Square Integrable function or quadratically integrable function $\mathfrak{L}_2$ function

A function $y(x)$ is said to be square integrable or $\mathfrak{L}_2$ function on the interval $(a,b)$ if $$\displaystyle {\int_a^b} {|y(x)|}^2 dx <\infty$$ or $$\displaystyle {\int_a^b} y(x) \bar{y}(x) dx <\infty$$.

For further reading, I suggest this Wikipedia page.

$y(x)$ is then also called ‘regular function’.

The kernel $K(x,t)$ , a function of two variables is an $\mathfrak{L_2}$ – function if atleast one of the following is true:

  • $\int_{x=a}^b \int_{t=a}^b |K(x,t)|^2 dx dt <\infty$
  • $\int_{t=a}^b |K(x,t)|^2 dx <\infty$
  • $\int_{x=a}^b |K(x,t)|^2 dt <\infty$

Inner Product of two $\mathfrak{L}_2$ functions

The inner product or scalar product $(\phi, \psi)$ of two complex $\mathfrak{L}_2$ functions $\phi$ and $\psi$ of a real variable $x$ ; $a \le x \le b$ is defined as

$(\phi, \psi) = \int_a^b \phi(x) \bar{\psi}(x) dx$  .

Where $\bar{\psi}(x)$ is the complex conjugate of  $\psi(x)$.

When $(\phi, \psi) =0$, or $\int_a^b \phi(x) \bar{\psi}(x) dx =0$ then $\phi$ and $\psi $ are called orthogonal to each other.

Norm of a function

The norm of a complex- function $y(x)$ of a single real variable $x$ is given by

$|| y(x) || = \sqrt{\int_a^b y(x) \bar{y(x)} dx}=\sqrt{\int_a^b |y(x)|^2 dx}$

Where $\bar{y(x)}$ represents the complex conjugate of $y(x)$.

The norm of operations between any two functions $\phi$ and $\psi$  follows Schwarz and Minkowski’s triangle inequalities, provided

$|| \phi \cdot \psi || \le ||\phi|| \cdot ||\psi||$ —- Schwarz’s Inequality

$|| \phi +\psi || \le ||\phi|| + ||\psi||$ ——-Triangle Inequality/Minkowski Inequality


 Solution of Integral Equations by Trial Method

A solution of an equation is the value of the unknown function which satisfies the complete equation. The same definition is followed by the solution of an integral equation too. First of all we will handle problems in which a value of the unknown function is given and we are asked to verify whether it’s a solution of the integral equation or not. The following example will make it clear:

  • Show that $y(x)= {(1+x^2)}^{-3/2}$ is a solution of $$y(x) = \dfrac{1}{1+x^2} – \int_0^x \dfrac{t}{1+x^2} y(t) dt$$.

This is a Volterra’s equation of second kind with lower limit $a=0$ and upper limit being the variable $x$.

Solution: Given $$y(x) = \dfrac{1}{1+x^2} – \int_0^x \dfrac{t}{1+x^2} y(t) dt \ldots (1)$$

where $y(x)= {(1+x^2)}^{-3/2} \ldots (2)$

and therefore, $y(t)= {(1+t^2)}^{-3/2} \ldots (3)$ (replacing x by t).

The Right Hand Side of (1)

$=\dfrac{1}{1+x^2} – \int_0^x \dfrac{t}{1+x^2} y(t) dt$

$=\dfrac{1}{1+x^2} – \int_0^x \dfrac{t}{1+x^2} {(1+t^2)}^{-3/2} dt$ [putting the value of $y(t)$ from (3)]

$=\dfrac{1}{1+x^2} -\dfrac{1}{1+x^2} \int_0^x \dfrac{t}{{(1+t^2)}^{3/2}} dt$

since $\dfrac{1}{1+x^2}$ is independent quantity as the integration is done with respect to $t$ i.e., dt only, therefore $\dfrac{1}{1+x^2}$ can be excluded outside the integration sign.

$=\dfrac{1}{1+x^2} +\dfrac{1}{1+x^2} \left({\dfrac{1}{\sqrt{1+x^2}} -1}\right)$

       Since $\int_0^x \dfrac{t}{{1+t^2}^{3/2}} dt $=$1-\dfrac{1}{\sqrt{1+x^2}}$



=The Left Hand Side of (2)

Hence, $y(x) ={(1+x^2)}^{-3/2}$ is a solution of (1). $\Box$

Trial method isn’t exactly the way an integral equation can be solved, it is however very important for learning and pedagogy point of views. In upcoming articles, we’ll learn various other techniques to solve an integral equation. But, for now, in next two parts of this series, we shall be reading how ordinary differential equations can be converted into integral equation and vice-versa.

1 comment
Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

You May Also Like

What is Real Analysis?

Real analysis is the branch of Mathematics in which we study the development on the set of real numbers. We reach on real numbers through a series of successive extensions and generalizations starting from the natural numbers. In fact, starting from the set of natural numbers we pass on successively to the set of integers, the set of rational numbers…

Four way valid expression

People really like to twist the numbers and digits bringing fun into life. For example, someone asks, “how much is two and two?” : the answer should be four according to basic (decimal based) arithmetic. But the same  with base three (in ternary number system) equals to 11. Two and Two also equals to Twenty Two. Similarly there are many ways you can…

8 big online communities a math major should join

Online communities are the groups of web savvy individuals who share communal interests. A community can be developed with just a single topic or by a bunch of philosophies. A better community binds its members through substantial debates. Mathematics is a very popular communal interest and there are hundreds of online communities formed in both Q&A and debate styles. Some…

The Mystery of the Missing Money – One Rupee

Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend asked her to sell then…

How many apples did each automattician eat?

Four friends Matt, James, Ian and Barry, who all knew each other from being members of the Automattic, called Automatticians, sat around a table that had a dish with 11 apples in it. The chat was intense, and they ended up eating all the apples. Everybody had at least one apple, and everyone know that fact, and each automattician knew…