Show that an integral is well defined

**Ase** · April 20th 2010, 07:43 AM

Hi

I am trying to show that for f belonging to L^2(-pi;pi) the integral that defines the complex Fourier Coefficients is well defined. In other words what I need to show is that

int_from -pi to pi(|f(x)*exp(-i*k*x)|dx) < infinity (limited)

I was thinking that since f belongs to L^2(-pi;pi) then the integral of this will be finite. Further more since the inteval is limited (-pi;pi) and k belonging to the set of integers the complex exponential function would also be finite.

Am I right?

**Laurent** · April 20th 2010, 07:51 AM

Originally Posted by Ase

Hi

I am trying to show that for f belonging to L^2(-pi;pi) the integral that defines the complex Fourier Coefficients is well defined. In other words what I need to show is that

int_from -pi to pi(|f(x)*exp(-i*k*x)|dx) < infinity (limited)

I was thinking that since f belongs to L^2(-pi;pi) then the integral of this will be finite. Further more since the inteval is limited (-pi;pi) and k belonging to the set of integers the complex exponential function would also be finite.

Am I right?

You didn't prove anything. Basically, since $|e^{-ikx}|=1$ , you have to prove that if $\int_{-\pi}^\pi |f(x)|^2 dx<\infty$ , then $\int_{-\pi}^\pi |f(x)| dx<\infty$ . The advice is to use Cauchy-Schwarz inequality (or the elementary inequality $a\leq\frac{a^2+1}{2}$ ).

**Ase** · April 20th 2010, 08:10 AM

Hi Laurent

Thanks a lot. I will get right at it.

**surjective** · April 20th 2010, 12:52 PM

For $f \in L^{2}(-\pi,\pi)$ I would like to show that:

$\int_{-\pi}^{\pi}|f(x)e^{-ikx}|< \infty$

I have done the following:

$\int_{-\pi}^{\pi}|f(x)e^{-ikx}|dx= \int_{-\pi}^{\pi}|f(x)(\cos(kx)-i\sin(kx)|dx\leq \int_{-\pi}^{\pi}|f(x)dx|$

How do I continue (I tried to use Cauchy-Sjcwarz inequality as suggested but it did not lead my anywhere)

Thanks a bunch.

**Laurent** · April 20th 2010, 12:58 PM

Originally Posted by surjective

For $f \in L^{2}(-\pi,\pi)$ I would like to show that:

$\int_{-\pi}^{\pi}|f(x)e^{-ikx}|< \infty$

I have done the following:

$\int_{-\pi}^{\pi}|f(x)e^{-ikx}|dx= \int_{-\pi}^{\pi}|f(x)(\cos(kx)-i\sin(kx)|dx\leq \int_{-\pi}^{\pi}|f(x)dx|$

How do I continue (I tried to use Cauchy-Sjcwarz inequality as suggested but it did not lead my anywhere)

Thanks a bunch.

To apply Cauchy-Schwarz, write $\int f(x) dx = \int f(x)g(x) dx$ where $g(x)=1$ .

Alternatively, use the simple inequality I gave in the previous post: $\int f\leq \frac{1}{2}\Big((\int f^2)+(\int 1)\Big)=\frac{1}{2}\Big((\int f^2) + 2\pi\Big)<\infty$ .

**surjective** · April 20th 2010, 01:24 PM

Here is what I have done:

$<br /> \int_{-\pi}^{\pi}|f(x)|dx=\int_{-\pi}^{\pi}|f(x)g(x)|dx \leq \int_{-\pi}^{\pi}|f(x)||g(x)|dx \leq \int_{-\pi}^{\pi}|f(x)||f(x)|dx < \infty<br />$

Correct?

**Laurent** · April 20th 2010, 01:29 PM

Originally Posted by surjective

Here is what I have done:

$<br /> \int_{-\pi}^{\pi}|f(x)|dx=\int_{-\pi}^{\pi}|f(x)g(x)|dx \leq \int_{-\pi}^{\pi}|f(x)||g(x)|dx \leq \int_{-\pi}^{\pi}|f(x)||f(x)|dx < \infty<br />$

Correct?

You should take a look at what Cauchy-Schwarz inequality for integrals is; because I can't see how you're applying it here.

**surjective** · April 20th 2010, 02:12 PM

Hello,

Let me try one more time:

$\int_{-\pi}^{\pi}|f(x)|dx= \int_{-\pi}^{\pi}|f(x)g(x)|dx \leq \left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}|g(x)|^{2}dx \right)^{\frac{1}{2}} \leq$

$\left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}} < \infty$

What do you think?

**Laurent** · April 21st 2010, 12:15 AM

Originally Posted by surjective

Hello,

Let me try one more time:

$\int_{-\pi}^{\pi}|f(x)|dx= \int_{-\pi}^{\pi}|f(x)g(x)|dx \leq \left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}|g(x)|^{2}dx \right)^{\frac{1}{2}} \leq$

$\left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}1 dx \right)^{\frac{1}{2}} =\sqrt{2\pi}\left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}< \infty$

What do you think?

I did the corrections above.

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