In mathematics, Fejér's theorem,
[Leopold Fejér]
Untersuchungen über Fouriersche Reihen
''Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
''
vol. 58
1904, 51-69. named after
Hungarian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Lipót Fejér, states the following:
Explanation of Fejér's Theorem's
Explicitly, we can write the Fourier series of ''f'' as
where the nth partial sum of the Fourier series of ''f'' may be written as
:
where the Fourier coefficients
are
:
Then, we can define
:
with ''F''
''n'' being the ''n''th order
Fejér kernel.
Then, Fejér's theorem asserts that
with uniform convergence. With the convergence written out explicitly, the above statement becomes
Proof of Fejér's Theorem
We first prove the following lemma:
Proof: Recall the definition of
, the
Dirichlet Kernel:
We substitute the integral form of the Fourier coefficients into the formula for
above
Using a change of variables we get
This completes the proof of Lemma 1.
We next prove the following lemma:
Proof: Recall the definition of the
Fejér Kernel
As in the case of Lemma 1, we substitute the integral form of the Fourier coefficients into the formula for
This completes the proof of Lemma 2.
We next prove the 3rd Lemma:
Proof: a) Given that
is the mean of
, the integral of which is 1, by linearity, the integral of
is also equal to 1.
b) As
is a geometric sum, we get a simple formula for
and then for
,using
De Moivre's formula :
c) For all fixed
,
This shows that the integral converges to zero, as
goes to infinity.
This completes the proof of Lemma 3.
We are now ready to prove Fejér's Theorem. First, let us recall the statement we are trying to prove
We want to find an expression for
. We begin by invoking Lemma 2:
By Lemma 3a we know that
Applying the triangle inequality yields
and by Lemma 3b, we get
We now split the integral into two parts, integrating over the two regions
and
.
The motivation for doing so is that we want to prove that
. We can do this by proving that each integral above, integral 1 and integral 2, goes to zero. This is precisely what we'll do in the next step.
We first note that the function ''f'' is continuous on
Ï€,Ï€ We invoke
the theorem that every periodic function on
Ï€,Ï€that is continuous is also bounded and uniformily continuous. This means that
. Hence we can rewrite the integral 1 as follows
Because
and
By Lemma 3a we then get for all n
This gives the desired bound for integral 1 which we can exploit in final step.
For integral 2, we note that since ''f'' is bounded, we can write this bound as
We are now ready to prove that
. We begin by writing
Thus,
By Lemma 3c we know that the integral goes to 0 as n goes to infinity, and because epsilon is arbitrary, we can set it equal to 0. Hence
, which completes the proof.
Modifications and Generalisations of Fejér's Theorem
In fact, Fejér's theorem can be modified to hold for pointwise convergence.
Sadly however, the theorem does not work in a general sense when we replace the sequence
with
. This is because there exist functions whose Fourier series fails to converge at some point.
However, the set of points at which a function in
diverges has to be measure zero. This fact, called Lusins conjecture or
Carleson's theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the ( Lebesgue) pointwise almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the ...
, was proven in 1966 by L. Carleson.
We can however prove a corollary relating which goes as follows:
A more general form of the theorem applies to functions which are not necessarily continuous . Suppose that ''f'' is in ''L''
1(-Ï€,Ï€). If the left and right limits ''f''(''x''
0±0) of ''f''(''x'') exist at ''x''
0, or if both limits are infinite of the same sign, then
:
Existence or divergence to infinity of the Cesà ro mean is also implied. By a theorem of
Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σ
''n'' is replaced with
(C, α) mean of the Fourier series .
References
* .
{{DEFAULTSORT:Fejer's theorem
Fourier series
Theorems in approximation theory