INTRODUCTION
The study of a special type of series whose terms are trigonometric functions of a variable was started in the 18th century when J.B.J. Fourier (1768-1830) was successful in his attempt to prove that an arbitrary function given in the interval can be expressed under certain conditions in a trigonometric series of the form
and subsequently, such type of special series is given the name ‘Fourier Series’. It has great importance in the study of many branches of science including social sciences. The knowledge of Fourier series provides a powerful technique for solving problems of a periodic nature which occur in situations like theory of conduction of heat, electrical and mechanical vibrations, propagation of electromagnetic waves, acoustics, optics, etc. Fourier was the first person to initiate an elaborate study of trigonometric series and afterwards Dirichlet, Parseval, Bessel and many other mathematician made major contributions to the study of conditions of a function to be represented in a trigonometric series, convergence, integration, and differentiation of the Fourier series. In this article we are going to study Fourier series and their properties.
Before starting the article of Fourier series, it is essential to study the behaviour of periodic functions, piecewise monotone and piecewise continuous functions and some properties related to them.
PERIODIC, PIECEWISE MONOTONE AND PIECEWISE CONTINUOUS FUNCTIONS
Definition: Let , a function
is said to be periodic if there exists a positive constant
such that
for all
; the least number
possessing this property (if such a number exists) is called the period of the function.
is called periodic of period or fundamental period
.
is periodic of period
implies also
for all
.
The trigonometric functions ,
are periodic of period
and
,
are periodic of period
.
is periodic of period
Every constant function is trivially periodic with no definite Fundamental period.
The following observations are quite clear from the definitions of periodic functions.
- If
is defined on an interval of length
then
can be extended to the whole of
by making
periodic of period
.
- The graph of a periodic function over
will be replicas of a portion over an interval of length
if
is period of the function.
Hence if a function is defined on
it can be made to be a periodic function on
of period
. In fact, sum of two periodic function of same period is also a periodic function of that period.
If is periodic of period
then the following results are quite easy to prove:
Definition: A function is said to be piecewise monotone if there exists a partition of
into open sub-intervals in each of which
is monotone.
Every monotone function is piecewise monotone but the converse is not true.
For example
is piecewise monotone on [0, 1], but is not monotone on [0, 1].
Definition: A function is said to be piecewise continuous if there exists a partition of
into open sub-intervals in each of which
is continuous.
Every continuous function is piecewise continuous but the converse is not true.
For example, the function defined on
by
is piecewise continuous, since it is continuous on
,
,
, but is not continuous on
;
and
are points of discontinuities of
on
.
Theorem 2.1.
Definition (Fourier Series): Let be an integrable function on
or if
is unbounded on
, let the improper integral
be absolutely convergent.
Then the trigonometric series
is called the Fourier series in corresponding to the function
, where
, called Fourier Coefficients, are given by
Definition: A sequence of Riemann integrable functions defined on
is called orthogonal if
and it is said to be orthonormal if
From Theorem 2.1, it easily follows that are all orthonormal on
. The function
is orthonormal on
FOURIER SERIES EXPANSION
Not every function can be expanded into Fourier series. There are certain conditions which are to be satisfied by a function for its expansion into Fourier series. The most familiar one is Dirichlet’s conditions.
Dirichlet’s Conditions
Let be a given function.
is said to satisfy the Dirichlet’s conditions in
if it satisfies one of the following two conditions:
is bounded on
and the interval
can be broken into a finite number of open sunintervals of
, in each of which
is monotone i.e., in other words
is bounded and piecewise monotone on
;
has a finite number of points of infinite discontinuities on
but when arbitrarily small neighbourhoods of these points of discontinuities are excluded from
,
is bounded and piecewise monotone in the remainder of the interval
and also the improper integral
is absolutely convergent.
Remark: Dirichlet’s conditions can also be put in a slightly different form:
is said to satisfy Dirichlet’s condition on
if
is a function of bounded variation on
or when
has a finite number of points of finite discontinuities on
and when arbitrarily small neighbourhoods of these points are excluded from
,
is a function of bounded variation on the reminder of
and
is absolutely convergent on
.
Theorem 3.1.
If the series converges uniformly to
on
then it is the Fourier series for
in
.
Proof. Since the series converges uniformly to
on
,
is the sum function of the series and we can write
(1)
By integrating term-by-term on , we get using Theorem 2.1,
Multiplying (1) by on both sides and integrating term-by-term (which is valid, since
is a bounded function and after multiplication the series is still uniformly convergent on
, we have
Multiplying (1) by on both sides and integrating term-by-term, we have
Hence, from definition the series (1) represents the Fourier series for on
.
Note: Fourier series exists for every Riemann integrable function. But that series may or may not converge to .
Theorem 3.2:
If be bounded, integrable and monotone on
then
Proof. (i) Since is monotone on
, applying second mean value theorem (Weierstrass form) of integral calculus, there exists a
such that
Since is convergent, by Cauchy’s condition of convergence
Since is
Hence
for
.
Proof: – (ii) For the integra .
We write .
Since is bounded and monotone on
,
is also bounded and monotone on
for
and
.
By the second mean value theorem,
Since is convergent, there exists
such that
for all
.
Therefore .
Since , for any positive
there exists a positive
such that
whenever
.
Hence whenever
by
,
Hence of , there exists
such that
whenever
This implies
\ben*}
\left\vert\int_{0}^{a}\varphi(x)\dfrac{\sin nx}{x}dx\right\vert & \leq\left\vert\int_{0}^{h}\varphi(x)\dfrac{\sin nx}{x}dx\right\vert+\left\vert\int_{h}^{a}\varphi(x)\dfrac{\sin nx}{x}dx\right\vert\\
& <\dfrac{\varepsilon}{2}+\dfrac{\varepsilon}{2}=\varepsilon\text{ when and
.}
\end{align*}
Thus
Corollary 1. If is bounded integrable and monotone on
where
,
then
Proof. where
is bounded and integrable on
.
is monotone increasing on
and hence on
where
.
Then by Theorem 3.2
Corollary 2. If be bounded and integrable and monotone on
for
then
Proof. From (ii) of Theorem 3.2,
Since we have
Remark: the integrals in Theorem 3.2 are known as Dirichlet’s Integrals.
Theorem 3.3
If be bounded, integrable and monotone on
and
(not necessarily in the same sense), where
, then
where stand for Fourier coefficients of
.
Proof. From definition of we have
Theorem 3.4 (Fourier-Dirichlet):
If is bounded, integrable, periodic pf period
and it is piecewise monotone on
then the Fourier series for
in
is equal to
at
in
, when
is continuous at
,
at
in
, when
and
exist,
at
, when
and
exist.
Proof. The Fourier series for (since it satisfies Dirichlet’s condition in
) in
is
where
and
Let,
Then
In the first integral, we will put and in the second integral, we are going to put
.
Then
Since satisfies Dirichlet’s conditions in
, if we consider
as functions of
then they satisfy Dirichlet’s conditions in
and in
respectively. If
exist then
and
have the limits when
tends to zero.
Hence, if , we have using Dirichlet’s integrals,
Hence (ii) proved.
If is continuous at
where
then
.
Hence,
which proves (i).
At ,
If and
exist, using Dirichlet’s integrals,
We have,
Similarly, it can be shown that
Remark: The Theorem 3.4 establishes the point-wise convergence of the Fourier series for to the function
and not the uniform convergence. The Fourier series expression of a function is unique.
Theorem 3.5. (Parseval’s Identity):
If the Fourier series for in
converges uniformly to
on
then
Where are the Fourier coefficients for
in
.
Proof. The Fourier series of in
is
Where
Since the series (1) converges uniformly to on
, then we can write (1) as
Multiplying both sides of (2) by and then integrating term-by-term (which is valid, since
is bounded on
or if
unbounded,
is absolutely convergent) on
we have
Theorem 3.6. (Bessel’s Inequality)
If is bounded and integrable on
or if
is unbounded on
but
is convergent, then
Where are the Fourier coefficients for
in
.
Proof. The Fourier series expression for on
is
Where are given by
for all .
Let
Since , we have
From (1)
Again from (1),
Hence using (2), (3) and (4) we have the inequality
Note: Taking in Bessel’s inequality we obtain
Since and hence
is integrable on
, the series
SOME EXAMPLES ON FOURIER SERIES REPRESENTATION OF FUNCTIONS IN ![Rendered by QuickLaTeX.com \left[-\pi,\ \pi\right]](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2080%2026'%3E%3C/svg%3E)
Example 4.1. We are going to solve the Fourier series representation of in
where
and hence we will deduce that .
Solution: is continuous on
and hence it is bounded and integrable on
. Since
for
,
is monotone increasing on
. Thus
satisfies Dirichlet’s conditions on
so that
can be represented by a Fourier series
Hence the Fourier series expansion for in
is
Since is continuous on
, so we can write
Example 4.2. We are going to find the Fourier series expansion of the function
Hence we will deduce that
Solution: Since is an even function, its derivative
is odd and hence
is symmetric about origin.
in
and
. Hence
is piecewise monotone in
. Hence
satisfies Dirichlet’s conditions in
. Therefore
can be represented by the Fourier series
For since
is an odd function.
Hence, Fourier series corresponding to in
is
Since is continuous on
,
Example 4.3. Obtain the Fourier series expansion of in
, where
Then we are going to show that the sum of the series
Solution: is piecewise monotone on
.
is continuous on
and hence is integrable on
. Thus
satisfies Dirichlet’s conditions in
.
So, can be represented by the Fourier series
where
For
Thus, the Fourier series expansion of in
is
i.e.,
Since is continuous on
,
Therefore,
Observation. If a function is not defined at
then
can be defined arbitrarily at those points. However, it is preferable to define
at those points in such a way that makes
continuous on
.
Example 4.4. We will find Fourier series of the periodic function with period
defined as
Then we will calculate the sum of the series at .
And we will prove that .
Solution: For the sake of continuity, we define
is piecewise monotone in
. It is bounded and periodic with period
.
is continuous on
except at
. Hence
is integrable on
. Thus
satisfies Dirichlet’s conditions on
. Hence
can be represented by Fourier series, which is
Where
Thus, Fourier series for in
is
At , the Fourier series for
converges to
Hence sum of the series at is
.
Since is continuous at
, we have
Example 4.5. If on
, we will prove that the Fourier series of
is given by
Then we will deduce that
and
Solution: is piecewise monotone on
.
is continuous on
and hence is integrable on
. Therefore
satisfies Dirichlet’s conditions on
. Hence
can be represented by Fourier series in
which is
where
Since is odd function, for
Hence the Fourier series for in
is
Since is continuous on
, we can write
Hence therefore,
Since
and
hence by Weierstrass M-test,
The 5series
[Remark: We can use comparison test for the test of convergence]
This implies the infinite series
converges uniformly to in
. Then by Parseval’s Identity (Theorem 3.5)
Example 4.6. Find the Fourier series of the periodic function with period
defined by
The we will find the sum of the series at and then we are going to show that
Solution: We define at
and at
as
, to make
continuous at
is piecewise monotone on
. It is continuous on
except at
and
and is bounded on
. Hence
is a bounded periodic and integrable function on
which validates the expansion of
into Fourier series, which is
Where
Thus, the Fourier series for in
is