Measure of non-periodicity of almost periodic functions

In summary, the Fourier transform of an almost periodic function can be evaluated to determine if the function is periodic or not, but the evaluation requires some assumptions about the type of noise.
  • #1
reterty
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As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a periodic function (not including the constant component of its Fourier series), with respect to the argument for the main period, is equal to zero. In the theory of almost periodic functions, the concept of an almost period is introduced. So, a similar integral of an almost periodic function for almost a period will be different from zero. Its value divided by this almost period and the largest of the amplitudes of the harmonics of the Fourier series will be a dimensionless quantity characterizing the degree of non-periodicity of this almost periodic function. Is my criterion correct and useful?
 
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  • #2
The problem with "almost periodic" is that those functions are essentially undefined with such a broad and simple description.

In general, you will have to define over what conditions and how you will do the evaluation. The Fourier transform assumes periodicity based on the limits you choose to integrate over. It can not tell you about any periodicity on the order of 1 day, if you only collect data for 1 minute. So, I think just defining your window and looking at the Fourier transform is the only thing we can do. Then for different circumstances, you'll get different spectral data out, which may still be hard to interpret.

In practice, this subject is most commonly described as "jitter" of electronic signals. It is an extremely well studied and hugely important subject. The treatment tends to be statistical in nature. People invariably end up making some (powerful) assumptions about the type of deviation, like "gaussian noise", for example, to allow them to analyze the more general cases. Do some searching about jitter for more information. IRL, we would look at the spectral width of the "almost periodic" frequency out of a Fourier series or spectrum analyzer. This is also called "phase noise".
 

1. What is the measure of non-periodicity of almost periodic functions?

The measure of non-periodicity of almost periodic functions is a mathematical concept that quantifies how closely a function resembles a periodic function. It is a measure of the deviation of the function from being strictly periodic.

2. How is the measure of non-periodicity calculated?

The measure of non-periodicity is typically calculated using the Fourier transform of the function. It involves taking the absolute value of the Fourier coefficients and summing them over all frequencies. The resulting value is then normalized by dividing it by the maximum value of the Fourier coefficients.

3. What is the significance of the measure of non-periodicity in mathematics?

The measure of non-periodicity is an important concept in the study of almost periodic functions, which have applications in various fields such as physics, engineering, and signal processing. It helps in characterizing the behavior of these functions and understanding their properties.

4. Can the measure of non-periodicity be used to classify functions as periodic or non-periodic?

No, the measure of non-periodicity alone cannot be used to classify functions as periodic or non-periodic. A function can have a non-zero measure of non-periodicity and still be considered periodic. However, a function with a measure of non-periodicity equal to zero is always periodic.

5. How does the measure of non-periodicity relate to the concept of almost periodicity?

The measure of non-periodicity is closely related to the concept of almost periodicity. It is a quantitative measure of how close a function is to being almost periodic. A function with a small measure of non-periodicity is considered to be more almost periodic, while a function with a large measure of non-periodicity is less almost periodic.

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