2 Fourier transform perspective
The preceeding calculation was not very enlightening, but at least it gives precise numbers. There is a more enlightening approach, beginning with our previous observation that is the Fourier transform of the indicator function of , up to some factors.
To get this going, let us first get our conventions straight. We define our Fourier transforms by
Then for any function , we have
Why is this useful? In general, it is difficult to say anything about the integral of products. However, the Fourier transform of a product is the convolution of the Fourier transforms, which is an operation we understand pretty well.
With our convention, we have
Here for any , the function is given by
Note that the area under is always . Fourier transforms take products to convolutions, and convolving with is pretty simple:
In words, the value of at is the average of the values of in .
With this in mind, we can look at
We start with the function , which is depicted above. Convolving with gives a piecewise linear function
Crucially, when , the value at is unchanged, since the function is constant on . The resulting function is constantly on the interval .
When we further convolve with , if , the resulting function is constantly on the interval . In general, this tells us that as long as , the integral will still be , and gets smaller afterwards.