# Finding a function from Fourier coefficients

• Jul 4th 2006, 04:05 AM
DMT
Finding a function from Fourier coefficients
Anyone have a clue what function f(x) (however weird it needs to be) will satisfy the following integral for any integer n>0:

$\int^{\pi}_{0} f(x)sin(nx)dx=(-1)^{n+1}$
• Jul 4th 2006, 06:07 AM
CaptainBlack
Quote:

Originally Posted by DMT
Anyone have a clue what function f(x) (however weird it needs to be) will satisfy the following integral for any integer n>0:

$\int^{\pi}_{0} f(x)sin(nx)dx=(-1)^{n+1}$

In retrospect I don't think this is going to be much use, but here it is
anyway:

Consider the formal Fourier series:

$g(x)=k+\frac{1}{\pi}\sum_1^{\infty} \frac{(-1)^n}{n}\cos(nx)$

Now if I have done this right your $f$ is the formal derivative of $g$.

RonL
• Jul 4th 2006, 06:43 AM
malaygoel
Quote:

Originally Posted by DMT
Anyone have a clue what function f(x) (however weird it needs to be) will satisfy the following integral for any integer n>0:

$\int^{\pi}_{0} f(x)sin(nx)dx=(-1)^{n+1}$

I think f(x) is
$n - \frac{1}{\pi sin(nx)}$

Keep Smiling
Malay
• Jul 4th 2006, 06:52 AM
DMT
Well, the point was to find an f(x) that gave a specific Fourier series of which those were the coefficients. I did find it finally though, I think.

I'm pretty sure it's just a version of an alternating delta function, with peaks at pi and -pi and then made 2pi periodic. I think the peaks go to infinity with infinite terms, but I'm not sure. This became a lot clearer with the similar cosine series.
• Jul 4th 2006, 07:23 AM
CaptainBlack
Quote:

Originally Posted by DMT
Well, the point was to find an f(x) that gave a specific Fourier series of which those were the coefficients. I did find it finally though, I think.

I'm pretty sure it's just a version of an alternating delta function, with peaks at pi and -pi and then made 2pi periodic. I think the peaks go to infinity with infinite terms, but I'm not sure. This became a lot clearer with the similar cosine series.

Humm - <scratches his chin.>

I'm not sure, .. how about the derivative of the delta function at pi.

(made periodic, factors of 2 or 1/2 inserted as needed etc etc ..)

RonL
• Jul 4th 2006, 10:01 AM
CaptainBlack
Quote:

Originally Posted by CaptainBlack
Humm - <scratches his chin.>

I'm not sure, .. how about the derivative of the delta function at pi.

(made periodic, factors of 2 or 1/2 inserted as needed etc etc ..)

RonL

Notes on numerical experiments:

the partial sums look like they may be heading for the derivative of
the delta function, its the term wise integrals that do not look like they
are heading for the delta or the unit step functions :confused:

However since the main property of the derivative of the delta is:

$
\int_{R} \delta'(x-a) f(x) dx = f'(a)
$

for any sufficiently well behaved function $f$ and with slight
abuse of notation.

It does look as though its what we need (if we slightly modify the
statment of the problem). On second thoughts no, it will introduce
an extra factor of 1/n where we don't want/need it :mad:

RonL
• Jul 4th 2006, 03:28 PM
DMT
Thansk for looking at this, but I have what I need now so you don't need to bother anymore.
• Jul 5th 2006, 02:50 AM
CaptainBlack
Quote:

Originally Posted by DMT
Thansk for looking at this, but I have what I need now so you don't need to bother anymore.

You have now stired my interest, and so can't stop being bothered :(
Anyway I think I know what it is (more or less) now, so I will now go
back to sleep.

RonL