Carleson's theorem is a fundamental result in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
establishing the
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
(
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
)
almost everywhere convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set and ...
of
Fourier series of
functions, proved by . The name is also often used to refer to the extension of the result by to functions for (also known as the ''Carleson–Hunt theorem'') and the analogous results for pointwise almost everywhere convergence of
Fourier integral
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s, which can be shown to be equivalent by transference methods.
Statement of the theorem
The result, in the form of its extension by Hunt, can be formally stated as follows:
The analogous result for Fourier integrals can be formally stated as follows:
History
A fundamental question about Fourier series, asked by Fourier himself at the beginning of the 19th century, is whether the Fourier series of a continuous function converges
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
to the function.
By strengthening the continuity assumption slightly one can easily show that the Fourier series converges everywhere. For example, if a function has
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
then its Fourier series converges everywhere to the local average of the function. In particular, if a function is continuously differentiable then its Fourier series converges to it everywhere. This was proven by Dirichlet, who expressed his belief that he would soon be able to extend his result to cover all continuous functions. Another way to obtain convergence everywhere is to change the summation method. For example,
Fejér's theorem In mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen''vol. 58 1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following:
Explanation of Fejér's Theorem's
Ex ...
shows that if one replaces ordinary summation by
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
then the Fourier series of any continuous function converges uniformly to the function. Further, it is easy to show that the Fourier series of any function converges to it in norm.
After Dirichlet's result, several experts, including Dirichlet, Riemann, Weierstrass and Dedekind, stated their belief that the Fourier series of any continuous function would converge everywhere. This was disproved by
Paul du Bois-Reymond, who showed in 1876 that there is
a continuous function whose Fourier series diverges at one point.
The almost-everywhere convergence of Fourier series for functions was postulated by , and the problem was known as ''Luzin's conjecture'' (up until its proof by ). showed that the analogue of Carleson's result for is false by finding such a function whose Fourier series diverges almost everywhere (improved slightly in 1926 to diverging everywhere). Before Carleson's result, the best known estimate for the partial sums of the Fourier series of a function in was
proved by Kolmogorov–Seliverstov–Plessner for , by
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
for , and by Littlewood–Paley for . This result had not been improved for several decades, leading some experts to suspect that it was the best possible and that Luzin's conjecture was false. Kolmogorov's counterexample in was unbounded in any interval, but it was thought to be only a matter of time before a continuous counterexample was found. Carleson said in an interview with that he started by trying to find a continuous counterexample and at one point thought he had a method that would construct one, but realized eventually that his approach could not work. He then tried instead to prove Luzin's conjecture since the failure of his counterexample convinced him that it was probably true.
Carleson's original proof is exceptionally hard to read, and although several authors have simplified the argument there are still no easy proofs of his theorem.
Expositions of the original paper include , , , and .
published a new proof of Hunt's extension which proceeded by bounding a
maximal operator Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability p ...
. This, in turn, inspired a much simplified proof of the ''L''
2 result by , explained in more detail in . The books and also give proofs of Carleson's theorem.
showed that for any set of measure 0 there is a continuous periodic function whose Fourier series diverges at all points of the set (and possibly elsewhere). When combined with Carleson's theorem this shows that there is a continuous function whose Fourier series diverges at all points of a given set of reals if and only if the set has measure 0.
The extension of Carleson's theorem to for was stated to be a "rather obvious" extension of the case in Carleson's paper, and was proved by . Carleson's result was improved further by
to the space and by to the space . (Here is if and otherwise, and if is a function then stands for the space of functions such that is integrable.)
improved Kolmogorov's counterexample by finding functions with everywhere-divergent Fourier series in a space slightly larger than .
One can ask if there is in some sense a largest natural space of functions whose Fourier series converge almost everywhere. The simplest candidate for such a space that is consistent with the results of Antonov and Konyagin is .
The extension of Carleson's theorem to Fourier series and integrals in several variables is made more complicated as there are many different ways in which one can sum the coefficients; for example, one can sum over increasing balls, or increasing rectangles. Convergence of rectangular partial sums (and indeed general polygonal partial sums) follows from the one-dimensional case, but the spherical summation problem is still open for .
The Carleson operator
The Carleson operator is the non-linear operator defined by
It is relatively easy to show that the Carleson–Hunt theorem follows from the
boundedness of the Carleson operator from to itself for .
However, proving that it is bounded is difficult, and this was actually what Carleson proved.
See also
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Convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in th ...
References
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* (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)
* "This monograph is a detailed and essentially self-contained treatment of the work of Carleson and Hunt."
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*{{Citation , last1=Zygmund , first1=A. , title=Trigonometric Series. Vol. I, II , orig-year=1935 , title-link=Trigonometric Series , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pre ...
, edition=3rd , series=Cambridge Mathematical Library , isbn=978-0-521-89053-3 , mr=1963498 , year=2002
Theorems in Fourier analysis