We have seen the theory of Hilbert spaces there is a class of functions
for which the Fourier series is convergent. This convergence is in the sense of the norm generated by this space, called the quadratic norm, ie we have obtained.

Fourier series convergence to the midjumpvalue in jumpdiscontinuities points.

But it is important to note that this convergence is in the sense of the norm.
Moreover we have found that given a sequence {u} of elements in space
There exists a function of our space of functions for which {u} are the Fourier coefficients and of trigonometric series so formed, converges to the function in the norm sense of this space
We now want to extend a little more this space of functions and find conditions for puntual and uniform convergence of Fourier series.
It is a fact, in a finite measure space M, as a closed interval, it can possible to shown that if p < q then
Specifically, for the definition of Fourier coefficients is necessary only makes sense that the function be in the space
because if
f ∈ L
1(T) => ∫
10 f(x) dx < ∞ then
f
n = ∫
10f(x) e
2πinx dx
≤ ∫
10f(x) e
2πinx dx =
∫
10f(x) dx =  f 
L1 < ∞
Thus the Fourier coefficients are well defined in this space
Lets consider now the normed space of functions E = (L
1(T),  f 
L1)
 f 
L1 = ∫
10 f(x) dx
It is possible to show that E is Banach space (normed and complete).
Lets considert now consider the sequence space
l
∞(Z) = {{a
n} : sup a
n < ∞}
We can define the normed space
H = (l
∞(Z),  {a
n} 
l∞)
Where the norm is now
 {a
n} 
l∞ = sup a
n
The space H so defined is also a Banach space.
We consider now the application
In this case, it can be shown that the application T, is linear, continuous and injective but not surjective (as we had in the analog case of square integrable functions, see the section on Hilbert spaces).
To see that it is only necessary
Lemma (Riemann  Lebesgue)
From this it follows the assertion that T is not surjective, for example take the series
Theorem of punctually Convergence 1 (Hölder's Criterion)
Let the function
Piecewise continuous, such that satisfies
the
Rightleft Hölder condition then
The following is a theorem similar, except that the Dini condition is slightly weaker than
Hölder condition
Theorem of punctually Convergence 1 (Dini's Criterion)
Let the function
and lets
then
Observation 1: differentiable functions verified trivially of Hölder' and Dini's Criterion because
Thus, f differentiable => Fourier series of f converges punctually
Observation 2: also this function satiesfies the Hölder' and Dini's Criterion although it is not differentiable in some pontis
Observation 3: There exist
such that its Fourier series diverges almost everywhere
Observation 4: Since 1968 we know that if
Fourier series of f converges almost everywhere (p = 2 is a Carlesson's result, shown in 1966)
Theorem 1: Uniformly convergence of Fourier series
If we have
Then the Fourier series converges uniformly and the convergenceratio is
Finally we will see a uniform convergence theorem in the context of
Sobolev Spaces , This theorem provides us a intermediate function's space in which we can delete continuous or differentiable functions conditions, but to understand it, we see first the following lemma which returns the Fourier coefficients of the derivatives of a function (obviously, if it's differentiable) in terms of Fourier coefficients:
Lema
Theorem 2: uniformly convergence of Fourier Series in the context of Sobolev spaces
Then the Fourier series of f converges uniformly convergence with ratio
What this important theorem says join with the previous Lemma is that the uniformly convergent of the Fourier series of a function is only necessary that the function has only a little more than "half derivative".
In summary, we have uniform convergence of Fourier Series of a function, it is only necessary