Integrability of the Fourier transforms

In this chapter, we first obtain certain multidimensional results for the integrability of the Fourier transform (see (5.6)) $$ \hat{f} (x) = \int_{\mathbb{R}^n}\, f(u)e^{-i \langle x, u \rangle}\, du,$$ which extend some theorems from Part I. Let us mention that various results of that kind can be found in the survey paper [128]. For example, in Section 7 of that paper many results are extensions of Zygmund’s test for the absolute convergence of Fourier series of a function of bounded variation. Concerning the results we are going to present here, our objective is not to generalize “everything”. On the contrary, we restrict ourselves to a very modest task: to generalized very few results, mostly to demonstrate how the indicator notation works and to give some background for discretization results in Chapter 8. Moreover, one of the goals is to show how “bad” is the direct generalization of some results as compared with those in the next Chapter 7. The latter extends the advanced results from Chapter 4.