ABSTRACT
A major application of the Fourier transform is in the study of systems. We may think of a system as a device that accepts functions as input and produces functions as output. For example, the differentiation system accepts a differentiable function f(x) as input and produces its derivative function f ′(x) as output. If the input is the function f(x) = 5f1(x)+3f2(x), then the output is 5f ′1(x) + 3f
′ 2(x); the differentiation system is linear.
We shall describe systems algebraically by h = Tf , where f is any input function, h is the resulting output function from the system, and T is the operator that represents the operation performed by the system on any input. For the differentiation system we would write the differentiation operator as Tf = f ′.
