Convergence of Fourier series Guide, Meaning , Facts, Information and Description
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, spaces, summability methods and Cesŕro mean.
We shall consider f an integrable function on the interval
[0,2π]. For such an f we define the Fourier coefficients
Before continuing we need to introduce Dirichlet's kernel. Taking the formula for , inserting it into the formula for and doing some algebra will give that
Preliminaries
by the formula
It is common to describe the connection between f and its Fourier series
by
The notation 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
The question we will be interested is: do the functions
(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.
where * stands for convolution and is the Dirichlet kernel which has an explicit formula,
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.
The simplest case is that of . If f is square-integrable then
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 . In
other words, for f in ,
converges to f in the 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
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 converges almost everywhere. Later on Hunt
generalized this to 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 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.
We say about a function f that it has an absolutely converging Fourier series if
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.
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 is Cesŕro summable to some a if
To discuss summability of Fourier series, we must replace with an appropriate notion. Hence we define
The order of growth of Dirichlet's kernel is logarithmic, i.e.
This estimate entails quantitive versions of some of the previous results. For any continuous function f and any t one has
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
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).
This is an Article on Convergence of Fourier series. Page Contains Information, Facts Details or Explanation Guide About Convergence of Fourier series Convergence at a given point.
Norm convergence
i.e. converges to f in the norm of . It
is easy to see that the opposite is true too: if the limit above is zero,
f must be in . So this is an if and only if
condition.is easy, because even divergence at a single point
implies divergence in norm, so the examples discussed
above can be used. The construction of an example of divergence in
was done by Kolmogorov.
Convergence almost everywhere
Absolute convergence
Obviously, if this condition holds then converges
absolutely for every t and on the other hand, it is enough that
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.Summability
It is not difficult to see that if a series converges to some a then it is also summable to it.
and ask: does converge to f? is no longer
associated with Dirichlet's kernel, but with Fejér's kernel, namely
where is Fejér's kernel,
The main difference is that Fejér's kernel is a positive kernel. This implies much better convegence properties
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 converges to some value x, it is also summable to it, so from the first summability property above, x = f(t).Order of growth
See Big O notation for the notation O(1). It should be noted that the actual value is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for some constant c we have
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.
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,
The equivalent problem for divergence everywhere is open. Sergei Konyagin managed to construct an integrable function such that for every t one has
It is not known whether this example is best possible. The only bound from the other direction known is log n.Multiple dimensions
which are known as "square partial sums". Replacing the sum above with
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 while for circular partial sums it is of the order of . References
Textbooks
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
This is the first proof that the Fourier series of a continuous function might diverge. In German
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.
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 spaces; two attempts at simplifying the proof; and a book that gives a self contained exposition of it.
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.
The Konyagin paper proves the divergence result discussed above. A simpler proof that gives only log log n can be found in Kahane's book.