Orthogonality

We all (basic school math) are familiar with the basic concept of orthogonality, where one line are orthogonal to another in the xy-plane, as is shown below.

We also use the words perpendicular or normal. A basic definition is that the angle between two orthogonal lines are 90 degrees. So if we add the blue and orange angles above we get 180 degrees, which indicates a straight line.

Furthermore, orthogonality between lines and planes in higher dimensions are defined in a similar way.

Now lets consider the idea of orthogonal polynomials. A polynomial is a function with the general form
f(x) = a_1 x^n + a_2 x^(n-1) . . . a_(n-1) x + a_n
where the x^i 's are the variables and the a_i 's are the coefficients, where n can be any positive integer value. As long as n is greater than 1 we are no longer consider straight lines and a whole new definition of orthogonality is needed, especially since we want to consider polynomials of any degree.

Orthogonal polynomials is on its own a field of mathematics. They have various properties and a lot of applications in engineering, physics and biology. I completed my Masters dissertation in this field, considering the convexity of the zeros of these types of polynomials.

A lot of research has been done concerning orthogonal polynomials and they are divided into certain classes, depending on certain attributes. They form part of the so called special functions.

So now the question, how do we define orthogonality between polynomials? We make use of an inner product formula, where two polynomials are orthogonal with respect to a certain measure (or a weight function).

Read more on this topic at: http://sydney.edu.au/science/chemistry/~mjtj/CHEM3117/Resources/poly_etc.pdf