the condition for Fourier series convergence

the condition for Fourier series convergence

 Jack Yong Li    Feb 08,2014

    I just finished the learning of Fourier series, in many book it says if the Fourier series meet one of the conditions, (1) Direchlet-Jordan (2) Dini-Lipschitz conditions. the Fourier series will convergence, I think even it doesn't  satisfy these 2 conditions, it can convergence. these 2 conditions is not necessary for Fourier series convergence,even both of f(x+) and f(x-) are infinite, but the sum of the f(x+) and f(x-) exist, the Foruier series should convergence.

  So the condition for the Fourier series convergence should be:

   f(x) is a Riemann integrable or absolute integrable function on [-，+] and if f(x+)+f(x-) exist on [-，+],the f(x)'s Forier Series convergence to (f(x+)+f(x-))

Proof

   f(x) is a Riemann integrable or absolute integrable function on [-，+],the partial sum of Fourier series should be

from Euler-Fourier formula

   

     =

     =

we have the equation

                 (θ≠0)

assume that t≠x, the partial Sum should be 

(let t-x=u)

(the definite integral of periodic function has the same value in any period length )

=

=(let t=-u)+

 =

0  is  the Singularity，assume that ε＞0，εcan go to very close to 0，but not equal to 0，we can change the partial sum of Fourier series to:

=

from Riemann lemma ,we know when M goes to +∞，→0，So the Sum equation can be write into:

=0................(1)

from Reimman lemma ,we can build this equation

=0,and then times f(x-)+f(x+)to this equation,

=

so the equation (1) can be changed into：

=

               ................(2)

letε→0，so the integrable length of

goes to 0, also u→0,and f(x+u)=f(x+),f(x-u)=f(x-)

the equation can be changed to:

=

=

=

=

finally =

 f(x) is a Riemann integrable or absolute integrable function on [-，+] and if f(x+)+f(x-) exist on [-，+],the f(x)'s Forier Series convergence to (f(x+)+f(x-))，it is not necessary to meet the Direchlet-Jordan  and  Dini-Lipschitz  conditions