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| author | siveshs <siveshs@gmail.com> | 2010-07-03 05:11:59 +0000 |
|---|---|---|
| committer | bnewbold <bnewbold@adelie.robocracy.org> | 2010-07-03 05:11:59 +0000 |
| commit | 9100ef01aafae71e51d01a65b7a1878caffb2e18 (patch) | |
| tree | a1923bbfd8e8429e608efea8df95d8302d2c9dac /Fourier Series.page | |
| parent | cfcb736f78224babc4131c3687ecaaf379c81f61 (diff) | |
| download | afterklein-wiki-9100ef01aafae71e51d01a65b7a1878caffb2e18.tar.gz afterklein-wiki-9100ef01aafae71e51d01a65b7a1878caffb2e18.zip | |
editing
Diffstat (limited to 'Fourier Series.page')
| -rw-r--r-- | Fourier Series.page | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/Fourier Series.page b/Fourier Series.page index bca75a0..251c509 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -1,4 +1,4 @@ -#<b>Why Fourier series is possible?</b> +#<b>Why the Fourier decomposition is possible?</b> We first begin with a few basic identities on the size of sets. Then, we will show that the set of possible functions representing sets is not larger than the set of available functions. This at best indicates that the Fourier series is not altogether impossible. ## To show that $(0,1) \sim \mathbb R$ @@ -8,7 +8,7 @@ We first begin with a few basic identities on the size of sets. Then, we will sh ## Proof that no. of available functions is greater than number of functions required to define the periodic function --> don't have the notes for this -#<b>Why Fourier series is plausible?</b> +#<b>Why Fourier decomposition is plausible?</b> To show that Fourier series is plausible, let us consider some arbitrary trignometric functions and see if it is possible to express them as the sum of sines and cosines: $1.\quad\sin^2(x) = ?$ @@ -84,7 +84,7 @@ Summing these two functions we get the following: <center></center> -#<b>What is the Fourier Series actually?</b> +#<b>Why does the Fourier decomposition actually work?</b> ##Initial Hypothesis Now, to prove that the Fourier series is indeed true, we begin with the following hypothesis: Let $f : \mathbb I \rightarrow \mathbb C$ be a continuous, periodic function where $I$ is some time interval(period of the function). Then it can be expressed as : @@ -193,5 +193,5 @@ Now, we know that the entire function space can be described by the defined basi --> don't quite remember this part -#<b>Why is Fourier series useful? </b> +#<b>Why the Fourier decomposition is useful? </b> Applications will be covered on Monday July 5, 2010. See you all soon!
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