Introduction to Hyperfunctions and Their Integral Transforms

After a short overview of generalized functions and of the different ways they can be defined, the concept of a hyperfunction is established, followed by an introduction to the most simple and familiar hyperfunctions. Then the elementary operational properties of hyperfunctions are presented. The so-called finite part hyperfunctions are introduced, followed by the important notion of the (definite) integral of a hyperfunction. The chapter closes with a definition of more familiar hyperfunctions frequently used in applications. The books of [19, Imai] and [20, Kaneko] are the principal references for this chapter.

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.

We assume that the reader will have some degree of familiarity with the subject of classical Laplace transformation such as presented for example in [14, Graf]. After a discussion on loop integrals of Hankel type and their ramifications, some facts about the classical two-sided Laplace transformation are recalled. Then, the two subclasses of hyperfunctions, the right-sided and left-sided originals and their Laplace transforms, are defined by using loop integrals. The Laplace transform of a hyperfunction with an arbitrary support is handled by decomposing it into a sum of a left-sided and right-sided original (canonical splitting); it is then shown that its practical computation can be reduced to the evaluation of two right-sided Laplace transforms. Many concrete examples of Laplace transforms of hyperfunctions are presented. The operational rules of Laplace transforms of hyperfunctions are clearly stated. The subject of inverse Laplace transforms and convolutions follows. Fractional integrals and derivatives of right-sided hyperfunctions are briefly over-viewed. The application track with Volterra integral equations and convolution integral equations over an infinite range concludes the chapter. This chapter represents the core of the integral transformations part of the book. The following chapters on Fourier and Mellin transformations are heavily based on the results of this chapter. Another similar approach, due to Komatsu introducing the so-called Laplace hyperfunctions, is sketched in the Appendix.

Because the convolution of two hyperfunctions is an established concept, we may use it to define the Hilbert transform of a hyperfunction. On the other hand, the classical Hilbert transform is also linked to the Cauchy-type integral. These two parents have unfortunately led to the fact that there is no adopted standard definition of the Hilbert transform. After some hesitation I have chosen the definition which relays on the Cauchy-type integral.

First, the classical Mellin transformation is introduced and the connection with the two-sided Laplace transformation is established. Several Mellin transforms of ordinary functions are then computed. In order to define the Mellin transform of a hyperfunction, the established connection with Laplace transformation is exploited. This connection is also used to establish all operational rules that govern the Mellin transformation. The two types of Mellin convolutions as well as the Mellin transform of a product and Parseval’s formula are then treated. Some simple applications conclude the chapter.

First we show that the conventional Hankel transform pair arises in a natural way when, in the two-dimensional Fourier transformation, polar coordinates are introduced. Unfortunately, no firm convention about the definition of the Hankel transform pair is established. We shall use the most widespread one. In order to lay the groundwork for the theory of Hankel transformation of hyperfunctions, we present a concise exposition of the various cylinder functions, the integrals of Lommel and MacRobert’s proof of the inversion formula. The Hankel transform of a hyperfunction defined on the positive part of the real line is then defined by using the Hankel functions for the kernel. Along the line of MacRobert’s proof and using the integrals of Lommel, we then prove that the defined Hankel transform of a hyperfunction is a self-reciprocal transformation when restricted to the strictly positive part of the real axis. The operational rules known for the Hankel transformation of ordinary functions are then carried over to the Hankel transformation of hyperfunctions. The chapter closes with a few applications about problems of mathematical physics.