Constant Function Zero and Orthogonal Functions

Date: 02/08/2002 at 10:56:01
From: Neeraj
Subject: Constant function zero and orthogonal functions

We're currently learning about orthogonal functions. Two functions are 
said to be orthogonal over a given interval if integrating their 
product (over that interval) equals zero. Here's where I'm confused:

I) The constant function f(x) = 0 will give zero no matter what 
function g(x) it is integrated with. Does this mean that the constant 
function zero is orthogonal to all functions?-----(1)

II) Also, what could be the geometrical interpretation of orthogonal 
functions? Orthogonal implies perpendicular. Does this mean that at 
their point of intersection, tangents to orthogonal functions are 
mutually perpendicular? If that is so, then from (1) above, zero 
should act as a normal to any curve.

Date: 02/08/2002 at 15:23:52
From: Doctor Peterson
Subject: Re: Constant function zero and orthogonal functions

Hi, Neeraj.

The concept of orthogonal functions is based on seeing functions as 
vectors. The integral of the product of two functions is similar to 
the dot product of two vectors, in that you are summing the products 
of corresponding "coordinates" of the two "vectors." 

This may be easier to see if you think of a two-dimensional vector as 
a function from the set {1,2} to the real numbers, where f(1) is the x 
coordinate of the vector and f(2) is the y-coordinate. Then the dot 
product of vectors f and g is the sum f(1)g(1) + f(2)g(2). When we 
have an infinite domain (making the function an infinite-dimensional 
vector), we have to integrate rather than just sum, but the idea 
remains the same.

Thus the "right angle" is not seen in the graphs of the functions, but 
in an infinite-dimensional space, each "point" of which represents a 
function. Orthogonal functions serve much the same role in this space 
as orthogonal vectors do.

Now, the zero function will be orthogonal to all functions in exactly 
the same sense that the zero vector is orthogonal to all vectors; the 
dot product is zero because the length is zero, but it really says 
nothing about the direction of the other vector, since the zero vector 
has no direction. Likewise, the zero function is trivially orthogonal 
to all functions just because it has no "direction".

You can read about these ideas here in the Dr. Math archives:

   Inner Product and L-2 Distance
   http://mathforum.org/dr.math/problems/queenie.11.14.01.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/