Fourier Transforms Associated with the Line Group

Abstract

In the preceding chapter we have regarded the finite Fourier transform as a transform of one function space into another. This emphasis is a useful one in order to give a unified approach to Fourier analysis on different groups. This chapter is devoted to the study of the line group. Parallel to Sec. 4.1, Sec. 5.1 is concerned with the operational rules of the Fourier transform in L¹. The inversion theory will follow by the theory of singular integrals presented in Chapter 3. Included are results on generalized derivatives (Peano and Riemann) and connections with Fourier transforms and moments of positive functions. The relation between Fourier transforms and Fourier coefficients given by the Poisson summation formula is developed in Sec. 5.1.5. Sec. 5.2 is devoted to the definition of the Fourier transform for functions in L^p, 1 < p ≤ 2, including the Titchmarsh inequality (Theorem 5.2.9), Parseval’s formula (Prop. 5.2.13), and Plancherel’s theorem (Theorem 5.2.23). The operational rules are developed, together with the central Theorem 5.2.21. Sec. 5.3 is concerned with a thorough investigation of the Fourier-Stieltjes transform with its basic properties. We specifically mention the Levy inversion formula (Theorem 5.3.9) and the uniqueness theorem (Prop. 5.3.11).