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.


Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word. How to write better comments?
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
Read More

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…

Albert Einstein and His introduction to the Concept of Relativity

Albert Einstein This name need not be explained. Albert Einstein is considered to be one of the best physicists in the human history. The twentieth century has undoubtedly been the most significant for the advance of science, in general, and Physics, in particular. And Einstein is the most luminated star of the 20th century. He literally created cm upheaval by…
Painted Diary
Read More

This Prime Generating Product generates successive prime factors

Any integer greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself. The basic ideology involved in this post is flawed and the post has now been moved to Archives. – The Editor Prime Generating Formulas We all know how hard it is to predict a formula for prime numbers! They have…
cropped Fotolia  M.jpg
Read More

Examination Strategies : Tactics & Tips

Every student or graduate knows how hard the first experience of passing exams is. Preliminary preparation starves the nervous system and the physical condition of the human body, however, the exam itself is always a stressful situation, which requires a candidate a great manifestation of mental and physical abilities. Therefore, just the knowledge of a subject is not enough for…

Getting Started with Measure Theory

Last year, I managed to successfully finish Metric Spaces, Basic Topology and other Analysis topics. Starting from the next semester I’ll be learning more pure mathematical topics, like Functional Analysis, Combinatorics and more. The plan is to lead myself to Combinatorics by majoring Functional Analysis and Topology. But before all those, I’ll be studying measure theory and probability this July – August. Probability…
Read More

Social Networks for Math Majors

Math or Mathematics is not as difficult as it is thought to be. Mathematical Patterns, Structures, Geometry and its use in everyday life make it beautiful. ‘Math majors’ term generally include Math students, Math professors and researchers or Mathematicians. Internet has always been a tonic for learners and whole internet is supposed to be a social network, in which one…