![]() | ||
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.
Contents
- Preliminaries
- Magnitude of Fourier coefficients
- Pointwise convergence
- Uniform convergence
- Absolute convergence
- Norm convergence
- Convergence almost everywhere
- Summability
- Order of growth
- Multiple dimensions
- References
Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean.
Preliminaries
Consider ƒ an integrable function on the interval [0,2π]. For such an ƒ the Fourier coefficients
It is common to describe the connection between ƒ and its Fourier series by
The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined:
The question here is: Do the functions
Before continuing, the Dirichlet kernel must be introduced. Taking the formula for
where ∗ stands for the periodic convolution and
The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely
a fact that plays a crucial role in the discussion. The norm of Dn in L1(T) coincides with the norm of the convolution operator with Dn, acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional ƒ → (Snƒ)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n → ∞.
Magnitude of Fourier coefficients
In applications, it is often useful to know the size of the Fourier coefficient.
If
for
If
If
If
and therefore, if
Pointwise convergence
There are many known sufficient conditions for the Fourier series of a function to converge 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 converges to the average of the left and right limits (but see Gibbs phenomenon).
The Dirichlet–Dini Criterion states that: if ƒ is 2π–periodic, locally integrable and satisfies
then (Snƒ)(x0) converges to ℓ. This implies that for any function ƒ of any Hölder class α > 0, the Fourier series converges everywhere to ƒ(x).
It is also known that for any periodic function of bounded variation, the Fourier series converges everywhere. See also Dini test. In general, the most common criteria for pointwise convergence of a periodic function f are as follows:
There exist continuous functions whose Fourier series converges pointwise but not uniformly; see Antoni Zygmund, Trigonometric Series, vol. 1, Chapter 8, Theorem 1.13, p. 300.
However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L1(T) and the Banach–Steinhaus uniform boundedness principle. As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive. It shows that the family of continuous functions whose Fourier series converges at a given x is of first Baire category, in the Banach space of continuous functions on the circle. So in some sense pointwise convergence is atypical, and for most continuous functions the Fourier series does not converge at a given point. However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere.
Uniform convergence
Suppose
for a constant
This theorem, first proved by D Jackson, tells, for example, that if
If
Absolute convergence
A function ƒ has an absolutely converging Fourier series if
Obviously, if this condition holds then
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 ƒ has absolutely converging Fourier series and is never zero, then 1/ƒ has absolutely converging Fourier series. The original proof of Wiener's theorem was difficult; a simplification using the theory of Banach algebras was given by Israel Gelfand. Finally, a short elementary proof was given by Donald J. Newman in 1975.
If
for
If ƒ is of bounded variation and belongs to a α-Hölder class for some α > 0, it belongs to the Wiener algebra.
Norm convergence
The simplest case is that of L2, which is a direct transcription of general Hilbert space results. According to the Riesz–Fischer theorem, if ƒ is square-integrable then
i.e.,
If 2 in the exponents above is replaced with some p, the question becomes much harder. It turns out that the convergence still holds if 1 < p < ∞. In other words, for ƒ in Lp,
If the partial summation operator SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ p < ∞.
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 L2 converges almost everywhere. This result is now known as Carleson's theorem. Later on Richard Hunt generalized this to Lp 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, Andrey Kolmogorov, as a student at the age of 19, in his very first scientific work, constructed an example of a function in L1 whose Fourier series diverges almost everywhere (later improved to diverge everywhere).
It might be interesting to note that Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a continuous function ƒ such that the Fourier series of ƒ fails to converge on any point of E.
Summability
Does the sequence 0,1,0,1,0,1,... (the partial sums of Grandi's series) converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence
It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it.
To discuss summability of Fourier series, we must replace
and ask: does
where
The main difference is that Fejér's kernel is a positive kernel. Fejér's theorem states that the above sequence of partial sums converge uniformly to ƒ. This implies much better convergence properties
Results about summability can also imply results about regular convergence. For example, we learn that if ƒ is continuous at t, then the Fourier series of ƒ cannot converge to a value different from ƒ(t). It may either converge to ƒ(t) or diverge. This is because, if
Order of growth
The order of growth of Dirichlet's kernel is logarithmic, i.e.
See Big O notation for the notation O(1). It should be noted that the actual value
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 quantitative versions of some of the previous results. For any continuous function f and any t one has
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
Upon examining 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
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
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 circular partial sums. Almost everywhere convergence of "square partial sums" (as well as more general polygonal partial sums) in multiple dimensions was established around 1970 by Charles Fefferman.