convergence of Fourier series
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the question of whether the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
converges to a given
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is researched by a field known as classical harmonic analysis, a branch of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of
pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
,
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
,
absolute convergence In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
, ''L''''p'' spaces,
summability method In mathematics, a divergent series is an infinite series that is not Convergent series, convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit of a sequence, limit. If a series converges, t ...
s and the Cesàro mean.


Preliminaries

Consider ''f'' an
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
function on the interval . For such an ''f'' the Fourier coefficients \widehat(n) are defined by the formula :\widehat(n)=\frac\int_0^f(t)e^\,\mathrmt, \quad n \in \Z. It is common to describe the connection between ''f'' and its Fourier series by :f \sim \sum_n \widehat(n) e^. The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined: :S_N(f;t) = \sum_^N \widehat(n) e^. The question of whether a Fourier series converges 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? Before continuing, the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
must be introduced. Taking the formula for \widehat(n), inserting it into the formula for S_N and doing some algebra gives that :S_N(f)=f * D_N where ∗ stands for the periodic
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
and D_N is the Dirichlet kernel, which has an explicit formula, :D_n(t)=\frac. The Dirichlet kernel is ''not'' a positive kernel, and in fact, its norm diverges, namely :\int , D_n(t), \,\mathrmt \to \infty a fact that plays a crucial role in the discussion. The norm of ''D''''n'' in ''L''1(T) coincides with the norm of the convolution operator with ''D''''n'', acting on the space ''C''(T) of periodic continuous functions, or with the norm of the linear functional ''f'' → (''S''''n''''f'')(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 f is an
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
function, :\left, \widehat f(n)\\le for K a constant that only depends on f. If f is a
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 ...
function, :\left, \widehat f(n)\\le . If f\in C^p :\left, \widehat(n)\\le . If f\in C^p and f^ has
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...
\omega_p, :\left, \widehat(n)\\le and therefore, if f is in the α-
Hölder class Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
:\left, \widehat(n)\\le .


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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
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 In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...
). The Dirichlet–Dini Criterion states that: if ''ƒ'' is 2–periodic, locally integrable and satisfies :\int_0^ \left, \frac2 - \ell \ \frac < \infty, then (S''n''''f'')(''x''0) converges to ℓ. This implies that for any function ''f'' of any
Hölder class Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
''α'' > 0, the Fourier series converges everywhere to ''f''(''x''). It is also known that for any periodic function of
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 ...
, the Fourier series converges everywhere. See also
Dini test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition Let ...
. In general, the most common criteria for pointwise convergence of a periodic function ''f'' are as follows: * If ''f'' satisfies a Holder condition, then its Fourier series converges uniformly. * If ''f'' is of bounded variation, then its Fourier series converges everywhere. * If ''f'' is continuous and its Fourier coefficients are absolutely summable, then the Fourier series converges uniformly. There exist continuous functions whose Fourier series converges pointwise but not uniformly; see Antoni Zygmund, ''
Trigonometric Series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...
'', vol. 1, Chapter 8, Theorem 1.13, p. 300. However, the Fourier series of a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in ''L''1(T) and the Banach–Steinhaus
uniform boundedness principle In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
. As typical for existence arguments invoking the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
, 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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 Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the res ...
shows that for a given continuous function the Fourier series converges almost everywhere. It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function ''f'' defined for all ''x'' in by :f(x) = \sum_^ \frac \sin\left \left( 2^ +1 \right) \frac\right


Uniform convergence

Suppose f\in C^p, and f^ has
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...
\omega; then the partial sums of the Fourier series converge to the function with speed :, f(x)-(S_Nf)(x), \le K \omega(2\pi/N) for a constant K that does not depend upon f , nor p, nor N. This theorem, first proved by D Jackson, tells, for example, that if f satisfies the \alpha-
Hölder condition In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that : , f(x) - f(y) , \leq C\, ...
, then :, f(x)-(S_Nf)(x), \le K . If f is 2\pi periodic and absolutely continuous on ,2\pi/math>, then the Fourier series of f converges uniformly, but not necessarily absolutely, to f.Stromberg (1981), Exercise 6 (d) on p. 519 and Exercise 7 (c) on p. 520.


Absolute convergence

A function ''ƒ'' has an absolutely converging Fourier series if :\, f\, _A:=\sum_^\infty , \widehat(n), <\infty. Obviously, if this condition holds then (S_N f)(t) converges absolutely for every ''t'' and on the other hand, it is enough that (S_N f)(t) converges absolutely for even one ''t'', then this condition holds. 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 so everywhere. The family of all functions with absolutely converging Fourier series is a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
(the operation of multiplication in the algebra is a simple multiplication of functions). It is called the
Wiener algebra In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by , is the space of absolutely convergent Fourier series. Here denotes the circle group. Banach algebra structure The norm of a function is given by :\, f\, =\s ...
, after
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
, 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 Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
. Finally, a short elementary proof was given by
Donald J. Newman Donald Joseph (D. J.) Newman (July 27, 1930 – March 28, 2007) was an American mathematician. He gave simple proofs of the prime number theorem and the Hardy-Ramanujan partition formula. He excelled on multiple occasions at the annual Putnam com ...
in 1975. If f belongs to a α-Hölder class for α > 1/2 then :\, f\, _A\le c_\alpha \, f\, _,\qquad \, f\, _K:=\sum_^ , n, , \widehat(n), ^2\le c_\alpha \, f\, ^2_ for \, f\, _ the constant in the
Hölder condition In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that : , f(x) - f(y) , \leq C\, ...
, c_\alpha a constant only dependent on \alpha; \, f\, _K is the norm of the Krein algebra. Notice that the 1/2 here is essential—there are 1/2-Hölder functions, which do not belong to the Wiener algebra. Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only O(1/n^\alpha) and then not summable. If ''ƒ'' is of
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 ...
''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 ''L''2, which is a direct transcription of general
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
results. According to the
Riesz–Fischer theorem In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz ...
, if ''ƒ'' is
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
then :\lim_\int_0^\left, f(x)-S_N(f) (x) \^2\,\mathrmx=0 ''i.e.'',  S_N f converges to ''ƒ'' in the norm of ''L''2. It is easy to see that the converse is also true: if the limit above is zero, ''ƒ'' must be in ''L''2. So this is an
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
condition. 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 ''L''p,  S_N(f) converges to ''ƒ'' in the ''L''''p'' norm. The original proof uses properties of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s and
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...
s, and another proof, due to
Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Aust ...
relies upon the Riesz–Thorin interpolation theorem. For ''p'' = 1 and infinity, the result is not true. The construction of an example of divergence in ''L''1 was first done by
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
(see below). For infinity, the result is a corollary of the
uniform boundedness principle In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
. If the partial summation operator ''SN'' is replaced by a suitable
summability kernel In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability ...
(for example the ''Fejér sum'' obtained by convolution with the
Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lip ...
), 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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
was posed by Nikolai Lusin in the 1920s. It was resolved positively in 1966 by
Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
. His result, now known as
Carleson's theorem Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the res ...
, tells the Fourier expansion of any function in ''L''2 converges almost everywhere. Later on, Richard Hunt generalized this to ''L''''p'' for any ''p'' > 1. Contrariwise,
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, as a student at the age of 19, in his very first scientific work, constructed an example of a function in ''L''1 whose Fourier series diverges almost everywhere (later improved to diverge everywhere).
Jean-Pierre Kahane Jean-Pierre Kahane (11 December 1926 – 21 June 2017) was a French mathematician with contributions to harmonic analysis. Career Kahane attended the École normale supérieure and obtained the ''agrégation'' of mathematics in 1949. He then wor ...
and
Yitzhak Katznelson Yitzhak Katznelson ( he, יצחק כצנלסון; born 1934) is an Israeli mathematician. Katznelson was born in Jerusalem. He received his doctoral degree from the University of Paris in 1956. He is a professor of mathematics at Stanford Uni ...
proved that for any given set ''E'' of
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
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 In mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a diverge ...
) converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence a_n is Cesàro summable to some ''a'' if :\lim_\frac\sum_^n a_k=a. 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 S_N with an appropriate notion. Hence we define :K_N(f;t)=\frac\sum_^ S_n(f;t), \quad N \ge 1, 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\sum_^ D_n. The main difference is that Fejér's kernel is a positive kernel.
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 Exp ...
states that the above sequence of partial sums converge uniformly to ''ƒ''. This implies much better convergence properties * If ''ƒ'' is continuous at ''t'' then the Fourier series of ''ƒ'' is summable at ''t'' to ''ƒ''(''t''). If ''ƒ'' is continuous, its Fourier series is uniformly summable (i.e. K_N f converges uniformly to ''ƒ''). * For any integrable ''ƒ'', K_N f converges to ''ƒ'' 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 ''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 S_N(f;t) converges to some value ''x'', it is also summable to it, so from the first summability property above, ''x'' = ''ƒ''(''t'').


Order of growth

The order of growth of Dirichlet's kernel is logarithmic, i.e. :\int , D_N(t), \,\mathrmt = \frac\log N+O(1). See
Big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
for the notation ''O''(1). The actual value 4/\pi^2 is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for ''some'' constant ''c'' we have :\int , D_N(t), \,\mathrmt > 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 quantitative versions of some of the previous results. For any continuous function ''f'' and any ''t'' one has :\lim_ \frac=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_ \frac=\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_ \frac=\infty. 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 :S_N(f;t_1,t_2)=\sum_\widehat(n_1,n_2)e^ which are known as "square partial sums". Replacing the sum above with :\sum_ 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. 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 Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contrib ...
.


Notes


References


Textbooks

*Dunham Jackson ''The theory of Approximation'', AMS Colloquium Publication Volume XI, New York 1930. * 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 In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...
'', Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. * Yitzhak Katznelson, ''An introduction to harmonic analysis'', Third edition. Cambridge University Press, Cambridge, 2004. * Karl R. Stromberg, ''Introduction to classical analysis'', Wadsworth International Group, 1981. :''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. Zygmund's book was greatly expanded in its second publishing in 1959, however.''


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 Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, "Une série de Fourier–Lebesgue divergente presque partout", ''
Fundamenta Mathematicae ''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical syst ...
'' 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 divergence everywhere. In French. *
Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
, "On convergence and growth of partial sums of Fourier series", ''
Acta Math. ''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research jo ...
'' 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 ''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Spr ...
911, Springer-Verlag, Berlin-New York, 1982. :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. *
Dunham Jackson Dunham Jackson (July 24, 1888 in Bridgewater, Massachusetts – November 6, 1946) was a mathematician who worked within approximation theory, notably with trigonometrical and orthogonal polynomials. He is known for Jackson's inequality. He ...
, ''Fourier Series and Orthogonal Polynomials'', 1963 * D. J. Newman, "A simple proof of Wiener's 1/f theorem", Proc. Amer. Math. Soc. 48 (1975), 264–265. *
Jean-Pierre Kahane Jean-Pierre Kahane (11 December 1926 – 21 June 2017) was a French mathematician with contributions to harmonic analysis. Career Kahane attended the École normale supérieure and obtained the ''agrégation'' of mathematics in 1949. He then wor ...
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Yitzhak Katznelson Yitzhak Katznelson ( he, יצחק כצנלסון; born 1934) is an Israeli mathematician. Katznelson was born in Jerusalem. He received his doctoral degree from the University of Paris in 1956. He is a professor of mathematics at Stanford Uni ...
, "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 trigonometric Fourier series everywhere", ''C. R. Acad. Sci. Paris'' 329 (1999), 693–697. * Jean-Pierre Kahane, ''Some random series of functions'', second edition. Cambridge University Press, 1993. :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. Fourier series