrated (sure you know how it is)! but thanks, I am just reading what you have written and it is a great help, thank you so much
 
  • #7
LCKurtz said:
You have

[tex]f_N(\frac{\pi}{2N}) = \frac 2 \pi\int_0^{\frac \pi {2N}}\frac{\sin(2Nt)}{\sin(t)}\, dt[/tex]

Letting u = 2Nt, du = 2Ndt gives

[tex]\frac 2 \pi\int_0^\pi \frac {\sin u}{\sin(\frac 1 {2N}u)}\frac 1 {2N}\, du[/tex]

Now if N is large, u/(2N) is small and you can use the fact that for small angles, sin(θ) ≈ θ
in the denominator. That will get you your equation.
I thought that it might involve some substitution like that but I didn't think it would cancel out and work that well. Thanks! great help.

The limits of the integral all work out nicely too, thanks!

Any ideas for the last part of the question?

If you have a look at page 5 of this document it has some suggestions.

http://www.scribd.com/doc/20565822/Fourier-Series

These are the conditions:

http://en.wikipedia.org/wiki/Dirichlet_conditions
 
Last edited:

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

2. How do Fourier series relate to Gibbs phenomenon?

Gibbs phenomenon is the phenomenon of oscillations or overshoots occurring near discontinuities of a Fourier series. These overshoots are caused by the inability of a finite number of terms in the series to accurately represent a discontinuous function.

3. What applications do Fourier series have?

Fourier series have numerous applications in mathematics, physics, engineering, and signal processing. They are commonly used in the analysis and synthesis of periodic signals, as well as in solving partial differential equations.

4. Can Gibbs phenomenon be eliminated?

No, Gibbs phenomenon cannot be eliminated completely. However, it can be reduced by using more terms in the Fourier series or by using a different method of approximating the function, such as using a different basis function.

5. Are there any real-world examples of Gibbs phenomenon?

Yes, Gibbs phenomenon can be observed in real-world situations such as the ringing that occurs when a piano key is struck, the distortion in the shape of a square wave on an oscilloscope, and the jagged edges of a digital image when zoomed in.

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