Recent content by 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.

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...

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...

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...

That is an interesting approach! I was wondering how you came up with the counter example.

Verty, you're a genius! I've spent days on this problem and you solved it in one night. I clearly need to get better at coming up with counter examples! Thank you so much for your help!

Isn't the following a counter example to your claim, though? "There are examples of functions whose sum does not have a period that is the larger of the two periods. For example if f(x)=cos(x)-cos((2/3)x) and g(x)=cos((2/3)x), then their sum is cos(x). The fundamental period of f is 6*pi and...

I made a little bit of progress. I was able to prove that if the fundamental period of f+g is less than R, then it must be of the form R/n where n is an integer such that n>1, m does not divide n, and n does not divide m. To do this, I wrote f(x) as f(x) = (f+g)(x) - g(x) and replaced x with...

Hmmm, I will see what I can do with this. Thank you! When you say something like shift(f), do you mean like f(x+A) or f(x)+A for some value A?

Fair enough. I have been trying to prove that sum of f and g must be equal to R if f and g are orthogonal. My method is to use proof by contradiction. I start by assuming that f+g has a fundamental period T that is less than R. Since T is the fundamental period of f+g and f+g has a period R, the...

Any hint as to what that shortcut may be?

They are not the same concept. If there exists a value T such that f(x+T)=f(x) for all x, then f has a period of T. If P is the smallest value such that f(x+P)=f(x) for all x, then P is the fundamental period of f. For example, cos(x) has a period of 4*pi, but it's fundamental period is 2*pi...

It is easy to prove that the period is T, but it does not prove it is the fundamental period.

I have been trying to find examples of orthogonal functions whose sum has a fundamental period less than R. I have had no luck so far.

Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...