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March 6th, 2018, 03:27 PM   #1
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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 bit-wise 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.
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March 6th, 2018, 05:33 PM   #2
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Bit-wise?

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.
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Last edited by skipjack; March 7th, 2018 at 03:58 AM.
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March 7th, 2018, 04:14 AM   #3
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This article may help.
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March 7th, 2018, 04:32 AM   #4
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Quote:
Originally Posted by v8archie View Post
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.
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.
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March 7th, 2018, 06:35 AM   #5
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This is what I was referring to:
Quote:
Originally Posted by Tolstov: Fourier Series - Ch. 1.10
The Fourier Series of a piecewise smooth (continuous or discontinuous) function $f(x)$... converges for all values of $x$. The sum of the series equals $f(x)$ at every point of continuity and equals... the arithmetical mean of the right-hand and left-hand limits at every point of discontinuity.
Thus, if we can find the coefficents (which requires the evaluation of an integral - although not just the integrability of the function), we get point-wise convergence.
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March 7th, 2018, 06:39 AM   #6
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Does the book define "smooth"?
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March 7th, 2018, 06:45 AM   #7
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Yeah - that indicates a continuous derivative. But this isn't a very strict requirement given that we only need it to hold piece-wise.
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March 7th, 2018, 06:46 AM   #8
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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.
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March 7th, 2018, 08:06 AM   #9
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Here's a good explanation of what I mean.

https://www.intmath.com/fourier-seri...-functions.php
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March 7th, 2018, 01:26 PM   #10
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Quote:
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?
A) Are these theoretical waves ie do you have a mathematical formula that describes them?

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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