- #1

Dschumanji

- 153

- 1

## 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 periodic function g with fundamental period R/m where m is a positive integer not equal to 1. Also suppose that the integral of the product of f and g over the interval [0, R] is equal to 0 (f and g are orthogonal). Can the fundamental period of f+g be less than R?

## Homework Equations

## The Attempt at a Solution

If m is equal to 1 or you do not assume orthogonality, then it is simple to produce examples where f+g has a fundamental period less than R. I can't seem to find an example of f and g being orthogonal and their sum having a fundamental period less than R. All of the examples I can think of must have a fundamental period of R. This is leading me to believe that f+g cannot have a fundamental period that is less than R if f and g are orthogonal. All of my attempts to prove that the fundamental period must be R have failed. I feel that there is maybe not enough information given to form a proof.