Number of Fourier coefficients necessary to approximate $delta(x)$ to given error
I am new to Fourier analysis, so I apologize if this question is very basic. For any integer $n$, I wish to construct a finite Fourier expansion $F_n$ with the following properties:
- $F_n(0) geq 1$
$F_n(x) < frac{1}{n}$ for $frac{2pi}{n} leq x < 2pi$
I know that, for any $n$, the infinite Fourier expansion of the Dirac delta satisfies the given properties. However, I have not been able to find any analyses for exactly how fast the series converges.
Question: As a function of $n$, how many terms are necessary to construct a Fourier expansion $F_n$ with the given properties?
sequences-and-series fourier-analysis fourier-series
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
add a comment |
I am new to Fourier analysis, so I apologize if this question is very basic. For any integer $n$, I wish to construct a finite Fourier expansion $F_n$ with the following properties:
- $F_n(0) geq 1$
$F_n(x) < frac{1}{n}$ for $frac{2pi}{n} leq x < 2pi$
I know that, for any $n$, the infinite Fourier expansion of the Dirac delta satisfies the given properties. However, I have not been able to find any analyses for exactly how fast the series converges.
Question: As a function of $n$, how many terms are necessary to construct a Fourier expansion $F_n$ with the given properties?
sequences-and-series fourier-analysis fourier-series
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01
add a comment |
I am new to Fourier analysis, so I apologize if this question is very basic. For any integer $n$, I wish to construct a finite Fourier expansion $F_n$ with the following properties:
- $F_n(0) geq 1$
$F_n(x) < frac{1}{n}$ for $frac{2pi}{n} leq x < 2pi$
I know that, for any $n$, the infinite Fourier expansion of the Dirac delta satisfies the given properties. However, I have not been able to find any analyses for exactly how fast the series converges.
Question: As a function of $n$, how many terms are necessary to construct a Fourier expansion $F_n$ with the given properties?
sequences-and-series fourier-analysis fourier-series
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
I am new to Fourier analysis, so I apologize if this question is very basic. For any integer $n$, I wish to construct a finite Fourier expansion $F_n$ with the following properties:
- $F_n(0) geq 1$
$F_n(x) < frac{1}{n}$ for $frac{2pi}{n} leq x < 2pi$
I know that, for any $n$, the infinite Fourier expansion of the Dirac delta satisfies the given properties. However, I have not been able to find any analyses for exactly how fast the series converges.
Question: As a function of $n$, how many terms are necessary to construct a Fourier expansion $F_n$ with the given properties?
sequences-and-series fourier-analysis fourier-series
sequences-and-series fourier-analysis fourier-series
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
edited Nov 22 '18 at 2:11
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
asked Nov 22 '18 at 2:00
Elliot Gorokhovsky
1,0921723
1,0921723
Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01
add a comment |
Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01
Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01
Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01
add a comment |
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Maybe this could be of help: en.wikipedia.org/wiki/Fejér_kernel
– Christian Blatter
Nov 22 '18 at 11:01