Fourier Analysis on ℝ n

In this chapter we review basic elements of Fourier analysis on ℝ n . Consequently, we introduce spaces of distributions, putting emphasis on the space of tempered distributions S′(ℝ n ). Finally, we discuss Sobolev spaces and approximation of functions and distributions by smooth functions. Throughout, we fix the measure on ℝ n to be Lebesgue measure. For convenience, we may repeat a few definitions in the context of ℝ n although they may have already appeared in Chapter C on measure theory. From this point of view, the present chapter can be read essentially independently. The notation used in this chapter and also in Chapter 2 is 〈ξ〉 = (1 + |ξ|2)1/2 where |ξ| = (ξ12 + ξ n 2)1/2, ξ ∈ ℝ n .