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 classic harmonic analysis, a branch of pure mathematics. Convergence is not necessarily a given in the general case, and there are criteria which need to be met in order for convergence to occur. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ... This article is about functions in mathematics. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ...

Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, L^p spaces, summability methods and Cesàro mean. Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a divergent series is a series that does not converge. ... In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ...

Preliminaries

Consider f an integrable function on the interval [0,2π]. For such an f the Fourier coefficients defined by the formula Integrability is a mathematical concept used in different areas. ...

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, the partial sums need to be defined:

The question we will be interested in 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? This is the main problem discussed in this article.

Before continuing the Dirichlet's kernel needs to be introduced. Taking the formula for , inserting it into the formula for S_N and doing some algebra will give that In mathematical analysis, the Dirichlet kernel is the collection of functions It is named after Johann Peter Gustav Lejeune Dirichlet. ...

where * stands for convolution and D_N is the Dirichlet kernel which has an explicit formula, In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely

a fact that will play a crucial role in the discussion.

Convergence at a given point.

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 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. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Approximation of square wave in 5 steps Approximation of square wave in 25 steps Approximation of square wave in 125 steps In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, (also known as ringing artifacts) is the peculiar manner in which the Fourier series of a... In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is the supremum running over all partitions P = { x1, ..., xn } of the interval [a, b]. In effect, the total variation is the vertical component of... 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. ...

However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel and the Banach-Steinhaus uniform boundedness principle. As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive. It shows that 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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ... The Baire category theorem is an important tool in general topology and functional analysis. ... In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has enough points for certain limit processes. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

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 f is square-integrable then In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, the Riesz-Fischer theorem in real analysis states that a function is square integrable if and only if the corresponding Fourier series converges uniformly in the space . ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...

i.e. S_N converges to f in the norm of L². It is easy to see that the converse is also true: if the limit above is zero, f must be in L². So this is an if and only if condition. ↔ ⇔ ≡ logical symbols representing iff. ...

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 In other words, for f in L^p, S_N(f) converges to f in the L^p norm. The original proof uses properties of holomorphic functions and Hardy spaces, and another proof, due to Salomon Bochner 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 L¹ was first done by Andrey Kolmogorov (see below). For infinity, the result is a more or less trivial corollary of the uniform boundedness principle. In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. ... Salomon Bochner (20 August 1899 - 2 May 1982) was a Polish-American mathematician, known for wide-ranging work in mathematical analysis, probability theory and differential geometry. ... In mathematics, the Riesz-Thorin theorem, often referred to as the ``Riesz-Thorin Interpolation Theorem or the ``Riesz-Thorin Convexity Theorem is a result about ``interpolation of operators. This should not be confused with somewhat different mathematical procedure of interpolation of functions. ... Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in different academic fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity). ... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ...

If partial summation operators S_N is replaced by a suitable summability 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. Later on Richard Hunt generalized this to L^p for any p > 1. Despite a number of attempts at simplifying the proof, it is still one of the most difficult results in analysis. In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... Nikolai Nikolaevich Luzin (Никола́й Никола́евич Лу́зин) (December 9, 1883 - January 28, 1950), Soviet/Russian mathematician. ... The 1920s is sometimes referred to as the Jazz Age or the Roaring Twenties, usually when speaking about the United States. ... Year 1966 (MCMLXVI) was a common year starting on Saturday (link will display full calendar) of the 1966 Gregorian calendar. ... Lennart Carleson (b. ... Richard Allen Hunt is an American mathematician. ...

Contrariwise, Andrey Kolmogorov, in his very first paper published when he was 21, constructed an example of a function in L¹ whose Fourier series diverges almost everywhere (later improved to divergence everywhere). Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in different academic fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity). ...

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 f such that the Fourier series of f fails to converge on any point of E. Jean-Pierre Kahane in 2006 Jean-Pierre Kahane (born in Paris on December 11, 1926) is a French mathematician. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ...

Absolute convergence

We say about a function f that it has an absolutely converging Fourier series if In mathematics, a series is a sum of a sequence of terms. ...

Obviously, if this condition holds then S_N(t) converges absolutely for every t and on the other hand, it is enough that S_N(t) 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.

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; a simplification using the theory of Banach algebras was given by Israel Gelfand. Finally, a short elementary proof was given by Donald Newman in 1975. In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... Norbert Wiener Norbert Wiener (November 26, 1894, Columbia, Missouri – March 18, 1964, Stockholm Sweden) was an American theoretical and applied mathematician. ... Israel Moiseevich Gelfand (Russian: ) (born in 1913) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ... Year 1975 (MCMLXXV) was a common year starting on Wednesday (link will display full calendar) of the Gregorian calendar. ...

Two useful tests that allow to check whether a function f belongs to the Wiener algebra are as follows: if f belongs 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.

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 a_n is Cesàro summable to some a if The infinite series 1 − 1 + 1 − 1 + · · · or is sometimes called Grandis series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. ... In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ...

It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it. In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ...

To discuss summability of Fourier series, we must replace S_N with an appropriate notion. Hence we define

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 In mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. ...

where F_N is Fejér's kernel,

The main difference is that Fejér's kernel is a positive kernel. This implies much better convergence properties

• If f is continuous at t then the Fourier series of f is summable at t to f(t). If f is continuous, its Fourier series is uniformly summable (i.e. K_N converges uniformly to f).
• For any integrable f, K_N converges to f in the L¹ norm.
• There is no Gibbs phenomenon.

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 S_N(f;t) converges to some value x, it is also summable to it, so from the first summability property above, x = f(t).

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 4 / π² is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for some constant c we have For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

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. See harmonic series (music) for the (related) musical concept. ...

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

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

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 log²N while for circular partial sums it 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 both square and circular partial sums).

References

Textbooks

• 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, Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. ISBN 0-521-89053-5
• Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0-486-63331-4

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, Une série de Fourier-Lebesgue divergente presque partout, Fundamenta math. 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, On convergence and growth of partial sums of Fourier series, Acta Math. 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 911, Springer-Verlag, Berlin-New York, 1982. ISBN 3-540-11198-0

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.

• D. J. Newman, A simple proof of Wiener's 1/f theorem, Proc. Amer. Math. Soc. 48 (1975), 264-265.
• Jean-Pierre Kahane and Yitzhak Katznelson, 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 trigonometrique Fourier series everywhere, C. R. Acad. Sci. Paris 329 (1999), 693-697.
• Jean-Pierre Kahane, Some random series of functions, second addition. Cambridge University Press, 1993. ISBN 0-521-45602-9

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.