A series in cosines and sines of multiple angles, that is, a series of the form
or, in complex form,
where
or, respectively,
are called the
coefficients of the trigonometric series.
Trigonometric series are first encountered in the work
of
L. Euler
(1744).
He obtained the expansions
In the middle of the
18th century,
in connection with the study
of problems on the free oscillations of strings, there arose
the question of the possibility of
"representing"
functions characterizing the initial position
of a string in the form of a sum of a
trigonometric series. This question raised fierce debates, continuing over several decades,
among the best analysts of that time, such as
D. Bernoulli,
J. d'Alembert,
J.L. Lagrange,
and Euler. These arguments related to the essence
of the notion of a function. At that time, functions were
usually related to their analytic specifications, which led to
the consideration of analytic or piecewise-analytic functions only. But here
the need arose to construct a trigonometric series
"representing"
a
function whose graph could be a rather arbitrary curve.
However, the significance of the arguments was even greater.
In fact, out of these discussions various questions arose connected with
many fundamentally important concepts and ideas of mathematical analysis
in general, such as the
"representation"
of functions by
Taylor series
and the
analytic continuation
of functions, the use of
divergent series,
interchange of limits, infinite systems of equations,
interpolation of functions by polynomials, etc.
Subsequently, as well as in this initial period, the theory of
trigonometric series served as a source of new ideas of mathematical analysis and influenced the
development of other branches of it. Investigations in trigonometric series played an important
role in the construction of the Riemann and Lebesgue integrals. The theory
of functions of a real variable originated and was then developed
in close connection with the theory of trigonometric series. As
generalizations of the theory of trigonometric series there emerged the
Fourier integral,
almost-periodic functions,
general orthogonal series,
and
abstract harmonic analysis.
Research into trigonometric series served as a starting point in the creation of
set theory. Trigonometric series are a powerful means for representing and studying functions.
The question introduced into the debates of the mathematicians in the
18th century
was solved in
1807
by
J. Fourier,
who gave formulas
for calculating the coefficients of the trigonometric series
(1)
that had to
"represent"
a function
on
:
and who applied them in the solution of problems of
heat conduction.
Formulas
(2)
have acquired the name
Fourier formulas,
although they were encountered earlier by
A. Clairaut
(
1754)
and
Euler
(
1777)
via term-by-term integration. The trigonometric series
(1)
whose coefficients are defined by
(2)
is called the
Fourier series
of
,
and the numbers
the
Fourier coefficients
of
.
The character of the results obtained depends on how the
"representation"
of the
function by a series is to be understood, and how the integral in
(2)
is to be understood. The modern form of the theory
of trigonometric series was obtained after the appearance of the
Lebesgue integral.
The theory of trigonometric series can conditionally be
divided into two main branches: the theory of
Fourier series,
in which it is supposed that the series
(1)
is the Fourier series
of some function, and the theory of general trigonometric series, where this hypothesis
is not made. The main results in the theory of general trigonometric
series are given below (here the measure of sets and measurability
of functions are to be understood in the sense of Lebesgue).
The first systematic study of trigonometric series in which it was not
supposed that these series are Fourier series, was the dissertation of
B. Riemann
(1853).
For this reason, the theory
of general trigonometric series is sometimes called the
Riemann theory of trigonometric series.
For the study of the properties of an arbitrary series
(1)
with coefficients converging to zero, Riemann considered the continuous function
that is the sum of the uniformly-convergent series
obtained after twice term-by-term integration of the series
(1).
If the series
(1)
converges at some point
to the number
,
then the second
symmetric derivative
of
exists at this point and is equal to
:
Since
this leads to the summation of
(1)
produced by the factors
,
called the
Riemann summation method.
By means of the function
,
Riemann formulated the
localization principle,
according to which the behaviour of the series
(1)
at a point
depends only on the behaviour of
in an arbitrarily small neighbourhood of this point.
If a trigonometric series converges on a set of
positive measure, then its coefficients converge to zero (the
Cantor–Lebesgue theorem).
Convergence to zero of the coefficients of a trigonometric
series also
follows
from convergence of the series on a set of the second category
(W. Young,
1909).
One of the central problems in the theory of general trigonometric series is that of
"representing"
an
arbitrary function by a trigonometric series. By strengthening results of
N.N. Luzin
(1915)
on the representation of functions by trigonometric series that are summable almost-everywhere
by the methods of Abel–Poisson and Riemann,
D.E. Men'shov
proved
(1940)
the following theorem, relating to the most important case when the representation of
is understood as convergence of the trigonometric series to
almost everywhere. There exists for any measurable, almost-everywhere finite
-periodic
function
a trigonometric series converging to it
almost-everywhere
(Men'shov's theorem).
It should be noted that even if the function
is integrable, one cannot in general choose the Fourier series of
for this series, since there exist Fourier series that are everywhere divergent.
Men'shov's theorem can be strengthened as follows: If a
-periodic
function
is measurable and finite almost everywhere, then there exists a continuous function
such that
almost everywhere and the differentiated Fourier series of
converges to
almost everywhere
(N.K. Bari,
1952).
It is not known
(1984)
whether the condition that
be finite almost-everywhere can be dropped in
Men'shov's theorem. In particular, it is not known
(1984)
whether there is a trigonometric series converging almost-everywhere to
.
Therefore the problem on the representation of functions that can take infinite
values on a set of positive measure has been considered for
the case when convergence almost-everywhere is replaced by a weaker
condition, namely, convergence in measure. Convergence in measure to functions
that can take infinite values is defined as follows: The sequence
of partial sums of a trigonometric series converges in measure to a function
if
where
converges to
almost everywhere, while the sequence
converges in measure to zero. In this formulation, the
question of the representation of functions has been
completely solved: there exists for any measurable
-periodic
function a trigonometric series converging to it in measure (Men'shov,
1948).
Much research has been devoted to the problem of uniqueness
of trigonometric series: Can two distinct trigonometric series converge to
the same function? In another formulation: If a trigonometric series converges
to zero, then does it follow that all coefficients of the series
are zero? Here one may have in mind convergence at all points
or at all points outside some set. The answers to these
questions depend essentially on the properties of
the set outside which convergence is presupposed.
The following terminology has been established. A set
is called a
set of uniqueness
(cf.
Uniqueness set),
or a
-set,
if the convergence everywhere on
of a trigonometric series to zero except, perhaps, at points of
,
implies that all coefficients of the series are zero. Otherwise
is called a
set of multiplicity,
or an
-set.
As
G. Cantor
showed
(1872),
the empty, as well as any finite, set is a
-set.
Any countable set is also a
-set
(Young,
1909).
On the other hand, every set of positive measure is an
-set.
The existence of
-sets
of measure zero was established by Men'shov
(1916),
who constructed the first example of a
perfect set
possessing these properties. This result is of prime importance in
the uniqueness problem. It follows from the existence of
-sets
of measure zero that in the representation
of functions by trigonometric series converging almost
everywhere, these series are automatically non-uniquely determined.
Perfect sets can also be
-sets
(Bari;
A. Rajchman,
1921).
In the uniqueness problem,
an important role is played by very precise characteristics of
sets of measure zero. The general question on
the classification of sets of measure zero into
-
and
-sets
remains open
(1984).
It has not even been solved for perfect sets.
The following problem touches on the uniqueness problem.
If a trigonometric series converges to a function
,
then does this series have to be the Fourier series of
?
P. du Bois-Reymond
(1877)
gave an affirmative answer to this question when
is Riemann integrable and the series converges to
at all points. Results of
Ch.J. de la Vallée-Poussin
(1912)
imply
that the answer is also affirmative in case the series is convergent
to a finite sum everywhere except on a countable set of points.
If a trigonometric series converges absolutely at some point
,
then the points of convergence of the series, as well as
the points of its absolute convergence, are distributed symmetrically about
(P. Fatou,
1906).
According to the
Denjoy–Luzin theorem,
absolute convergence on a set of positive measure of
a trigonometric series
(1)
implies that the series
is convergent, and hence that
(1)
is absolutely convergent for all
.
This property is also possessed by sets of the
second category and certain sets of measure zero.
This survey only covers one-dimensional trigonometric series
(1).
There
are separate results relating to general trigonometric series of several variables.
Here in many cases it is also necessary to find natural formulations of the problems.