Finding a function from Fourier coefficients

**DMT** · July 4th 2006, 04:05 AM

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

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

**CaptainBlack** · July 4th 2006, 06:07 AM

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

**malaygoel** · July 4th 2006, 06:43 AM

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

**DMT** · July 4th 2006, 06:52 AM

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.

**CaptainBlack** · July 4th 2006, 07:23 AM

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

**CaptainBlack** · July 4th 2006, 10:01 AM

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

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

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

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

RonL

**DMT** · July 4th 2006, 03:28 PM

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

**CaptainBlack** · July 5th 2006, 02:50 AM

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

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