Expressing Function Norm Using Fourier Coefficients

In summary, the conversation discusses the possibility of expressing the norm of a function in terms of Fourier coefficients. It is mentioned that the L^2-norm can be expressed as the square root of the sum of the squares of the coefficients. A sample question is given, asking for the Fourier coefficients of a specific function and how to express the norm in terms of these coefficients. It is clarified that the discussion is referring to the L^2 norm, which is a measure of energy. The summary concludes by stating that the L^2 norm of the given function is equal to the sum of the squares of its coefficients.
  • #1
8
0
Hi,

I was wondering if it is possible to express the norm of a function in terms of Fourier coefficient. If so, how do you go through it if given a particular function.

Thanks
 
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  • #2
What norm? The [tex]L^2[/tex]-norm can be expressed, and it is the square root of the sum of the squares of the coefficients.
 
  • #3
Here is a sample question:

What are the Fourier coefficients of the function f(x)=ae^(-ix)+b+ce^(ix)? And express the norm in terms of Fourier coefficients.

They don't mention if it is the L^2 norm or not.
 
  • #4
well, it's only the L2 norm (a measure of energy) in which the L2 norm of the time-domain function (over one period) is equal to the L2 of the frequency-domain data (the Fourier coefficients).

i think the L2 norm of your f(x) is a2 + b2 + c2.
 

1. What are Fourier coefficients?

Fourier coefficients are numerical values that represent the amplitude and phase of individual sinusoidal functions that make up a given periodic function. They are used in Fourier analysis to break down complex periodic functions into simpler components.

2. How are Fourier coefficients calculated?

Fourier coefficients are typically calculated using an integral function called the "Fourier transform" or its discrete version, the "Discrete Fourier transform". These functions take a periodic function as input and output the corresponding Fourier coefficients.

3. What is the significance of Fourier coefficients?

The significance of Fourier coefficients lies in their ability to break down complex periodic functions into simpler components, making it easier to analyze and manipulate. They are also used in various applications such as signal processing, image compression, and solving differential equations.

4. Can Fourier coefficients be negative?

Yes, Fourier coefficients can be negative. This indicates that the corresponding sinusoidal component has a phase shift of 180 degrees or π radians. In other words, the component is inverted or flipped on the x-axis compared to a positive coefficient.

5. How do Fourier coefficients relate to the Fourier series?

The Fourier series is a mathematical representation of a periodic function using a sum of sinusoidal functions with different frequencies and amplitudes. The Fourier coefficients are used to calculate the values of these sinusoidal components. Therefore, the Fourier coefficients are an essential part of the Fourier series.

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