Analytic Properties

Having presented the elementary properties of hyperfunctions in the first chapter, we shall now enter into more subtle topics. First, sequences and series of hyperfunctions are investigated, then Cauchy-type integrals that play an important part in the theory of hyperfunctions are discussed. The basic question of any theory of generalized functions, namely, how ordinary functions can be embedded in their realm, is investigated (projection of an ordinary function). The book of [12, Gakhov] has been very helpful for the treatment of this question. The subject of the projection or restriction of a hyperfunction to a smaller interval is then exposed. The important notions of holomorphic hyperfunction, analytic and micro-analytic hyperfunctions are discussed, and the more technical concepts such as support, singular support and singular spectrum are introduced. The product of two generalized functions is always a difficult point in any theory about generalized functions. Generally, the product of two generalized function cannot be defined. We shall discuss under what circumstances the product of two hyperfunctions makes sense. The sections on periodic hyperfunctions and their Fourier series and the important subject of convolution of hyperfunctions form the ending material of this chapter. Also, the track of applications to integral and differential equations starts here.