Orthogonal functions

Introduction

In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function, f(t), as a complicated entity in a f(), in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).

Vectors

Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be from the Complex numbers for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector ${\vec {\mathbf {v}}}=<1,4,3>$ means that the vector ${\vec {\mathbf {v}}}$ is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of ${\vec {\mathbf {v}}}$ in the second direction is 4. This is often written as $v_{y}=4$.

Notation

We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say $v_{2}=4$ instead of $v_{y}=4$. Instead of writing ${\vec {\mathbf {v}}}=<1,4,3>$ we can write ${\vec {\mathbf {v}}}=\sum _{k=1}^{3}v_{k}{\hat {\mathbf {a}}}_{k}$ where the ${\hat {\mathbf {a}}}_{k}$ denotes a unit vector (or basis vector) in the kth direction. The idea of basis vectors was implicit in the notation ${\vec {\mathbf {v}}}=<1,4,3>$.

Changing Basis Sets

Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors. Let us remind ourselves how we do the transformation from a one coordinate system to another.

Inner Products

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Orthogonality for Vectors

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Functions and Vectors, an Analogy

Independent and Dependent Variables

We may think of the number of the direction, $k$, as the independent variable of a vector and the component in that direction, $v_{k}$ as the dependent variable of the vector ${\vec {\mathbf {v}}}$ in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from $-\infty$ to $\infty$, and $k\in {1,2,3}$.

Changing Basis Sets with Functions

• Fourier Series
• Reconstructing Bandlimited Signals from Sample Points