Period length of product of two periodic function, Fourier Series

In summary, the conversation discusses the period length of a product of two periodic functions and whether it can be determined based on the individual periods of the functions. It is mentioned that the period of the product may be the larger period of the two functions, but this has not been proven. The speaker is only interested in the product of functions, as they have already proven the properties for the sum of functions.
  • #1
keenPenguin
21
3
Hello,

I am currently dealing with Fourier Series, and a question about the period length of a product of two periodic functions is bothering me. If you have two different periodic functions with different (or the same) period and you sum them up, what happens to the period? Can this generally be said? Also (and this is what I'm especially interested in), what happens to the period of the product of these functions?

Another question: Fourier coefficients, no matter if real or complex, all go over any periodic interval of a function. Actually what happens if you (deliberately) chose an interval which is two times or three times the period? Does it (generally) still work?

kP
 
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  • #2
Pardon me for archeological tendencies but I found this question very relevant:

keenPenguin said:
Hello,
...
If you have two different periodic functions with different (or the same) period and you sum them up, what happens to the period? Can this generally be said? Also (and this is what I'm especially interested in), what happens to the period of the product of these functions?
...

I got feeling that period for product of periodic functions is the bigger period among the two functions, but I was unable to prove it.

I'm only interested on product of functions. [For sum of functions I waas able to prove the needed properties]
 

FAQ: Period length of product of two periodic function, Fourier Series

1. What is the period of a product of two periodic functions?

The period of a product of two periodic functions is the smallest positive value of T for which the product function repeats itself. This means that the product function will have the same value at t and t + T, where t is any real number.

2. How is the period of a product of two periodic functions calculated?

The period of a product of two periodic functions can be calculated by finding the least common multiple (LCM) of the individual periods of the two functions. This can be done by using the Euclidean algorithm or by finding the prime factors of the two periods and multiplying the highest powers of each prime together.

3. Can the period of a product of two periodic functions be infinite?

No, the period of a product of two periodic functions cannot be infinite. Since the product function repeats itself, it must have a finite period. However, the period can be arbitrarily large, depending on the individual periods of the two functions.

4. What is the relationship between the period of a product of two periodic functions and the periods of the individual functions?

The period of a product of two periodic functions is a multiple of the individual periods of the two functions. This means that the product function will repeat itself at intervals of the individual periods. Additionally, if the two individual periods are rational multiples of each other, the period of the product function will be the LCM of the two individual periods.

5. How does the period of a product of two periodic functions affect the Fourier series representation?

The period of a product of two periodic functions affects the Fourier series representation in that the fundamental frequency will be inversely proportional to the period. This means that as the period increases, the fundamental frequency decreases and vice versa. Additionally, the Fourier series coefficients will also be affected by the period, as they are calculated using the period and the individual periods of the two functions.

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