# Fourier series

*of a function in a system of functions which are orthonormal on an interval *

The series

whose coefficients are determined by

(1) |

These coefficients are called the Fourier coefficients of . In general it is assumed that is square integrable on . For many systems this requirement can be relaxed by replacing it by another which ensures the existence of all the integrals in (1).

The Fourier series in the trigonometric system is defined for every function that is integrable on . It is the series

(2) |

with coefficients

(3) |

Fourier series for functions in several variables are constructed analogously. A further generalization leads to Fourier coefficients and Fourier series for elements of a Hilbert space.

The theory of Fourier series in the trigonometric system has been most thoroughly developed, and these were the first examples of Fourier series. If one has in mind Fourier series in the trigonometric system, it is usual to talk simply of Fourier series, without indicating the system by which they are constructed.

Fourier series form a considerable part of the theory of trigonometric series. Fourier series first appeared in the papers of J. Fourier (1807) devoted to an investigation of the problems of heat conduction. He suggested representing a function given on by the trigonometric series (2) with coefficients determined by (3). Such a choice of coefficients is natural from many points of view. For example, if the series (2) converges uniformly to , then term-by-term integration leads to the expressions for the coefficients and given in (3). These formulas had been obtained already by L. Euler (1777) by term-by-term integration.

Using (3) the Fourier series (2) can be constructed for every function that is integrable over . Integrability of the function can be understood in various senses, for example integrability according to Riemann or Lebesgue. Depending on this, one speaks of Fourier–Riemann series, Fourier–Lebesgue series, etc. The concepts of the Riemann and the Lebesgue integral themselves arose to a considerable extent in connection with research on Fourier series. The modern presentation of the theory of Fourier series was developed after the construction of the Lebesgue integral, and since then it has developed mainly as the theory of Fourier–Lebesgue series. Below it is assumed that the function has period and is Lebesgue integrable over the period.

In the theory of Fourier series one studies the relation between the properties of functions and the properties of their Fourier series; in particular, one investigates questions on the representation of functions by Fourier series.

The proof of a minimum property of the partial sums of Fourier series goes back to the work of F. Bessel (1828): Given an , then among all the trigonometric polynomials of order ,

the smallest value of the integral

is attained for the partial sum of the Fourier series (2) of :

This smallest value is equal to

This implies the Bessel inequality

which is satisfied for every function in .

The system of trigonometric functions is a closed system (cf. Closed system of elements (functions)), that is, if , then the Parseval equality

is valid, where are the Fourier coefficients of . In particular, for functions in the series

(4) |

is convergent. The converse assertion also holds: If for a system of numbers the series (4) converges, then these numbers are the Fourier coefficients of a certain function (F. Riesz and E. Fischer, 1907).

The Fourier coefficients of any integrable function tend to zero. This statement is called the Riemann–Lebesgue theorem. B. Riemann proved it for Fourier–Riemann series and e dimension','../l/l057830.htm','Lebesgue function','../l/l057840.htm','Lebesgue inequality','../l/l057850.htm','Lebesgue integral','../l/l057860.htm','Lebesgue measure','../l/l057870.htm','Lebesgue summation method','../l/l057940.htm','Lebesgue theorem','../l/l057950.htm','Measure','../m/m063240.htm','Metric space','../m/m063680.htm','Metric theory of functions','../m/m063700.htm','Orthogonal series','../o/o070370.htm','Perron method','../p/p072370.htm','Potential theory','../p/p074140.htm','Regular boundary point','../r/r080680.htm','Singular integral','../s/s085570.htm','Suslin theorem','../s/s091480.htm','Urysohn–Brouwer lemma','../u/u095860.htm','Vitali variation','../v/v096790.htm')" style="background-color:yellow;">H. Lebesgue for Fourier–Lebesgue series.

If the function is absolutely continuous, then the Fourier series for the derivative can be obtained by term-by-term differentation of the Fourier series for . This implies that if the derivative of order of a function is absolutely continuous, then the estimates

are valid for the Fourier coefficients of .

The first convergence criterion for Fourier series was obtained by P.G.L. Dirichlet in 1829. His result (the Dirichlet theorem) can be formulated as follows: If a function has a finite number of maxima and minima over the period and is everywhere continuous, except at a finite number of points where it may have discontinuities of the first kind, then the Fourier series of converges for all , and, moreover, at points of continuity it converges to and at points of discontinuity it converges to . Subsequently, this assertion was extended to arbitrary functions of bounded variation (C. Jordan, 1881).

According to the localization principle proved by Riemann (1853), the convergence or divergence of the Fourier series of a function at a point , and the value of the sum when it converges, depends only on the behaviour of in an arbitrarily small neighbourhood of .

Many different convergence criteria for Fourier series at a point are known. R. Lipschitz (1864) established that the Fourier series of a function converges at a point if is satisfied for all sufficiently small , where and are certain positive constants (the Lipschitz criterion). The Dini criterion is more general: The Fourier series of a function converges to at a point if the integral

converges, where . The value is usually taken for . For example, if the Fourier series of converges at a point where this function is continuous, then the sum of the series is necessarily equal to .

Lebesgue (1905) proved that if

as , then the Fourier series of converges to at . This Lebesgue criterion is stronger than all those given above and stronger than the de la Vallée-Poussin criterion and the Young criterion. But verifying it is usually difficult.

A convergence criterion of another type is given by the Hardy–Littlewood theorem (1932): The Fourier series of a function converges at a point if the following conditions are satisfied:

1)

as ; and

2) the estimates

are valid for the Fourier coefficients of .

Besides convergence criteria for Fourier series at a point, criteria for uniform convergence have been studied also. Let a function have period and be continuous. Then its Fourier series converges uniformly to it on the whole real line if the modulus of continuity (cf. Continuity, modulus of) of satisfies the condition

(the Dini–Lipschitz criterion) or if has bounded variation (the Jordan criterion).

From this one can obtain criteria for uniform convergence of Fourier series on a certain interval if the localization principle for uniform convergence is used. The latter is formulated as follows. If two functions are equal on an interval , then on each strictly interior interval , , either the Fourier series of these functions are both uniformly convergent or neither is uniformly convergent. In other words, the uniform convergence of the Fourier series of a function on an interval depends only on the behaviour of in an arbitrarily small extension of this interval.

P. du Bois-Reymond (1876) established that the continuity of a function at a certain point does not guarantee that its Fourier series converges at this point. Later it was proved that the Fourier series of a continuous function may diverge on an everywhere-dense set of measure zero that is of the second category.

If nothing is assumed about the function except that it is integrable, then its Fourier series may turn out to be divergent almost-everywhere, or even everywhere. The first examples of such functions were constructed by A.N. Kolmogorov (1923, 1926). Later it was shown that this may be true both for the Fourier series of the function itself and for the function conjugate to it.

As early as 1915, N.N. Luzin made the conjecture that the Fourier series of every -function converges almost-everywhere. For a long time only partial results were obtained in this direction. The general form of the problem turned out to be very difficult and it was only in 1966 that L. Carleson proved the validity of this conjecture (see Carleson theorem). The Fourier series of -functions when also converge almost-everywhere. Kolmogorov's example shows that it is impossible to strengthen this result any further in terms of the spaces .

Since the partial sums of a Fourier series do not always converge, one also considers the summation of Fourier series by some average of the partial sums and uses this to represent the function. One of the simplest examples are the Fejér sums (cf. Fejér sum), which are the arithmetical means of the partial sums of the Fourier series:

For every integrable function the sums converge to almost-everywhere and, moreover, converge at every point where is continuous; if is continuous everywhere, then they converge uniformly.

According to the Denjoy–Luzin theorem, if the trigonometric series (2) at every converges absolutely on a set of positive measure, then the series

(5) |

converges, and hence the series (2) converges absolutely for all . Thus, the absolute convergence of (2) is equivalent to convergence of (5).

e theorem','../l/l057530.htm','Lebesgue constants','../l/l057800.htm','Limit theorems','../l/l058920.htm','Lyapunov theorem','../l/l061200.htm','Markov–Bernstein-type inequalities','../m/m110060.htm','Orthogonal polynomials','../o/o070340.htm')" style="background-color:yellow;">S.N. Bernstein [S.N. Bernshtein] (1934) proved that if the modulus of continuity of a function satisfies

then the Fourier series of converges absolutely. It is impossible to weaken this condition: If is a modulus of continuity of function type such that the series

diverges, then a function can be found with modulus of continuity satisfying and whose Fourier series does not converge absolutely.

In particular, the Fourier series of functions satisfying a Lipschitz condition of order converge absolutely. When , absolute convergence need not hold (Bernshtein, 1914).

If is a function of bounded variation and if its modulus of continuity satisfies

(6) |

then the Fourier series of converges absolutely (see [9]). Condition (6) cannot be weakened (see [10]).

In contrast to the above, the following theorem gives a criterion for the absolute convergence for an individual function. A necessary and sufficient condition for the absolute convergence of the Fourier series of a function is that the series

converges, where is the best approximation to in the metric of by trigonometric polynomials containing harmonics (see [11]).

The series (2) can be considered as the real part of the power series

The imaginary part

(7) |

is called the series conjugate to the series (2).

Let and let (2) be its Fourier series. Then for almost-all the function

exists (I.I. Privalov, 1919). The function is called the conjugate function to ; it need not be integrable. However, if , then the Fourier series of is the series (7) (V.I. Smirnov, 1928).

In many cases one can deduce some property or other of the conjugate series (7) from the properties of the function or its Fourier series (2), for example, convergence in the metric of , convergence or summability at a point, or almost-everywhere, etc.

Properties of Fourier series under special assumptions on their coefficients have also been studied. For example, lacunary trigonometric series, when the only non-zero coefficients are those indexed by numbers forming a lacunary sequence, that is, . Another example of special series are series with monotone coefficients.

All that has been said above concerns Fourier series of the form (2). For Fourier series in a rearranged trigonometric system certain properties of the Fourier series in the trigonometric system, taken in the usual order, do not hold. For example, there is a continuous function such that its Fourier series after a certain rearrangement diverges almost-everywhere (see [12]–[15]).

The theory of Fourier series for functions in several variables (multiple Fourier series) has been developed to a lesser extent. Some of the multi-dimensional results are analogous to the one-dimensional results. But there are crucial differences.

Let be a point of the -dimensional space, let be an -dimensional vector with integer coordinates and let . For a function with period in each variable and Lebesgue integrable over the -dimensional cube , the Fourier series in the trigonometric system is

(8) |

where the summation is over all and

are the Fourier coefficients of . The Fourier series (8) is written in complex form. Writing it in trigonometric form as a series in the products of multiple cosines and sines is rather more clumsy.

Various definitions of the partial sums of the series (8) are possible; for example, partial sums over rectangles

and over circles

(9) |

where is the radius and .

The circular partial sums (9) are not so suitable for representing functions as are their Riesz means

For Riesz means of order of Fourier series of -functions the localization principle is valid; this is not so for smaller (S. Bochner, 1936). The Riesz means of circular partial sums of critical order play an essential role also in other questions about Fourier series of functions in several variables.

There is a continuous function in two variables with a Fourier series that does not converge over rectangles at any interior point of the square (see [16]).

Certain results about Fourier series in the trigonometric system can be generalized considerably; for example, they can be carried over in a corresponding way to the spectral decompositions corresponding to self-adjoint elliptic differential operators.

#### References

[1] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |

[2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

[3] | G.H. Hardy, W.W. Rogosinsky, "Fourier series" , Cambridge Univ. Press (1965) |

[4] | N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) |

[5] | H. Lebesgue, "Leçons sur les séries trigonométriques" , Gauthier-Villars (1906) |

[6] | A.B. Paplauskas, "Trigonometric series from Euler to Lebesgue" , Moscow (1966) (In Russian) |

[7] | P.L. Ul'yanov, "Solved and unsolved problems in the theory of trigonometric and orthogonal series" Russian Math. Surveys , 19 : 1 (1964) pp. 1–62 Uspekhi Mat. Nauk , 19 : 1 (1964) pp. 3–69 |

[8] | Sh.A. Alimov, V.A. Il'in, E.M. Nikishin, "Convergence problems of multiple trigonometric series and spectral decomposition. I" Russian Math. Surveys , 31 : 6 (1976) pp. 29–86 Uspekhi Mat. Nauk , 31 : 6 (1976) pp. 28–83 |

[9] | R. Salem, "On a theorem of Zygmund" Duke Math. J. , 10 (1943) pp. 23–31 |

[10] | S.V. Bochkarev, "On a problem of Zygmund" Math. USSR Izv. , 7 : 3 (1973) pp. 629–637 Izv. Akad. Nauk SSSR , 37 (1973) pp. 630–638 |

[11] | S.B. Stechkin, "On absolute convergence of orthogonal series" Dokl. Akad. Nauk SSSR , 102 (1955) pp. 37–40 (In Russian) |

[12] | A.N. [A.N. Kolmogorov] Kolmogoroff, D.E. [D.E. Menshov] Menschoff, "Sur la convergence des séries de fonctions orthogonales" Math. Z. , 26 (1927) pp. 432–441 |

[13] | Z. Zahorski, "Une série de Fourier permutée d'une fonction de classe divergente partout" C.R. Acad. Sci. Paris , 251 (1960) pp. 501–503 |

[14] | P.L. Ul'yanov, "Divergent Fourier series" Russian Math. Surveys , 16 : 3 (1961) pp. 1–75 Uspekhi Mat. Nauk , 16 : 3 (1961) pp. 61–142 |

[15] | A.M. Olevskii, "Divergent Fourier series for continuous functions" Soviet Math. Dokl. , 2 (1961) pp. 1382–1386 Dokl. Akad. Nauk SSSR , 141 (1961) pp. 28–31 |

[16] | C. Fefferman, "On the divergence of multiple Fourier series" Bull. Amer. Math. Soc. , 77 (1971) pp. 191–195 |

#### Comments

A closed system is also called a complete system (cf. Complete system of functions). Instead of Riemann–Lebesgue theorem one often uses Riemann–Lebesgue lemma.

For multiple Fourier series see, e.g., Chapt. 7 of [a5].

#### References

[a1] | R.E. Edwards, "Fourier series. A modern introduction" , 1–2 , Springer (1979–1982) |

[a2] | J.-P. Kahane, "Séries de Fourier absolument convergentes" , Springer (1970) |

[a3] | Y. Katznelson, "An introduction to harmonic analysis" , Wiley (1968) |

[a4] | H. Dym, H.P. McKean, "Fourier series and integrals" , Acad. Press (1972) |

[a5] | E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971) |

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Fourier series.

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