Fourier series like to converge to the midpoint of the jump of a function under certain
conditions. The condition given for convergence in the following theorem is due to Dini. It is a
generalization of the usual theorem presented in elementary books on Fourier series methods.
[3].
Recall
lt→ixm+ f (t) ≡ f (x+), and lt→imx − f (t) ≡ f (x − )
Theorem 11.4.1Let f be a periodic function of period 2π which is in R
([− π,π])
.Suppose at some x, f
(x+ )
and f
(x− )
both exist and that the function
| |
y → ||f-(x-−-y)−-f (x−-)+-f (x+-y)-− f-(x+)||≡ h (y) (11.11)
| y |
(11.11)
is in R
((0,π])
which means
∫ π
εl→im0+ h (y)dy exists (11.12)
ε
(11.12)
Then
lim Snf (x) = f (x+)+-f-(x− ). (11.13)
n→ ∞ 2
(11.13)
Proof:
∫ π
Snf (x ) = Dn (x − y) f (y)dy
−π
Change variables x − y → y and use the periodicity of f and D_{n} along with the formula for
D_{n}
(y)
to write this as
∫
S f (x ) = πD (y)f (x − y)
n −π n
∫ π ∫ 0
= Dn (y)f (x− y)dy + Dn (y)f (x − y)dy
∫0π −π
= Dn (y)[f (x − y)+ f (x +y)]dy
0 (( ) )[ ]
∫ π1-sin--n-+-12-y-- f (x−-y)+-f-(x+-y)
= 0 π sin (y) 2 dy. (11.14)
2
Note the function
1sin ((n+ 1)y)
y → -------(y2)----,
π sin 2
while it is not defined at 0, is at least bounded and by L’Hospital’s rule,
so defining it to equal this value at 0 yields a continuous, hence Riemann integrable
function and so the above integral at least makes sense. Also from the property that
∫_{−π}^{π}D_{n}
(t)
dt = 1,
∫ π
f (x+)+ f (x− ) = Dn (y)[f (x+ )+ f (x− )]dy
−∫π
= 2 π D (y)[f (x+ )+ f (x− )]dy
0 n
∫ π 1sin ((n + 1)y)
= π----sin(y2)----[f (x+ )+ f (x− )]dy
0 2
∫ | |
= ε2||f-(x-−-y)−-f (x−-)+-f (x+-y)-− f-(x+)||-y(-) dy
ε1 | y |2 sin y2
∫ ε2||f (x − y)− f (x− )+ f (x+ y) − f (x+)||
≤ ||----------------y----------------||dy
ε1
∫ π| |
= ||f (x−-y)-− f-(x− )+-f-(x-+-y)−-f (x+-)||dy
ε1 | y |
∫ π ||f (x− y)− f (x − )+ f (x + y)− f (x+ )||
− ||---------------y-----------------||dy
ε2
Letting
{εk}
be any sequence of positive numbers converging to 0, this shows
{ | | }
∫ π ||f (x− y)− f (x− )+ f (x + y)− f (x+)|| ∞
||------------2-sin(y)-------------||dy
εk 2 k=1
is a Cauchy sequence because the difference between the k^{th} and the m^{th} terms,
ε_{k}< ε_{m}, is no larger than the difference between the k^{th} and m^{th} terms of the
sequence
{∫ π ||f (x− y)− f (x− )+ f (x + y)− f (x+)|| } ∞
||--------------------------------||dy
εk y k=1
θ ∫ π
2K δ- + 1 |f (x − y)− f (x− )+ f (x+ y) − f (x+)|dy.
θ δ δ
(Why?) This proves the corollary.
As pointed out by Apostol [3], where you can read more of this sort of thing, this is a very
remarkable result because even though the Fourier coeficients depend on the values of the
function on all of
[− π,π ]
, the convergence properties depend in this theorem on very local
behavior of the function.