How to approximate identity function using Fourier sine series
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I want to approximate identity function $g(x) = x$ for $x in [0,x_c]$ with $x_c<pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x in [0,pi]$, $f(x)$ is assumed to be a $2pi$-periodic function (that is, we can know everything about $f(x)$ by looking at its values for $x in [-pi,pi]$), and approximation needs to be close within precision of $k$ fractional digits in binary or decimal for $xin [0,x_c]$. $k$ is not fixed.
The question is, how many sine terms would be required to satisfy this, in function of $k$ (and $x_c$)?
real-analysis fourier-analysis interpolation
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
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add a comment |
$begingroup$
I want to approximate identity function $g(x) = x$ for $x in [0,x_c]$ with $x_c<pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x in [0,pi]$, $f(x)$ is assumed to be a $2pi$-periodic function (that is, we can know everything about $f(x)$ by looking at its values for $x in [-pi,pi]$), and approximation needs to be close within precision of $k$ fractional digits in binary or decimal for $xin [0,x_c]$. $k$ is not fixed.
The question is, how many sine terms would be required to satisfy this, in function of $k$ (and $x_c$)?
real-analysis fourier-analysis interpolation
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
$endgroup$
$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
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– denklo
Jan 11 at 9:28
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Is gibbq phenomen relevant? We have a continuouq function here.
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– Math_QED
Jan 11 at 9:50
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I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
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@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03
add a comment |
$begingroup$
I want to approximate identity function $g(x) = x$ for $x in [0,x_c]$ with $x_c<pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x in [0,pi]$, $f(x)$ is assumed to be a $2pi$-periodic function (that is, we can know everything about $f(x)$ by looking at its values for $x in [-pi,pi]$), and approximation needs to be close within precision of $k$ fractional digits in binary or decimal for $xin [0,x_c]$. $k$ is not fixed.
The question is, how many sine terms would be required to satisfy this, in function of $k$ (and $x_c$)?
real-analysis fourier-analysis interpolation
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
$endgroup$
I want to approximate identity function $g(x) = x$ for $x in [0,x_c]$ with $x_c<pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x in [0,pi]$, $f(x)$ is assumed to be a $2pi$-periodic function (that is, we can know everything about $f(x)$ by looking at its values for $x in [-pi,pi]$), and approximation needs to be close within precision of $k$ fractional digits in binary or decimal for $xin [0,x_c]$. $k$ is not fixed.
The question is, how many sine terms would be required to satisfy this, in function of $k$ (and $x_c$)?
real-analysis fourier-analysis interpolation
real-analysis fourier-analysis interpolation
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
edited Jan 11 at 9:55
Sayako Nama
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
asked Jan 11 at 8:58
Sayako NamaSayako Nama
11
11
$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
$endgroup$
– denklo
Jan 11 at 9:28
$begingroup$
Is gibbq phenomen relevant? We have a continuouq function here.
$endgroup$
– Math_QED
Jan 11 at 9:50
$begingroup$
I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
$begingroup$
@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03
add a comment |
$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
$endgroup$
– denklo
Jan 11 at 9:28
$begingroup$
Is gibbq phenomen relevant? We have a continuouq function here.
$endgroup$
– Math_QED
Jan 11 at 9:50
$begingroup$
I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
$begingroup$
@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03
$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
$endgroup$
– denklo
Jan 11 at 9:28
$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
$endgroup$
– denklo
Jan 11 at 9:28
$begingroup$
Is gibbq phenomen relevant? We have a continuouq function here.
$endgroup$
– Math_QED
Jan 11 at 9:50
$begingroup$
Is gibbq phenomen relevant? We have a continuouq function here.
$endgroup$
– Math_QED
Jan 11 at 9:50
$begingroup$
I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
$begingroup$
I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
$begingroup$
@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03
add a comment |
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$begingroup$
Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon
$endgroup$
– denklo
Jan 11 at 9:28
$begingroup$
Is gibbq phenomen relevant? We have a continuouq function here.
$endgroup$
– Math_QED
Jan 11 at 9:50
$begingroup$
I don't know, like this is not about approximating g(x)=x exactly for $x in [-pi,pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom.
$endgroup$
– Sayako Nama
Jan 11 at 9:58
$begingroup$
@SayakoNama Ok, no worries then :)
$endgroup$
– denklo
Jan 11 at 9:59
$begingroup$
Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x in [x_c,pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $xin [-pi,pi]$, which is rendered impossible unless infinite sums are made.
$endgroup$
– Sayako Nama
Jan 11 at 10:03