Bounded variation and sampling

Abstract

In this chapter, we shall consider certain problems where a function of bounded variation generates trigonometric series which are then compared with its Fourier integral. In fact, Chapter 4 in [198] is devoted to these problems. Here, the more modern term “sampling” is equivalent to the older “discretization”. Those who expected a sort of Whittaker–Kotel’nikov–Shannon type matter might be disappointed. The present chapter can be considered as a development of certain of the results in [198]. One type of these results, the Poisson summation formula, is old and classical. However, results related to bounded variation are specific and a bit off a field. The other one is more recent and aims to the comparison of Fourier integrals and trigonometric series related to functions and sequences with bounded variation.