Distance of functions and fourier coefficients

In summary, the distance between two functions is not given by simply taking the integral of their difference. Instead, it is calculated using the square root of that integral, which is known as the L2 norm. This distance is also the Fourier coefficient of each term in the Fourier expansion of a periodic function f(t) that is closest to f(t). This is because the finite Fourier series is the least squares approximation using this distance formula. Other distance formulas, such as the L1 and uniform norms, can also be used, but the L2 norm is preferred in Fourier Analysis.
  • #1
sridhar10chitta
28
0
There are two functions f(t) and g(t); t is the independent variable.
The distance between the two functions will be given by [1/2pi integral{f(t)-g(t)}^2 dt]^1/2 between -pi and +pi.
Apparently, this distance also is the Fourier coefficient of each term in the Fourier
expansion of a periodic function f(t) such that it is closest to f(t).

Why is this so ?
why is not the distance given by f(t)-g(t) simply ?
i.e 1/sqrt(2pi) integral{f(t)-g(t)}dt between -pi and +pi.
 
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  • #2
sridhar10chitta said:
There are two functions f(t) and g(t); t is the independent variable.
The distance between the two functions will be given by [1/2pi integral{f(t)-g(t)}^2 dt]^1/2 between -pi and +pi.
Apparently, this distance also is the Fourier coefficient of each term in the Fourier
expansion of a periodic function f(t) such that it is closest to f(t).

Why is this so ?
No! The distance between two functions is NOT given by that. It is, rather, given by the square root of that function. The way you have it, the triangle inequality would not be true.
In [itex]L_2[/itex], the space of square integrable functions, it can be show that the finite Fourier series is essentially the "least squares approximation" to the function using that distance (with the square root).

why is not the distance given by f(t)-g(t) simply ?
i.e 1/sqrt(2pi) integral{f(t)-g(t)}dt between -pi and +pi.
That wouldn't make much sense would it? For one thing, the "distance from f to g" would be the negative of the "distance from g to f"!

You CAN define "distance from f to g" to be [itex]\int |f(t)- g(t)|dt[/itex] where the integral is taken over whatever interval you are interested in. That is the "L1 norm. You can also define "distance from f to g" to be max|f(t)-g(t)| where the "max" is the largest value that takes on over whatever interval you are interested in. That's called the "uniform norm" because convergence defined in that norm is equivalent to uniform convergence. We use the "root square" distance formula in Fourier Analysis precisely because the finite Fourier series is the "least squares approximation" with that distance formula.
 
  • #3


The reason for using the Fourier coefficient as the distance between two functions is because it takes into account the entire periodic behavior of the functions, rather than just a single point or value. The Fourier coefficient is a measure of how much a particular frequency or component contributes to the overall behavior of the function. By calculating the distance between two functions using the Fourier coefficients, we are essentially comparing their entire periodic behavior and not just a single point or value.

On the other hand, simply calculating the distance as f(t)-g(t) would only give us the difference between the two functions at a particular point, which may not accurately represent the overall behavior of the functions. Additionally, using the Fourier coefficient as the distance also allows us to compare functions with different periods, as the coefficients take into account the entire periodic behavior of the functions.

Furthermore, the use of the Fourier coefficient also allows us to find the closest approximation of one function to another, as the Fourier expansion of a function with a finite number of terms is the closest approximation of that function by a trigonometric polynomial. This means that by finding the distance between two functions using the Fourier coefficients, we are essentially finding the closest approximation of one function to the other.

In summary, the use of Fourier coefficients as the distance between two functions takes into account the entire periodic behavior of the functions and allows for accurate comparison and approximation, making it a more suitable measure of distance than simply taking the difference between the two functions at a single point.
 

What is the distance of functions?

The distance of functions refers to the measure of how different two functions are from each other. It can be calculated using various metrics, such as the L2 norm or the supremum norm.

What are Fourier coefficients?

Fourier coefficients are the coefficients of the Fourier series expansion of a function. They represent the amount of each component frequency present in the function.

How are the distance of functions and Fourier coefficients related?

The distance of functions can be calculated using the Fourier coefficients of the functions. The closer the Fourier coefficients of two functions are, the smaller their distance will be.

Can the distance of functions be used to compare functions of different types?

Yes, the distance of functions can be used to compare functions of different types as long as they have the same domain and range. It is a useful tool for evaluating the similarity between functions and their approximations.

How is the distance of functions useful in signal processing?

The distance of functions is useful in signal processing for analyzing and comparing different signals. It can help identify similarities and differences between signals, and can be used for signal reconstruction and noise reduction.

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