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March 6th, 2018, 03:27 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 169 Thanks: 2  General Fourier Series vs Fourier Cosine/Sine series
I am currently learning the very impressive Fourier series. I am still new to this topic, but I am burning with a couple of questions... 1. Would the general form of Fourier series be enough to model any bitwise or periodic curves? Why do we need Fourier cosine and sine series? What are they good for? 2. Let's say I want to break down a repeating sound pulses into basic building blocks of sine waves and cosine waves. Generally, how should I do it? Last edited by skipjack; March 7th, 2018 at 03:57 AM. 
March 6th, 2018, 05:33 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra 
Bitwise? Cosine and sine series are subsets of the set of Fourier series, not distinct from it. Cosine series model even period functions. Sine series model odd periodic functions. I can't recall the detail off the top of my head, but Fourier series are pointwise convergent for pretty much any integrable, periodic function. Last edited by skipjack; March 7th, 2018 at 03:58 AM. 
March 7th, 2018, 04:14 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,719 Thanks: 1536  This article may help.

March 7th, 2018, 04:32 AM  #4 
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92  That's definitely false. Even for continuous functions we don't have pointwise convergence everywhere in general. The question of pointswise convergence is very subtle. A very very deep theorem (Carleson) does say that we can expect almost everywhere convergence to a continuous function.

March 7th, 2018, 06:35 AM  #5  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra 
This is what I was referring to: Quote:
 
March 7th, 2018, 06:39 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,719 Thanks: 1536 
Does the book define "smooth"?

March 7th, 2018, 06:45 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra 
Yeah  that indicates a continuous derivative. But this isn't a very strict requirement given that we only need it to hold piecewise.

March 7th, 2018, 06:46 AM  #8 
Senior Member Joined: Jun 2015 From: England Posts: 766 Thanks: 223 
The sine is an odd function the cosine is an even function. You will not model an even function with a series comprising only odd functions and vice versa. So in the general case you need both, but if your target function is odd or even itself you can use just the simpler series. Note that the general function can be split into the sum of an odd function and an even function, each of which can be modelled by suitable series. 
March 7th, 2018, 08:06 AM  #9 
Senior Member Joined: Jun 2015 From: England Posts: 766 Thanks: 223  
March 7th, 2018, 01:26 PM  #10  
Senior Member Joined: Jun 2015 From: England Posts: 766 Thanks: 223  Quote:
B) Or are they real soundwaves that have been recorded in some way so you have numeric data.? If A) then yes you can use the integral formula from the link I gave. It may be necessary to perform numerical integration. If B) Then you need something like the Fast Fourier Transform or Harmonic Analysis. There are recognised set down procedures for these.  

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cosine or sine, fourier, general, series 
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