Section 1.1: Understanding Polynomial Spaces

The set of real polynomials ( mathbb{R}[x] ) functions as a vector space over ( mathbb{R} ). Its constituents are polynomials with coefficients that are real numbers, such as ( x^2 + ?x + 1 ) and ( x^? - x^2?? ). Due to the endless nature of polynomial powers, ( mathbb{R}[x] ) requires an infinite basis to encompass all its elements. The most apparent basis is ( {1, x, x^2, x^3, ldots} ), allowing any polynomial in ( mathbb{R}[x] ) to be expressed as a unique sum of these basis elements.

However, this isn't the only basis available for ( mathbb{R}[x] ), which is primarily why mathematicians prefer orthogonal bases. An orthogonal basis is characterized by each basis vector being perpendicular to every other basis vector except itself. Such bases simplify many mathematical operations, making them favorable for various applications, from graphs to scaffolding.

Now, to identify orthogonal polynomials in ( mathbb{R}[x] ), we first need to establish a scalar product. Let's define this concept before breaking it down further.

In ( mathbb{R}[x] ), we can multiply elements, which is a unique feature among vector spaces. We exploit this by integrating the product of two polynomials. The variables ( a ) and ( b ) restrict the polynomial's domain, ensuring inputs are numbers greater than ( a ) and less than ( b ) for a finite integral. Thus, we integrate their product over the interval ([a, b]).

Next, we introduce ( w(x) ), known as the weight function, allowing us to control the contribution of each section of the interval ([a, b]) to the scalar product. This approach resembles the dot product, where each polynomial functions as a mapping of numbers in ([a, b]) to another value, paralleling a vector with infinitely many entries.

Let's illustrate how to find orthogonal polynomials through an example. Consider ( a = 0 ) and ( b = 1 ) with a uniform weight function, ( w(x) = 1 ). Our scalar product now appears as follows:

Assuming ( f(x) = x ), we aim to find a quadratic polynomial orthogonal to ( f(x) ) with respect to our scalar product, expressed as ( <x, ax^2 + bx + c> = 0 ). We focus solely on ensuring orthogonality, allowing us to scale the polynomial to be monic and reducing our variables to ( <x, x^2 + ax + b> = 0 ). Now, let's carry out the calculation:

This calculation permits any values of ( a ) and ( b ) that satisfy the equation, leading us to choose ( a = 1 ) and ( b = -7/6 ). Consequently, our quadratic becomes ( x^2 + x - 7/6 ).

Let's visualize these two orthogonal polynomials over the interval ([0, 1]):

This visualization confirms their perpendicularity! The integrated product yields:

The areas above and below the x-axis appear balanced, aligning with our expectation of a zero integral over ([0, 1]).

Section 1.2: The Gram-Schmidt Process

Given the mathematical preference for orthogonal bases, it’s no surprise that there exists a method to convert any vector space basis into an orthogonal one. This technique, known as the Gram-Schmidt process, will not be elaborated upon here, but its application to the basis of ( mathbb{R}[x] ) is noteworthy.

The Gram-Schmidt process not only orthogonalizes the basis but also normalizes it, ensuring each new basis vector has a length of one. For ( mathbb{R}[x] ) and our scalar product, this results in the following expression for a new basis polynomial ( f(x) ):

Applying the Gram-Schmidt process to the obvious basis ( {1, x, x^2, x^3, ldots} ) yields the first four polynomials as follows:

Applications: In approximation theory, large matrix-vector equations are computed to find optimal polynomial approximations for functions. These matrices map ( mathbb{R}[x] ) to itself, and employing an orthonormal basis simplifies these matrices to the identity form, facilitating easier solutions. This is particularly beneficial, as calculators utilize Taylor series or Padé approximations rather than directly knowing exponential or trigonometric functions.

I hope you found this exploration of angles between polynomials enlightening.

Challenge: Find a linear polynomial that is orthogonal to ( x^2 - 9 ) with respect to our defined scalar product.