Convergence of Fourier series

In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis.

Classic harmonic analysis is a branch of pure mathematics. In other words, despite the fact that Fourier series has enormous practical applications, the questions discussed in this article are quite delicate, and do not seem to have much use for an engineer.

We will only repeat in this article properties of Fourier series that we will need. A reader looking for a general introduction would be better served by reading Fourier series first. We will also assume familiarity with various types of convergence. Useful background can be found in pointwise convergence, uniform convergence, absolute convergence, $L^{p}$ spaces, summability methods and Cesàro mean.

Preliminaries

We shall consider f an integrable function on the interval [0,2π]. For such an f we define the Fourier coefficients ${\widehat {f}}(n)$ by the formula

${\widehat {f}}(n)={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)e^{-int}\,dt.$

It is common to describe the connection between f and its Fourier series by

$f\sim \sum {\widehat {f}}(n)e^{int}.$

The notation $\sim$ here means that the sum represents the function in some sense. In order to investigate this more carefully, we need to define the partial sums

$S_{N}(f;t)=\sum _{n=-N}^{N}{\widehat {f}}(n)e^{int}.$

The question we will be interested is: do the functions $S_{N}(f)$ (which are functions of the variable t we omitted in the notation) converge to f and in which sense? Are there conditions on f ensuring this or that type of convergence? This is the main problem discussed in this article.

Before continuing we need to introduce Dirichlet's kernel. Taking the formula for ${\widehat {f}}(n)$, inserting it into the formula for $S_{N}$ and doing some algebra will give that

$S_{N}(f)=f*D_{N}$

where * stands for convolution and $D_{N}$ is the Dirichlet kernel which has an explicit formula,

$D_{n}(t)={\frac {\sin((n+{\frac {1}{2}})t)}{\sin(t/2)}}$

The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely

$\int |D_{n}(t)|\,dt\to \infty$

a fact that will play a crucial role in the discusion.

Convergence at a given point.

There are many known tests that ensure that the series converges at a given point x. For example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). It is also known that for any function of any Hölder class and any function of bounded variation the Fourier series converges everywhere. See also Dini test.

However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. The easiest proof uses the non-boundedness of Dirichlet's kernel and the Banach-Steinhaus uniform boundedness principle and thus is nonconstructive (that is, it shows that a continuous function whose Fourier series does not converge at 0 does exist without actually saying what that function might look like).

An interesting result claims that the family of continuous functions whose Fourier series converges at x is of first Baire category so in some sense this property is atypical, and for most functions the Fourier series does not converge.

Norm convergence

The simplest case is that of $L^{2}$. If f is square-integrable then

$\lim _{N\rightarrow \infty }\int _{0}^{2\pi }\left|f(x)-S_{N}(f)\right|^{2}\,dx=0$

i.e. $S_{N}$ converges to f in the norm of $L^{2}$. It is easy to see that the opposite is true too: if the limit above is zero, f must be in $L^{2}$. So this is an if and only if condition.

If 2 is in the exponents above is replaced with some p, the question becomes much harder. It turns out that it still holds if $1<p<\infty$. In other words, for f in $L^{p}$, $S_{N}(f)$ converges to f in the $L^{p}$ norm. The proof uses properties of holomorphic functions and Hardy spaces. For p = 1 and infinity, this is not true. The construction of an example in the case $p=\infty$ is easy, because even divergence at a single point implies divergence in $L^{\infty }$ norm, so the examples discussed above can be used. The construction of an example of divergence in $L^{1}$ was done by Kolmogorov.

Convergence almost everywhere

The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s and remained open until finally resolved positively in 1966 by Lennart Carleson. Indeed, Carleson showed that the Fourier expansion of any function in $L^{2}$ converges almost everywhere. Later on Hunt generalized this to $L^{p}$ for any p > 1. Despite a number of attempts at simplifying the proof, it is still one of the most difficult results in analysis.

Contrariwise, Kolmogorov, in his very first paper published when he was 21, constructed an example of a function in $L^{1}$ whose Fourier series diverges almost everywhere (later improved to divergence everywhere).

It might be interesting to note that Kahane and Katznelson proved that for any given set E of measure zero, there exists a continuous function f such that the Fourier series of f fails to converge on any point of E.

Absolute convergence

We say about a function f that it has an absolutely converging Fourier series if

$\sum _{n=-\infty }^{\infty }|{\widehat {f}}(n)|<\infty$

Obviously, if this condition holds then $S_{N}(t)$ converges absolutely for every t and on the other hand, it is enough that $S_{N}(t)$ converges absolutely for even one t, then this condition will hold. In other words, for absolute convergence there is no issue of where the sum converges absolutely — if it converges absolutely at one point then it does it everywhere.

The family of all functions with absolutely converging Fourier series is a Banach algebra (the operation of multiplication in the algebra is a simple multiplication of functions). It is called the Wiener algebra, after Norbert Wiener, who proved that if f has absolutely converging Fourier series and is never zero, then 1/f has absolutely converging Fourier series. The original proof of Wiener's theorem was difficult, and its simplification using the theory of Banach algebras is considered as one of its great achievements.

Two useful tests that allow to check whether a function f belongs to the Wiener algebra are as follows: if f belong to a α-Hölder class for α > ½ then it belongs to the Wiener algebra (the ½ here is essential — there are ½-Hölder functions which do not belong to the Wiener algebra). If f is of bounded variation and belongs to a α-Hölder class for any α, it belongs to the Wiener algebra.

Summability

Does the sequence 0,1,0,1,0,1,... converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any series $a_{n}$ is Cesàro summable to some a if

$\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}a_{n}=a$

It is not difficult to see that if a series converges to some a then it is also summable to it.

To discuss summability of Fourier series, we must replace $S_{N}$ with an appropriate notion. Hence we define

$K_{N}(f;t)={\frac {1}{N}}\sum _{n=1}^{N}S_{n}(f;t)$

and ask: does $K_{N}(f)$ converge to f? $K_{N}$ is no longer associated with Dirichlet's kernel, but with Fejér's kernel, namely

$K_{N}(f)=f*F_{N}$

where $F_{N}$ is Fejér's kernel,

$F_{N}={\frac {1}{N}}\sum _{n=1}^{N}D_{n}$

The main difference is that Fejér's kernel is a positive kernel. This implies much better convegence properties

• If f is continuous at t then the Fourier series of f is summable at t to f(t). If f is continuous, its Fourier series is uniformly summable (i.e. $K_{N}$ converges uniformly to f).
• For any integrable f, $K_{N}$ converges to f in the $L^{1}$ norm.
• There is no Gibbs phenomenon.

Results about summability can also imply results about regular convergence. For example, we learn that if f is continuous at t, then the Fourier series of f cannot converge to a value different from f(t). It may either converge to f(t) or diverge. This is because, if $S_{N}(f;t)$ converges to some value x, it is also summable to it, so from the first summability property above, x = f(t).

Order of growth

The order of growth of Dirichlet's kernel is logarithmic, i.e.

$\int |D_{N}(t)|\,dt={\frac {2}{\pi }}\log N+O(1)$

See Big O notation for the notation O(1). It should be noted that the actual value 2/π is both difficult to calculate and of almost no use. The fact that for some constant c we have

$\int |D_{N}(t)|\,dt>c\log N+O(1)$

is quite clear when one examines the graph of Dirichlet's kernel. The integral over the n-th peak is bigger than c/n and therefore the estimate for the harmonic sum gives the logarithmic estimate.

This estimate entails quantitive versions of some of the previous results. For any continuous function f and any t one has

$\lim _{N\to \infty }{\frac {S_{N}(f;t)}{\log N}}=0.$

However, for any order of growth ω(n) smaller than log, this no longer holds and it is possible to find a continuous function f such that for some t,

$\varlimsup _{N\to \infty }{\frac {S_{N}(f;t)}{\omega (N)}}=\infty .$

The equivalent problem for divergence everywhere is open. Sergei Konyagin managed to construct an integrable function such that for every t one has

$\varlimsup _{N\to \infty }{\frac {S_{N}(f;t)}{\sqrt {\log N}}}=\infty .$

It is not known whether this example is best possible. The only bound from the other direction known is log n.

Multiple dimensions

When discussing the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses. For example, in two dimensions, one may define

$S_{N}(f;t_{1},t_{2})=\sum _{|n_{1}|\leq N,|n_{2}|\leq N}{\widehat {f}}(n_{1},n_{2})e^{i(n_{1}t_{1}+n_{2}t_{2})}$

which are known as "square partial sums". Replacing the sum above with

$\sum _{n_{1}^{2}+n_{1}^{2}\leq N^{2}}$

lead to "circular partial sums". The difference between these two definitions is quite notable. For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of $\log ^{2}N$ while for circular partial sums it is of the order of ${\sqrt {N}}$.

Many of the results true for one dimension are wrong or unknown in multiple dimensions. In particular, the equivalent of Carleson's theorem is still open (for both square and circular partial sums).

References

Textbooks

• Nina K. Bary, A treatise on trigonometric series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book. The Macmillan Co., New York 1964.
• Antoni Zygmund, Trigonometric series, Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. ISBN 0521890535
• Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0486633314

The Katznelson book is the one using the most modern terminology and style of the three. The original publishing dates are: Zygmund in 1935, Bari in 1961 and Katznelson in 1968.

Articles referred to in the text

• Paul du Bois Reymond, Ueber die Fourierschen Reihen, Nachr. Kön. Ges. Wiss. Göttingen 21 (1873), 571-582.

This is the first proof that the Fourier series of a continuous function might diverge. In German

• Andrey Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout, Fundamenta math. 4 (1923), 324-328.
• Andrey Kolmogorov, Une série de Fourier-Lebesgue divergente partout, C. R. Acad. Sci. Paris 183 (1926), 1327-1328

The first is a construction of an integrable function whose Fourier series diverges almost everywhere. The second is a strengthening to divergance everywhere. In French.

• Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966) 135-157.
• Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), 235-255. Southern Illinois Univ. Press, Carbondale, Ill.
• Charles Louis Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551-571.
• Michael Lacey and Christoph Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7:4 (2000), 361-370.
• Ole G. Jørsboe and Leif Mejlbro, The Carleson-Hunt theorem on Fourier series. Lecture Notes in Mathematics 911, Springer-Verlag, Berlin-New York, 1982. ISBN 3540111980

This is the original paper of Carleson, where he proves that the Fourier expansion of any continuous function converges almost everywhere; the paper of hunt where he generalizes it to $L^{p}$ spaces; two attempts at simplifying the proof; and a book that gives a self contained exposition of it.

• Jean-Pierre Kahane and Yitzhak Katznelson, Sur les ensembles de divergence des séries trigonométriques, Studia Math. 26 (1966), 305-306

In this paper the authors show that for any set of zero measure there exists a continuous function on the circle whose Fourier series diverges on that set. In French.

• Sergei Vladimirovich Konyagin, On divergence of trigonometrique Fourier series everywhere, C. R. Acad. Sci. Paris 329 (1999), 693-697.
• Jean-Pierre Kahane, Some random series of functions, second addition. Cambridge University Press, 1993. ISBN 0521456029

The Konyagin paper proves the ${\sqrt {\log n}}$ divergence result discussed above. A simpler proof that gives only log log n can be found in Kahane's book.