1

How to show that \begin{align} \left| \frac{\int_0^\infty \cos(ax)e^{-x^4} dx}{\int_0^\infty \cos(bx)e^{-x^4} dx} \right| \le 1 \end{align} if $a\ge b \ge 0$.

This is what I did.

One has to show then that \begin{align} \frac{ \left|\int_0^\infty \cos(ax)e^{-x^4} dx \right| }{ \left|\int_0^\infty \cos(bx)e^{-x^4} dx \right|} \le 1 \end{align}

Or that \begin{align} \left|\int_0^\infty \cos(ax)e^{-x^4} dx \right| \le \left|\int_0^\infty \cos(bx)e^{-x^4} dx \right| \end{align} But how to show the last inequality?

Boby
  • 6,381

1 Answers1

4

You cannot, because your inequality does not hold: just take $a=4$ and $b=3.5$.

If we take the function $$ \psi: \xi \mapsto \int_{0}^{+\infty}\cos(\xi x)e^{-x^4}\,dx $$ this is its graph over $[0,10]$:

$\hspace{2cm}$enter image description here

and it is not monotonic in absolute value, even if fast-decaying.

Jack D'Aurizio
  • 361,689
  • Thanks. You see the reason I thought that this inequality was correct is because I know it is true when the exponential is of degree 1 or degree 2. That since $\int_0^\infty \cos(ax) e^{-x} dx=\frac{1}{1+a^2}$ and $\int_0^\infty \cos(ax) e^{-x^2} dx= 0.5*\sqrt{\pi} e^{a^2/4}$. With this we get \begin{align} \frac{\int_0^\infty \cos(ax) e^{-x} }{\int_0^\infty \cos(bx) e^{-x} } =\frac{1+b^2}{1+a^2} \ \frac{\int_0^\infty \cos(ax) e^{-x^2} }{\int_0^\infty \cos(bx) e^{-x^2} } =e^{\frac{b^2-a^2}{4}}\end{align} Both of which are less than 1 if $a>b$. So, why is it not the same for $x^4$? – Boby Jul 01 '16 at 16:36
  • @Boby: because the Fourier transform is slightly oscillating, while that does not happen in the other cases. – Jack D'Aurizio Jul 01 '16 at 16:43
  • Could you explain more along the lines of Fourier analysis? I am very curios. Also, is it correct to say that if we exponent $x^k$ then the inequality is true for all $0< k\le2$ ? – Boby Jul 01 '16 at 16:46
  • Interesting fact, it probably deserves a separate question. – Jack D'Aurizio Jul 01 '16 at 17:20
  • Ok. I will post a new question. – Boby Jul 01 '16 at 17:22
  • see this question http://math.stackexchange.com/questions/1846072/bounds-on-fk-a-b-frac-int-0-infty-cosa-x-e-xk-dx-int-0-infty You can also modify it any way you want. – Boby Jul 01 '16 at 17:28
  • @Boby: thank you. A very interesting (and probably quite hard) problem has born here. – Jack D'Aurizio Jul 01 '16 at 19:57
  • I am very glad you liked this problem. At first, I did not realize that Fourier analysis would be at the heart of this problem. Thank you for all of your ideas. It seems like you are the person with the correct experience and set of skills to study this. In case if you decided to pursue this problem, let me know if you get anywhere. I will be very interested in seeing your results. – Boby Jul 01 '16 at 20:04