In mathematics, Fejér's theorem,[1][2] named after Hungarian mathematician Lipót Fejér, states the following:[3]

Fejér's Theorem — Let be a continuous function with period , let be the nth partial sum of the Fourier series of , and let be the sequence of Cesàro means of the sequence , that is the sequence of arithmetic means of . Then the sequence converges uniformly to on as n tends to infinity.

Explanation of Fejér's Theorem's

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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

 

Proof of Fejér's Theorem

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We first prove the following lemma:

Lemma 1 — The nth partial sum of the Fourier series   may be written using the Dirichlet Kernel as:  

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:

Lemma 2 — The nth Cesaro sum   may be written using the Fejér Kernel as:  

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:

Lemma 3 — The Fejer Kernel has the following 3 properties:

  • a)  
  • b)  
  • c) For all fixed  ,  

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 an 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 [-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  . 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

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In fact, Fejér's theorem can be modified to hold for pointwise convergence.[3]

Modified Fejér's Theorem — Let   be continuous at  , then   converges pointwise as n goes to infinity.

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.[4] 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, was proven in 1966 by L. Carleson.[4] We can however prove a corollary relating which goes as follows:

Corollary — Let  . If   converges to s as n goes to infinity, then   converges to s as n goes to infinity.

A more general form of the theorem applies to functions which are not necessarily continuous (Zygmund 1968, Theorem III.3.4). Suppose that f is in L1(-π,π). If the left and right limits f(x0±0) of f(x) exist at x0, or if both limits are infinite of the same sign, then

 

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 σn is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).

References

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  1. ^ Lipót Fejér, « Sur les fonctions intégrables et bornées », C.R. Acad. Sci. Paris, 10 décembre 1900, 984-987, .
  2. ^ Leopold Fejér, Untersuchungen über Fouriersche Reihen, Mathematische Annalen, vol. 58, 1904, 51-69.
  3. ^ a b "Introduction", An Introduction to Hilbert Space, Cambridge University Press, pp. 1–3, 1988-07-21, retrieved 2022-11-14
  4. ^ a b Rogosinski, W. W.; Rogosinski, H. P. (December 1965). "An elementary companion to a theorem of J. Mercer". Journal d'Analyse Mathématique. 14 (1): 311–322. doi:10.1007/bf02806398. ISSN 0021-7670.