Fourier Analysis on ℝⁿ

Abstract

In this chapter we review basic elements of Fourier analysis on ℝⁿ. Consequently, we introduce spaces of distributions, putting emphasis on the space of tempered distributions S′(ℝⁿ). Finally, we discuss Sobolev spaces and approximation of functions and distributions by smooth functions. Throughout, we fix the measure on ℝⁿ to be Lebesgue measure. For convenience, we may repeat a few definitions in the context of ℝⁿ 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 ∈ ℝⁿ.