chapter  13
44 Pages

The Fourier Transform

In this chapter, we will study properties of the Fourier transform̂f (x) of a function f on Rn, n ≥ 1, defined formally (for the moment) as

̂f (x) = 1 (2π)n

f (y) e−ix·ydy, x ∈ Rn. (13.1)

Here x · y = ∑n1 xkyk is the usual dot product of x = (x1, . . . , xn) and y = (y1, . . . , yn), and i is the complex number i =

√−1 = eiπ/2. Both f and̂f may be complex-valued.