Hikosaburo Komatsu

Operational Calculus, Hyperfunctions and Ultradistributions

A review is given on a new foundation of the Heaviside operational calculus based on the Laplace transformation of hyperfunctions. Then the hyperfunctions with support bounded from below and the Mikusinski operators are compared. Neither of them is included in the other. The ultradistributions (in a wide sense) with support bounded from below are shown to form an important subclass of the intersection.

Irregularity of Hyperbolic Operators

Publisher Summary This chapter explores the Gevrey classes and the irregularity condition. The irregularity condition is necessary in general because a formally hyperbolic equation with real analytic coefficients is well-posed in a corresponding Gevrey class of functions and ultradistributions. The irregularity is a microlocal invariant. The irregularity of a microdifferential operator is also defined as analytic pseudodifferential operator that is relative to a microdifferential operator of simple characteristic. The chapter highlights the methods of Hamada and Ōuchi regarding the Cauchy problem in the complex domain for meromorphic data. It also describes the Heaviside calculus.

Linear ordinary differential equations with Gevrey coefficients

Ultradistribution solutions of Gevrey classes and distribution solutions are discussed for linear ordinary differential equations. The results previously established for equations with real analytic coefftcients are extended to the case where the coefficients are in the corresponding Gevrey class or infinitely differentiable under a natural condition on the irregularity at each singular point of the equation. We consider the linear differential operator

Ultradistributions and Hyperbolicity

There are infinitely many classes of generalized functions, called ultradistributions, between the distributions of L. Schwartz [34] and the hyperfunctions of M. Sato [32]. Each class of ultradistributions have similar properties as the distributions or the hyperfunctions. They form a sheaf on which linear differential operators act as sheaf homomorphisms. We have, among others, two structure theorems of ultradistributions, the structure theorem of ultradistributions with support in submanifold (which implies a Whitney type extension theorem for ultradifferentiable functions), and the kernel theorem for ultradistributions.

Seki’s Theory of Elimination as Compared with the Others’

This is an enlarged version of the authors’ papers published in Journal of Northwest University (Natural Science Edition), 33 Nos.3 and 4 (2003). Added is Section 3, where proofs are given of the main properties of determinants and resultants as originally introduced by Seki Takakazu (1642?–1708), Etienne Bezout (1739–83), James J. Sylvester (1814–97) and Arthur Cayley (1821–95).

Hyperfunctions, Ultradistributions and Microfunctions

As exemplified in the introduction of L. Schwartz’ book [24] individual generalized functions had been known before he introduced the distributions D′ (Ω) in 1948–51. Dirac’s δ function and Hadamard’s finite part of x + α , α < -1, are famous. More generally P. Levy [16] had determined the dual of C m ([a, b]). Yet the true history of generalized functions started with Schwartz.

An elementary theory of hyperfunctions and microfunctions

1. Sato’s definition of hyperfunctions. The hyperfunctions are a class of generalized functions introduced by M. Sato [34], [35], [36] in 1958–60, only ten years later than Schwartz’ distributions [40]. As we will see, hyperfunctions are natural and useful, but unfortunately they are not so commonly used as distributions. One reason seems to be that the mere definition of hyperfunctions needs a lot of preparations. In the one-dimensional case his definition is elementary. Let Ω be an open set in R. Then the space B(Ω) of hyperfunctions on Ω is defined to be the quotient space (1.1) B(Ω) = O(V \Ω)/O(V ) , where V is an open set in C containing Ω as a closed set, and O(V \Ω) (resp. O(V )) is the space of all holomorphic functions on V \Ω (resp. V ). The hyperfunction f(x) represented by F (z) ∈ O(V \Ω) is written (1.2) f(x) = F (x+ i0) − F (x− i0) and has the intuitive meaning of the difference of the “boundary values”of F (z) on Ω from above and below.

Dual Kōmura Spaces

A dual Kōmura space is by definition a locally convex space E in which for any absolutely convex bounded set A there is another B ⊃ A such that the natural mapping E A → E B is weakly compact. Fundamental properties and hereditarity are discussed together with motivations and applications.

The Necessity of the Irregularity Condition for Solvability in Gevrey Classes (s) and {s}

The author reviews briefly the classical theory of homogeneous solutions of linear ordinary differential equations near an irregular singular point and its application to the existence of ultradistribution solutions of Gevrey classes. Then he develops an analogous theory for formal solutions of linear partial differential equations near a characteristic surface of constant multiplicity. As a consequence he shows that the irregularity condition he introduced earlier in [13] and [14] is necessary in general in order that a formally hyperbolic equation with real analytic coefficients be well posed in a corresponding Gevrey class of functions and ultradistributions.

A Generalization of the Cauchy-Kowalevsky Theorem and Boundary Values of Solutions of Elliptic Equations

An analogue of the Cauchy-Kowalevsky theorem is given for ultradifferentiable functions, and its dual is used to characterize those solutions of a homogeneous elliptic equation on one side of a hyperplane that have boundary values in a class of ultradistributions.