Fourier coefficients and partial sum of Fejer

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SUMMARY

The discussion focuses on the application of Fourier coefficients and the Fejér partial sums for a continuously differentiable function f(t) defined on the circle T1. Participants demonstrate the relationship between the Fourier coefficients cr and the Fejér sums σn(f,t), particularly showing that the limit of the maximum deviation gn between f(t) and σn(f,t) approaches zero as n approaches infinity. The calculations confirm that as n increases, the approximation of f(t) by the Fejér sums becomes increasingly accurate, indicating that f is constant in this limit.

PREREQUISITES
  • Understanding of Fourier series and coefficients
  • Knowledge of Fejér's theorem and partial sums
  • Familiarity with limits and continuity in calculus
  • Basic proficiency in mathematical notation and analysis
NEXT STEPS
  • Study the properties of Fourier coefficients in depth
  • Explore Fejér's theorem and its implications for convergence
  • Investigate the relationship between Fourier series and continuous functions
  • Learn about uniform convergence and its significance in analysis
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Mathematicians, students studying Fourier analysis, and anyone interested in the convergence properties of series in functional analysis.

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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:
 
Last edited by a moderator:
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a. b. solved !

c. ...
 
Last edited:

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