Carleson's theorem
Encyclopedia
Carleson's theorem is a fundamental result in mathematical analysis
establishing the pointwise
(Lebesgue
) almost everywhere convergence of Fourier series
of L2 functions, proved by . The name is also often used to refer to the extension of the result by to Lp functions for p ∈ (1, ∞) (also known as the Carleson–Hunt theorem) and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods.
The analogous result for Fourier integrals can be formally stated as follows:
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
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 proved 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
shows that if one replaces ordinary summation by Cesàro summation
then the Fourier series of any continuous function converges pointwise everywhere to the function. Further, it is easy to show that the Fourier series of any L2 function converges to it in L2 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 L2 functions was conjectured by , and the problem was known as Luzin's conjecture (up until its proof by ). showed that the analogue of Carleson's result for L1 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 sn of the Fourier series of a function in Lp was
proved by Kolmogorov–Seliverstov–Plessner for p = 2, by G. H. Hardy
for p = 1, and by Littlewood–Paley for p > 1 . 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 L1 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. This, in turn, inspired a much simplified proof of the L2 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 Lp for p > 1 was stated to be a "rather obvious" extension of the case p = 2 in Carleson's paper, and was proved by . Carleson's result was improved further by
to the space Llog+(L)log+log+(L) and by to the space Llog+(L)log+log+log+(L). (Here log+(L) is log(L) if L>1 and 0 otherwise, and if φ is a function then
φ(L) stands for the space of functions f such that φ(f(x)) is integrable.)
improved Kolmogorov's counterexample by finding functions with everywhere-divergent Fourier series in a space slightly larger than Llog+(L)1/2.
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 Llog+(L).
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 L2.
A fundamental property of the Carleson operator is that it is a bounded (non-linear) map from Lp(R) to itself for 1 < p < ∞. The Carleson–Hunt theorem follows easily from this (and in fact from slightly weaker estimates).
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
establishing the pointwise
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values...
(Lebesgue
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
) almost everywhere convergence of Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of L2 functions, proved by . The name is also often used to refer to the extension of the result by to Lp functions for p ∈ (1, ∞) (also known as the Carleson–Hunt theorem) and the analogous results for pointwise almost everywhere convergence of Fourier integrals, 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:- Let ƒ be an Lp periodic functionPeriodic functionIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
for some p ∈ (1, ∞), with Fourier coefficients
. Then
- for almost every x.
The analogous result for Fourier integrals can be formally stated as follows:
- Let ƒ ∈ Lp(R) for some p ∈ (1, ∞) have Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. Then
- for almost every x ∈ R.
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 pointwisePointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
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
Bounded variation
In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
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 proved 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
Fejér's theorem
In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequence of Cesàro means of the sequence of partial sums of the Fourier series of f converges uniformly to f on...
shows that if one replaces ordinary summation by Cesàro summation
Cesàro summation
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
then the Fourier series of any continuous function converges pointwise everywhere to the function. Further, it is easy to show that the Fourier series of any L2 function converges to it in L2 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 L2 functions was conjectured by , and the problem was known as Luzin's conjecture (up until its proof by ). showed that the analogue of Carleson's result for L1 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 sn of the Fourier series of a function in Lp was
proved by Kolmogorov–Seliverstov–Plessner for p = 2, by G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
for p = 1, and by Littlewood–Paley for p > 1 . 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 L1 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. This, in turn, inspired a much simplified proof of the L2 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 Lp for p > 1 was stated to be a "rather obvious" extension of the case p = 2 in Carleson's paper, and was proved by . Carleson's result was improved further by
to the space Llog+(L)log+log+(L) and by to the space Llog+(L)log+log+log+(L). (Here log+(L) is log(L) if L>1 and 0 otherwise, and if φ is a function then
φ(L) stands for the space of functions f such that φ(f(x)) is integrable.)
improved Kolmogorov's counterexample by finding functions with everywhere-divergent Fourier series in a space slightly larger than Llog+(L)1/2.
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 Llog+(L).
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 L2.
The Carleson operator
The Carleson operator C is a non-linear operator defined byA fundamental property of the Carleson operator is that it is a bounded (non-linear) map from Lp(R) to itself for 1 < p < ∞. The Carleson–Hunt theorem follows easily from this (and in fact from slightly weaker estimates).