Derivatives of 2pi-periodic functions

[Charles49]

Suppose $$u(x)$$ is periodic with period $$2\pi$$. Also $$m\le u(x)\le M$$.

Then is it possible for some derivatives of $$u(x)$$ to be outside $$[m, M]$$? In other words, can any derivative be 2pi-periodic and have a different amplitude?

When $$u(x)$$ is a sine curve, then it is not true because, the frequency of $$u(x)$$ is exactly 1. The derivatives cannot change the amplitude since the frequency is exactly one and the chain rule doesn't affect the amplitude.

However, I think it is possible when you construct a periodic function which is a sum of infinite sine curves. I am not sure how to construct such an example which is smooth. The smoothness requirement ensures that it is not easy to construct such a function which can be decomposed into a finite number of sine curves.

Any thoughts?

 

[berkeman]

Suppose $$u(x)$$ is periodic with period $$2\pi$$. Also $$m\le u(x)\le M$$.

Then is it possible for some derivatives of $$u(x)$$ to be outside $$[m, M]$$? In other words, can any derivative be 2pi-periodic and have a different amplitude?

When $$u(x)$$ is a sine curve, then it is not true because, the frequency of $$u(x)$$ is exactly 1. The derivatives cannot change the amplitude since the frequency is exactly one and the chain rule doesn't affect the amplitude.

However, I think it is possible when you construct a periodic function which is a sum of infinite sine curves. I am not sure how to construct such an example which is smooth. The smoothness requirement ensures that it is not easy to construct such a function which can be decomposed into a finite number of sine curves.

Any thoughts?

Why are you constraining the frequency to be one?

 

[SteamKing]

You've got frequency and amplitude mixed up. For the sine function, the amplitude is 1.0 and the period is 2pi.

 

[AlephZero]

You don't need to worry about an infinite number of functions. Just think about something like

u(x) = sin(x) + sin(100x)
u'(x) = cos(x) + 100 cos(100 x)

 

[Charles49]

You don't need to worry about an infinite number of functions. Just think about something like

u(x) = sin(x) + sin(100x)
u'(x) = cos(x) + 100 cos(100 x)

Thanks, AlephZero, i feel stupid now...