Fourier coefficients and partial sum of Fejer

In summary, the conversation discusses a continuously differentiable function f(t) that is twice differentiable over the circle T1. The Fourier coefficients of f(t) and the partial sum of Fejer, σn(f,t), are also mentioned. The main goal is to demonstrate that the limit of gn, the maximum difference between f(t) and σn(f,t) as n approaches infinity, is equal to 0 when f is a constant function. The solution involves calculating the Fourier coefficients using the given formula and using them to solve for gn.
  • #1
Dassinia
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0

Homework Statement


f(t) a continuously differentiable function twice over the circle T1
cr its Fourier coefficients and σn(f,t) partial sum of Fejer.
a.Demonstrate that
http://imageshack.us/a/img94/5992/ds35.png b. Consider k as -n≤k≤n , using cr coefficients calculate
http://imageshack.us/a/img542/8306/7jz2.png

c. Consider a function gn= max | f(t)-σn(f,t) | t in the circle T1
show that lim n*gn=0 n->∞
f is constant

Homework Equations


The Attempt at a Solution


I've calculated
http://img9.imageshack.us/img9/7775/mz30.png
I end up with a +1.. the summation about f(t) is in relative numbers Z using Fourier but here it is |r|>n is it from here that I can "delete" the +1 ?

b. :confused:
c. :confused:
 
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  • #2
a. b. solved !

c. ...
 
Last edited:

1. What are Fourier coefficients?

Fourier coefficients are used in Fourier series to represent a periodic function as a sum of simple trigonometric functions. They represent the amplitude and phase of each individual frequency component in the Fourier series.

2. How are Fourier coefficients calculated?

The Fourier coefficients can be calculated using the formula c_n = (1/T) * ∫f(x)e^(-inxω_0)dx, where T is the period of the function, ω_0 is the fundamental frequency, and n is the frequency component. This integral is taken over one period of the function.

3. What is the significance of the Fejer partial sum?

The Fejer partial sum is a sequence of partial sums of a Fourier series, where each term is the average of the first n terms of the Fourier series. It is significant because it provides a better approximation of the original function compared to the partial sums of the Fourier series, and it converges pointwise to the original function.

4. How is the Fejer partial sum different from the Fourier partial sum?

The Fejer partial sum is different from the Fourier partial sum in that it is an average of the first n terms of the Fourier series, rather than just the first n terms. This results in a smoother and more accurate approximation of the original function.

5. What is the application of Fourier coefficients and the Fejer partial sum in real-world problems?

Fourier coefficients and the Fejer partial sum have numerous applications in various fields such as signal processing, image and sound compression, data analysis, and solving differential equations. They are also used in the study of periodic phenomena and in the development of efficient algorithms for solving problems in physics, engineering, and mathematics.

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