Another Fourier series question
TCL
point of T where f(c+)+f(c-) does NOT exist (not even equal to infty
or -infty).
Can the Fourier series of f still converge at c(to some number, infty
and -infty included)?
[By Fejer's theorem, if f(c+)+f(c-) exists, and if the Fourier series
S of f converges at c, then the sum S at c must be 1/2 (f(c+)
+f(c-) ). ]
-TCL
W^3
<45c3ed52-9514-438b...@t20g2000yqa.googlegroups.com>,
TCL <tl...@cox.net> wrote:
If f is odd, then the Fourier series of f at 0 is just a sum of 0's,
hence converges to 0. There are plenty of odd f's for which f(0+),
f(0-) don't even exist.
TCL
> In article
> <45c3ed52-9514-438b-9e8a-0ad3d7096...@t20g2000yqa.googlegroups.com>,
I made a mistake, let me restate my question:
Let f be a Lebesgue integrable function on the circle T. Let c be a
point of T where lim_{h->0}(f(c+h)+f(c-h)) does NOT exist (not even
equal to infty
or -infty).
Can the Fourier series of f still converge at c(to some number, infty
and -infty included)?
[By Fejer's theorem, if lim_{h->0}(f(c+h)+f(c-h)) exists, and if
the Fourier series
S of f converges at c, then the sum S at c must be 1/2 lim_{h->0}(f(c
+h)+f(c-h)). See Katznelson's An Introduction to Harmonic Analysis. ]
-TCL
Your example is not good for this restated version.
W^3
<7ab8b7e4-6058-4bff...@h19g2000yqb.googlegroups.com>,
TCL <tl...@cox.net> wrote:
Let E = U (a_n,b_n), where (a_n,b_n) is a pwdj sequence of intervals
-> 0+, with the intervals lying in (0,pi) and so small that (1/t)X_E
is in L^1. Set f(t) = X_E. Then f is bounded, hence is in L^1. Take c
= 0 and note lim_{h->0+}(f(h)+f(-h)) = lim_{h->0+}f(h) fails to exist.
But
S_n(f,0) =
(1/2pi)int_(0,pi) [sin(n+1/2)t]/[sin(t/2)] * t((1/t)X_E)dt.
Note that (t/[sin(t/2)])(1/t)X_E is in L^1. By the Riemann Lebesgue
Lemma, S_n(f,0) -> 0 and we're done.
TCL
>
> - Show quoted text -
Is there a theorem (more general than Fejer's) that implies your
example?
My many thanks to all your excellent replies !!
TCL
>
> - Show quoted text -
Ok, I see that your example follows from the Dirichlet-Dini Criterion.
So the theorem I am looking for is a theorem that implies both Fejer's
theorem and this Criterion.
Not sure if such theorems are known.
Maybe this is an open question.