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...