Re: [sage-devel] Issues with Fourier series

[Eric Gourgoulhon]

Le mercredi 9 août 2017 14:53:28 UTC+2, David Joyner a écrit :
>
>
> Thanks for looking into this, Eric. I agree with your comments that the 
> docs 
> (and possible the code) needs some improvement. 
>
>
OK, thanks. 
This is now  https://trac.sagemath.org/ticket/23672

Eric.

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Re: [sage-devel] Issues with Fourier series

[David Joyner]

On Wed, Aug 9, 2017 at 7:03 AM, Eric Gourgoulhon  wrote:
> Hi,
>
> It seems that the only way to compute with Sage the Fourier series of a
> periodic real function is through the methods fourier_series_* of
> piecewise-defined functions. Let us take a trivial example: the Fourier
> series of the cosine function:
>
> sage: f = piecewise([((0,2*pi), cos(x))])
>
> To get the cosine coefficient of order n of the Fourier series, one has to
> call the method fourier_series_cosine_coefficient with n and the half-period
> as argument:
>
> sage: f.fourier_series_cosine_coefficient(1, pi)
> 1
>
> So far so good. Now
>
> sage: f.fourier_series_partial_sum(1, pi)
> 0
>
> This is quite surprising at first glance and is actually the offset issue
> already discussed in #8603. The expect result is obtained via
>
> sage: f.fourier_series_partial_sum(2, pi)
> cos(x)
>
> To understand why one has to provide 2 as the first argument, let us have a
> look at the documentation:
>
> sage: f.fourier_series_partial_sum?
> Signature:  f.fourier_series_partial_sum(*args, **kwds)
> Docstring:
>Returns the partial sum
>
>   f(x) sim frac{a_0}{2} + sum_{n=1}^N [a_ncos(frac{npi
>   x}{L}) + b_nsin(frac{npi x}{L})],
>
>as a string.
>
> At the very least, this is very incomplete: there is no explicit relation
> between N and *args, nor between L and *args. One has to infer them from the
> provided examples. Moreover the specified output type is wrong: it is not a
> string but a symbolic expression:
>
> sage: type(f.fourier_series_partial_sum(2, pi))
> 
>
> The documentation of fourier_series_cosine_coefficient is better because it
> has an INPUT section:
>
> sage: f.fourier_series_cosine_coefficient?
> Signature:  f.fourier_series_cosine_coefficient(*args, **kwds)
> Docstring:
>Returns the n-th Fourier series coefficient of cos(npi x/L), a_n.
>
>INPUT:
>
>* "self" - the function f(x), defined over -L x L
>
>* "n" - an integer n=0
>
>* "L" - (the period)/2
>
>OUTPUT: a_n = frac{1}{L}int_{-L}^L f(x)cos(npi x/L)dx
>
> However , it  needs some correction: n=0 looks like a typo here.
>
> Besides the documentation, another issue is the relation between the period
> and the piecewise function domain. Since one has to provide the half-period
> L as an argument to the fourier_series_* methods, this suggests that L is
> independent from the domain provided to piecewise. Let us then define
>
> sage: f = piecewise([((0,4*pi), cos(x))])
>
> This is the same cosine function as before but defined on a larger domain.
> Let us ask for the Fourier coefficient of order 1  by providing pi as the
> half-period, as above:
>
> sage: f.fourier_series_cosine_coefficient(1, pi)
> 2
>
> Boom!
> Having a look at the code of fourier_series_cosine_coefficient, one sees
> clearly why there is such an error:
>
> sage: f.fourier_series_cosine_coefficient??
> from sage.all import cos, pi
> x = SR.var('x')
> result = 0
> for domain, f in parameters:
> for interval in domain:
> a = interval.lower()
> b = interval.upper()
> result += (f*cos(pi*x*n/L)/L).integrate(x, a, b)
> return SR(result).simplify_trig()
>
> The integral is taken over all the domain, while it should be taken over a
> single period only.
> A solution would be to deduce the half-period L from the domain (instead of
> getting it as an argument)  by writing something like
>
> L = (self.domain().sup() - self.domain().inf())/2
>
> I volunteer to write a ticket to improve the documentation and possibly the
> code. Any comment/suggestion prior to do so is welcome.
>

Thanks for looking into this, Eric. I agree with your comments that the docs
(and possible the code) needs some improvement.


> Eric.
>
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[sage-devel] Issues with Fourier series

[Eric Gourgoulhon]

Hi,

It seems that the only way to compute with Sage the Fourier series of a 
periodic real function is through the methods fourier_series_* of 
piecewise-defined 
functions 
.
 
Let us take a trivial example: the Fourier series of the cosine function:

sage: f = piecewise([((0,2*pi), cos(x))])

To get the cosine coefficient of order n of the Fourier series, one has to 
call the method fourier_series_cosine_coefficient with n and the 
half-period as argument:

sage: f.fourier_series_cosine_coefficient(1, pi)
1

So far so good. Now

sage: f.fourier_series_partial_sum(1, pi)
0

This is quite surprising at first glance and is actually the offset issue 
already discussed in #8603 . The 
expect result is obtained via

sage: f.fourier_series_partial_sum(2, pi)
cos(x)

To understand why one has to provide 2 as the first argument, let us have a 
look at the documentation:

sage: f.fourier_series_partial_sum?
Signature:  f.fourier_series_partial_sum(*args, **kwds)
Docstring: 
   Returns the partial sum

  f(x) sim frac{a_0}{2} + sum_{n=1}^N [a_ncos(frac{npi
  x}{L}) + b_nsin(frac{npi x}{L})],

   as a string.

At the very least, this is very incomplete: there is no explicit relation 
between N and *args, nor between L and *args. One has to infer them from 
the provided examples. Moreover the specified output type is wrong: it is 
not a string but a symbolic expression:

sage: type(f.fourier_series_partial_sum(2, pi))


The documentation of fourier_series_cosine_coefficient is better because it 
has an INPUT section:

sage: f.fourier_series_cosine_coefficient?
Signature:  f.fourier_series_cosine_coefficient(*args, **kwds)
Docstring: 
   Returns the n-th Fourier series coefficient of cos(npi x/L), a_n.

   INPUT:

   * "self" - the function f(x), defined over -L x L

   * "n" - an integer n=0

   * "L" - (the period)/2

   OUTPUT: a_n = frac{1}{L}int_{-L}^L f(x)cos(npi x/L)dx

However , it  needs some correction: n=0 looks like a typo here. 

Besides the documentation, another issue is the relation between the period 
and the piecewise function domain. Since one has to provide the half-period 
L as an argument to the fourier_series_* methods, this suggests that L is 
independent from the domain provided to piecewise. Let us then define

sage: f = piecewise([((0,4*pi), cos(x))])

This is the same cosine function as before but defined on a larger domain. 
Let us ask for the Fourier coefficient of order 1  by providing pi as the 
half-period, as above:

sage: f.fourier_series_cosine_coefficient(1, pi)
2

Boom!
Having a look at the code of fourier_series_cosine_coefficient, one sees 
clearly why there is such an error:

sage: f.fourier_series_cosine_coefficient??
from sage.all import cos, pi
x = SR.var('x')
result = 0
for domain, f in parameters:
for interval in domain:
a = interval.lower()
b = interval.upper()
result += (f*cos(pi*x*n/L)/L).integrate(x, a, b)
return SR(result).simplify_trig()

The integral is taken over all the domain, while it should be taken over a 
single period only. 
A solution would be to deduce the half-period L from the domain (instead of 
getting it as an argument)  by writing something like

L = (self.domain().sup() - self.domain().inf())/2

I volunteer to write a ticket to improve the documentation and possibly the 
code. Any comment/suggestion prior to do so is welcome.

Eric. 

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