Sum of Related Periodic Functions

[Dschumanji]

I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question:

Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c be an integer greater than 1. Is it possible for f(x)+f(cx) to have a fundamental period less than T?

Many simple examples would seem to indicate that the answer is no, but I can't find a proof and have failed to develop my own proof. I searched through many other books on counterexamples and can't seem to find an example that would indicate the answer is yes.

 

[RaulTheUCSCSlug]

I believe that the answer would be no as well. At least that's what I think it would be intuitively. Perhaps a proof could be made by taking the derivative where the integer is then seen as a constant and comparing said constant to the period or something like that. Perhaps a proof could be made based off of the definitions you have given? I'm not sure how exactly you want to go about your proof. Anything that you started or something else you had in mind?

 

[Dschumanji]

I believe that the answer would be no as well. At least that's what I think it would be intuitively. Perhaps a proof could be made by taking the derivative where the integer is then seen as a constant and comparing said constant to the period or something like that. Perhaps a proof could be made based off of the definitions you have given? I'm not sure how exactly you want to go about your proof. Anything that you started or something else you had in mind?

The function may or may not be differentiable. I have been trying to construct a proof (using contradiction) to show that the sum must have a fundamental period of T using only the information given. I have only gotten as far as showing that if the sum has a fundamental period less than T, then that period must be of the form T/d where d is an integer greater than 1, c does not divide d, and d does not divide c. I have hit a road block trying to show that d must be 1. It seems that there is not enough information to finish the proof. If that were the case then there should be a counterexample showing that the fundamental period of the sum can be less than T.

 

[Svein]

Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c be an integer greater than 1. Is it possible for f(x)+f(cx) to have a fundamental period less than T?

I am not exactly sure of what you are asking, but I will present a proof of something that I hope is what you are asking.

Assume f(x) has fundamental period T (i.e. f(x+T)= f(x) for all x). Of course f(x) is also periodic with period 2T, 3T,... but T is the smallest value that the period can have. Now, if f(x) is also periodic with period U ( f(x+U)= f(x) for all x) and U>T, then there must exist an integer n such that n⋅T≤U<(n+1)⋅T. If U>n⋅T, then f(x+(U-n⋅T)) = f((x+U)-n⋅T)=f(x-n⋅T)=f(x), so f would be periodic with period (U-n⋅T). But subtracting n⋅T from the inequality results in 0≤(U-n⋅T)<T, a contradiction (since T is assumed to be the smallest value for the period).

 

[Dschumanji]

I am not exactly sure of what you are asking, but I will present a proof of something that I hope is what you are asking.

Assume f(x) has fundamental period T (i.e. f(x+T)= f(x) for all x). Of course f(x) is also periodic with period 2T, 3T,... but T is the smallest value that the period can have. Now, if f(x) is also periodic with period U ( f(x+U)= f(x) for all x) and U>T, then there must exist an integer n such that n⋅T≤U<(n+1)⋅T. If U>n⋅T, then f(x+(U-n⋅T)) = f((x+U)-n⋅T)=f(x-n⋅T)=f(x), so f would be periodic with period (U-n⋅T). But subtracting n⋅T from the inequality results in 0≤(U-n⋅T)<T, a contradiction (since T is assumed to be the smallest value for the period).

Your proof shows that if a periodic function has a fundamental period T (the smallest period of the function), then any other period of the function must be an integer multiple of T. My question asks if a periodic function f(x) has a fundamental period T, can the sum of the function f(x) and the function f(cx) have a fundamental period less than T (note that c is an integer greater than 1). The important thing here is that we are adding a higher frequency version of f(x) to f(x).

 

[Dschumanji]

Either this topic is really boring or no one else has been able to find any new information. I have been searching through books online and have yet to come across any counterexamples or proofs.