
Math Discussion forum for anything math related. 

Thread Tools  Display Modes 
#1




Dirichlet conditions for the Fourier series
Hi Folks,
I have read about the three conditions that must be satisfied before the Fourier series can be successfully used to represent any given function. Can someone please confirm that my understanding of them is correct! 1) The function must have a single defined value for each value in its domain. For every value of x there should be only one value of y. Most of the electronic (This is my field) functions will satisfy this condition. One function that wouldn't satisfy this condition is the circle function (sometimes not reffered as a function because two values are obtained for x) because that would give two values for each value of x. 2) The function must not have any infinite discontinuity of range within its domain. This condition states that the function shouldn't have an infinite number of discontinuities on its domain. It's kind of hard to come up with a function that doesn't abide to this condition. One example would be: let f be a function of x so that its value is: 1 if x is a rational number 0 if x is an irrational number This function has an infinite number of discontinuities, because in any finite interval, there is an infinite number of rational numbers, and an infinite number of irrational numbers between them! So the function would have an infinite number of "jumps" between 0 and 1. I guess if a function did have a lot of jump the Fourier representaion would be skewed by the Gibbs Phenomina. The first and second differentials of the function must be piecewise continuous over the domain. a. within each interval, the function is continuous; and b. on the "borders" of each interval, the function can be discontinuous, as long as the limit coming from each side exists. For example, the tan(x) function is NOT piecewise continuous, because near pi/2 multiples it "Points towards" infinity. Had it pointed towards any real value, it would be piecewise continuous. Thank you in advance for you guidance and help. Jag. 
#2




My, you have been doing your homework dude.
But that's only two conditions, by my counting. 1) Exactly so. The standard parabola y^2=4ax also gets caught here. 2.) Condition correct but you only need consider the interval of interest not the whole domain of the function. 3) The third condition is that there must be only a finite number of maxima and minima on the function in the interval concerned. A function that violates this is y=sin(1/x) 0<x<=1 y=0 1<=x<=0 You may be interested to know that the fourier series at any point converges to the average of the convergenge approaching from the left and from the right. If these are equal the convergence is exact, 
#3




Quote:
Quote:
I found another definition of Dirichlet Conditions at http://mathworld.wolfram.com/Dirichl...onditions.html, a site associated with Wolfram, the creator of Mathematica. Mark 
#4




Right, since this is not a function. However, you can break it up into two pieces, each of which is a function, and each of which satisfies all the conditions.
y1 = +sqrt(4*a*x) y2 = sqrt(4*a*x) Mark 
#5




I think the link definition implies jagjit's condition1
Quote:
The Dirichlet conditions spring from a wider Dirichlet theorem about the convergence of series. 
#7




Thanks
Quote:
jagjits condition 1 as I suspected. It also implies there are functions that are not single valued or there would be no need to state this. It should be remembered that when Mr Dirichlet was around multivalued functions were admitted to the fraternity of functions. So this condition made sense. It has become fashionable to disallow multivalued functions in certain branches of mathematics, although they are certainly still acknowledged in complex analysis. It is really a matter of definition. The important thing, for which Jagjit should be congratulated, is that (s)he is looking at the conditions under which a theorem applies. These are all too often glossed over on the basis that 'you will never meet an exception in the exam' . So when one crops up in real life, confusion reigns. 
#8




Conceptually Fourier Series/Transform
Hi Folks,
Firstly I would like to thank all the good people who helped me on my last post (Dirchlet conditions). I recently heard where they played a recording of someone singing she didnt sound very good, however after the studio had done something to it the same song didnt sound too bad. It sounded a lot more crisp. I have only started out in this field, but has this got something to do with Fourier. I understand that it cant be the Fourier series as the signal must be periodic. It could be the Fourier transform though, could they decompose the audio signal segment into its composite sine and cosine waves and just reconstruct the signal using the inverse transform. When reconstructing could they not put back certain frequencies (certain notes that couldnt be reached). If that is possible then could they also put frequencies in that wernt there at all. I cant see how else they would do it. What I am saying is very abstract, I have heard that this kind of stuff falls into the DSP domain but is my thinking right. P.S. Happy Easter to everyone. Cheers Jag. 
Tags 
conditions, dirichlet, fourier, series 
Related Site Pages  
Section  Title  
Worksheet  Simultaneous equations for circuit analysis  
Worksheet  Mixedfrequency signals  
Worksheet  Phasor mathematics  
Textbook  Harmonics in polyphase power systems : Polyphase Ac Circuits  
Textbook  More on spectrum analysis : Mixedfrequency Ac Signals  
Textbook  Square wave signals : Mixedfrequency Ac Signals  
Textbook  Quirks : Using The spice Circuit Simulation Program  
Textbook  Solving simultaneous equations : Algebra Reference  
Textbook  Introduction : Boolean Algebra  
Textbook  Quantum physics : Solidstate Device Theory 
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Fourier Series to Fourier Transform  jag1972  Math  2  09132012 10:01 AM 
How to solve fourier series integrals  gutto@Mac  Math  2  11272007 06:23 PM 
Fourier Series and Filters  Mazaag  General Electronics Chat  32  11212007 06:54 PM 
Fourier Series  Mazaag  Math  12  03182007 05:50 PM 
Convolutions, Fourier Series...  eazhar  Homework Help  2  03172007 04:10 PM 
Thread Tools  
Display Modes  

