Constant Function Zero and Orthogonal FunctionsDate: 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/ |
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