Fourier Transform I

Part 2: Convolution

Another important notion that we will need in connection with the Fourier Transform is that of convolution. If f and g are both absolutely integrable over the real line, then the convolution of f and g is given by

Suppose f is a "random" absolutely integrable function. If g is a nonegative function with integral 1 and support concentrated near 0, then the convolution f * g at x may be thought of as an averaging of the function values of f near x.

1. Let f be the function given in the worksheet and let h_n be the first sequential approximation to the delta function.
   

   Compare the graph of the convolution f * h_n with the graph of f for n = 1, 2, 3, and 4.

2. Now calculate the convolution of f with the delta function and compare that with the graph of f itself.

3. Alter the the functions h_n by shifting the graph of each function one unit to the right. Calculate the convolution of f with the shifted function h_n for n = 1, 2, 3, and 4, and compare the graph of this convolution with the graph of f itself.

4. Calculate the convolution of f with the shifted delta function and compare that with the graph of f itself.

5. Pick another amount to shift h_n and the delta function, and repeat Steps 3 and 4.

6. Summarize what you have learned about convolution from these experiments.