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