Fejér's theorem

In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: 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 Fn being the nth order Fejér kernel. Then, Fejér's theorem asserts that with uniform convergence. With the convergence written out explicitly, the above statement becomes 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: 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 [-π,π]. We invoke the theorem that every periodic function on [-π,π] 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 .