Functions of Exponential Type and Bounded on the Real Space ( Fourier Transforms of Distribution of Compact Support )

Abstract

In this chapter we consider entire functions of at most normal type with respect to the order 1 (i.e. functions of exponential type) that are bounded for real values of the variables. The importance of this class of functions is lies in the fact that it contains the Fourier transforms of functions of compact support and belonging to L1 (ℝn). Besides, since the Fourier transforms of distributions of compact support are entire functions of polynomial growth on ℝn , by “suppression of growth” their study can often be reduced to that of corresponding functions which are bounded on ℝn . Side by side with the problem of regularity of growth, which is the main topic of this book, we study some problems on the connection between the behaviour of the functions under consideration on the whole space and their behaviour on certain discrete sets. Such problems are of interest for some applications, in particular, for completeness problems of systems of exponentials or monomials.