## 1. INTRODUCTION

The study of a special type of series whose terms are trigonometric functions of a variable was started in the 18^{th} 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 â€˜Fourier Seriesâ€™. 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.

## 2. PERIODIC, PIECEWISE MONOTONE AND PIECEWISE CONTINIOUS FUNCTIONS

**Definition:** Let , a function is said to be periodic if there exists a positive constant suchÂ 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 Â 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 Â 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 Â is **orthonormal** on

## 3. 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 **Dirichletâ€™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 .

**Â **

**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 **Cauchyâ€™s condition of convergence**

**Â **

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 **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

Thus

**Corollary 1. **If is bounded integrable and monotone on where

,

then

*Â *

** Proof.Â **Â 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 Dirichletâ€™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 Dirichletâ€™s conditions in, if we consider as functions of then they satisfy Dirichletâ€™s conditions in and in respectively. If exist then and have the limits when tends to zero.

Hence, if , we have using Dirichletâ€™s integrals,

**Hence (ii) proved.**

If is continuous at where then .

Hence,

**which proves (i).**

At ,

If and exist, using Dirichletâ€™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. (Parsevalâ€™s Identity):**

If the Fourier series for in converges uniformly to on then

Where are the Fourier coefficients for in .

**Proof.**Â 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. (Besselâ€™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.**Â 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 Besselâ€™s inequality we obtain

Since and hence is integrable on , the series

## 4. SOME EXAMPLES ON FOURIER SERIES REPRESENTATION OF FUNCTIONS IN

**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 Dirichletâ€™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 Dirichletâ€™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 **Dirichletâ€™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.**Â Â Â 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:**Â 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 Dirichletâ€™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 onÂ . is continuous onÂ and hence is integrable onÂ . Therefore satisfies Dirichletâ€™s conditions onÂ . Hence can be represented by Fourier series inÂ 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 **Parsevalâ€™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

i.e.

Since is periodic function of period , so . Since is not continuous at , the Fourier series for converges at .

Hence sum of the series at is equal to .

At , the Fourier series for converges to

Therefore

**Example 4.7.** Let be defined as follows:

We are going to obtain the Fourierâ€™s coefficients and the Fourier series for the function and hence we will find that sum of the series can be represented as

**Solution:** is piecewise monotone in . It is periodic with period . It is bounded on . is continuous in except at . Hence it is integrable on . Thus satisfies Dirichletâ€™s conditions on which validates Fourier series expansion of in .

The Fourier coefficients are

For ,

when is odd, . When