Fejér's theorem

In mathematics, Fejér's theorem,Leopold Fejér
Untersuchungen über Fouriersche Reihen
''Mathematische Annalen''
vol. 58
1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following:

Explanation of Fejér's Theorem's

Explicitly, we can write the Fourier series of ''f'' as
$f(x)= \sum_^ c_n \, e^$where the nth partial sum of the Fourier series of ''f'' may be written as
:$s_n(f,x)=\sum_^nc_ke^,$
where the Fourier coefficients $c_k$ are
:$c_k=\frac\int_^\pi f(t)e^dt.$
Then, we can define
:$\sigma_n(f,x)=\frac\sum_^s_k(f,x) = \frac\int_^\pi f(x-t)F_n(t)dt$
with ''F''_''n'' being the ''n''th order Fejér kernel.

Then, Fejér's theorem asserts that

$\lim_ \sigma_n (f, x) = f(x)$

with uniform convergence. With the convergence written out explicitly, the above statement becomes

$\forall \epsilon > 0 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), < \epsilon, \, \forall x \in \mathbb$

Proof of Fejér's Theorem

We first prove the following lemma:

Proof: Recall the definition of $D_n(x)$, the Dirichlet Kernel:$D_n(x) = \sum_^n e^.$We substitute the integral form of the Fourier coefficients into the formula for $s_n(f,x)$ above

s_n(f,x)=\sum_^n c_ke^ = \sum_^n frac\int_^\pi f(t)e^dt e^ = \frac \int_^\pi f(t) \sum_^n e^ \, dt = \frac \int_^\pi f(t) \, D_n(x-t) \, dt.Using a change of variables we get

$s_n(f,x) = \frac \int_^\pi f(x-t) \, D_n(t) \, dt.$

This completes the proof of Lemma 1.

We next prove the following lemma:

Proof: Recall the definition of the Fejér Kernel $F_n(x)$

$F_n(x) = \frac \sum_^D_k(x)$As in the case of Lemma 1, we substitute the integral form of the Fourier coefficients into the formula for $\sigma_n(f,x)$

\sigma_n(f,x)=\frac\sum_^s_k(f,x) = \frac\sum_^ \frac \int_^\pi f(x-t) \, D_k(t) \, dt = \frac \int_^\pi f(x-t) \, frac\sum_^ D_k(t)\, dt = \frac \int_^\pi f(x-t) \, F_n(t) \, dtThis completes the proof of Lemma 2.

We next prove the 3rd Lemma:

Proof: a) Given that $F_n$ is the mean of $D_n$, the integral of which is 1, by linearity, the integral of $F_n$ is also equal to 1.

b) As $D_n(x)$ is a geometric sum, we get a simple formula for $D_n(x)$ and then for $F_n(x)$,using De Moivre's formula :

$F_n(x) = \frac \sum_^\frac = \frac \frac \geq 0$

c) For all fixed $\delta > 0$,

$\int_ F_n (x) \, dx = \frac \int_ \frac \, dx \leq \frac \int_ \frac \, dx$

This shows that the integral converges to zero, as $n$ 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

$\forall \epsilon > 0 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), < \epsilon, \, \forall x \in \mathbb$

We want to find an expression for $, \sigma_n(f,x) - f(x) ,$. We begin by invoking Lemma 2:

$\sigma_n(f,x)= \frac \int_^\pi f(x-t) \, F_n(t) \, dt.$By Lemma 3a we know that

\sigma_n(f,x) - f(x) = \frac \int_^\pi f(x-t) \, F_n(t) \, dt - f(x) = \frac \int_^\pi f(x-t) \, F_n(t) \, dt - f(x) \frac \int_^\pi F_n(t) \, dt = \frac \int_^\pi f(x) \, F_n(t) \, dt=\frac \int_^\pi (x-t)-f(x)\, F_n(t) \, dt.

Applying the triangle inequality yields

, \sigma_n(f,x) - f(x) , = , \frac \int_^\pi (x-t)-f(x)\, F_n(t) \, dt, \leq \frac \int_^\pi , (x-t)-f(x)\, F_n(t), \, dt = \frac \int_^\pi , f(x-t)-f(x), \, , F_n(t), \, dt, and by Lemma 3b, we get

$, \sigma_n(f,x) - f(x) , = \frac \int_^\pi , f(x-t)-f(x), \, F_n(t) \, dt.$We now split the integral into two parts, integrating over the two regions $, t, \leq \delta$ and $\delta \leq , t, \leq \pi$.

$, \sigma_n(f,x) - f(x) , = \left( \frac \int_ , f(x-t)-f(x), \, F_n(t) \, dt \right) + \left( \frac \int_ , f(x-t)-f(x), \, F_n(t) \, dt \right)$The motivation for doing so is that we want to prove that $\lim_ , \sigma_n(f,x) - f(x) , =0$. 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 π,π We invoke the theorem that every periodic function on π,πthat is continuous is also bounded and uniformily continuous. This means that $\forall \epsilon > 0,\exist \delta > 0: , x-y, \leq \delta \implies , f(x)-f(y), \leq \epsilon$. Hence we can rewrite the integral 1 as follows

$\frac \int_ , f(x-t)-f(x), \, F_n(t) \, dt \leq \frac \int_ \epsilon \, F_n(t) \, dt = \frac\epsilon \int_ \, F_n(t) \, dt$Because $F_n(x) \geq 0, \forall x\in \mathbb$ and $\delta \leq \pi$$\frac\epsilon \int_ \, F_n(t) \, dt \leq \frac\epsilon \int_^\pi \, F_n(t) \, dt$By Lemma 3a we then get for all n

$\frac\epsilon \int_^\pi \, F_n(t) \, dt = \epsilon$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 $M=\sup_ , f(t),$

$\frac \int_ , f(x-t)-f(x), \, F_n(t) \, dt \leq \frac \int_ 2M \, F_n(t) \, dt = \frac \int_F_n(t) \, dt$We are now ready to prove that $\lim_ , \sigma_n(f,x) - f(x) , =0$. We begin by writing

$, \sigma_n(f,x) - f(x) , \leq \epsilon \, + \frac \int_F_n(t) \, dt$Thus,$\lim_ , \sigma_n(f,x) - f(x) , \leq \lim_ \epsilon \, + \lim_ \frac \int_F_n(t) \, dt$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 $\lim_ , \sigma_n(f,x) - f(x) , =0$, 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 $\sigma_n (f,x)$ with $s_n (f,x)$. 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 $L^2(-\pi, \pi)$ diverges has to be measure zero. This fact, called Lusins conjecture or Carleson's theorem, 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

:$\sigma_n(x_0) \to \frac\left(f(x_0+0)+f(x_0-0)\right).$

Existence or divergence to infinity of the Cesàro 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

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{{DEFAULTSORT:Fejer's theorem
Fourier series
Theorems in approximation theory