Theory Of Hyperfunctions Research Articles

Sato hyperfunctions via relative Dolbeault cohomology

The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of Grothendieck and Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

The Stokes Phenomenon for Some Moment Partial Differential Equations

We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.

Achievements by Professor Akira Kaneko

In this article, we review the achievements by Professor Akira Kaneko, who mainly contributed to the continuation of regular solutions to linear partial differential equations and to the theory of hyperfunctions.

Hyperfunctions, formal groups and generalized Lipschitz summation formulas

A construction relating the theory of hyperfunctions with the theory of formal groups and generalizations of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli-type polynomials, related to the Lazard formal group. Related families of one-dimensional hyperfunctions are also constructed.

Vector Valued Hyperfunctions and Boundary Values of Vector Valued Harmonic and Holomorphic Functions

We develop the theory of hyperfunctions with values in a locally convex nonnecessarily metrizable space E and find necessary conditions and sufficient conditions such that a reasonable theory of E -valued hyperfunctions exists. In particular, we show that it exists for various spaces of distributions but there is no such theory for the spaces of real analytic functions and distributions with compact support. We also show that vector valued hyperfunctions can be interpreted as boundary values of vector valued harmonic or holomorphic functions and, in many cases, as suitable cohomology groups.

Fourier analysis, distributions and hyperfunctions

The theory of hyperfunctions and Fourier hyperfunctions is a really natural extension of the Schwartz theory of distributions and tempered distributions. We show that the naturalness of hyperfunctions comparing our results in hyperfunctions and the corresponding results in distributions in such areas as the characterization of test function spaces in terms of Fourier transformations and Bochner-Schwartz theorems. We have one important guiding theme to compare distributions and hyperfunctions. Estimates related to distributions are tempered or polynomially increasing and estimates related to hyperfunctions are slowly increasing or infra-exponentially increasing.

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces $S_\alpha^0$ which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley--Wiener--Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation.

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.

A calculus approach to hyperfunctions. II

We consider any hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results via heat kernel.

Operational calculus: A theory of hyperfunctions

Operateur d'integration h et operateur de differentiation S (classes des hyperfonctions: C et C H ). Equations differentielles ordinaires lineaires a coefficients lineaires (la classe C/C des hyperfonctions). Operateur de decalage exp(−λS) et operateur de diffusion exp(−λS 1/2 ). Applications aux equations aux derivees partielles

A $p$-Adic Theory of Hyperfunctions, I

Publisher Summary This chapter presents a p-adic theory of hyperfunctions of several variables by using relative cohomologies of rigid analytic spaces. The chapter reviews the general theory of relative cohomologies of rigid analytic spaces, and discusses the relation between the usual topology of K and the Grothendieck topology of X. A lemma on the relative cohomologies on a polydisk is proven, and the duality is obtained. As for the theory of rigid analytic spaces, the terminology of Bosch-Gunter-Remmert is used. Projective limits, and various lemmas are presented in the chapter. The chapter also proves various theorems.

Lorentz Invariant String Regularized Dimensionally by Means of Hyperfunctions. II

In this sequel the anomalies of the Virasoro algebra and the Poincare algebra of the free string arc calculated in the hypothetical space-time in continuous dimension:;. The light-like gauge is employed. The role of the center-of-mass variables is analyzed in the framework established in the preceding paper. The method of dimensional regularization is applied on the basis of the theory of hyperfunctions, which provides us with a more general concept than the analytic continuation with respect to the dimensionality. It is concluded that the above anomalies disappear, thus removing the origin of the critical dimension derived in the light-like gauge.

Lorentz Invariant String Regularized Dimensionally by Means of Hyperfunctions. I

In this sequel the anomalies of the Virasoro algebra and the Poincare algebra of the free string arc calculated in the hypothetical space-time in continuous dimension:;. The light-like gauge is employed. The role of the center-of-mass variables is analyzed in the framework established in the preceding paper. The method of dimensional regularization is applied on the basis of the theory of hyperfunctions, which provides us with a more general concept than the analytic continuation with respect to the dimensionality. It is concluded that the above anomalies disappear, thus removing the origin of the critical dimension derived in the light-like gauge.

Current Commutator Anomaly Regularized Dimensionally by Means of Hyperfunctions

In the framework of dimensional regularization the Goto-Imamura-Schwingcr term in spinor quantum electrodynamics is shown to vanish for any dimensionality. The implications are discussed in detail. In the limit of the massless electron, the usual argc1ment based on the analytic continuation with respect to the dimensionality is inadequate. The more general mathematical basis is provided by the theory of hyperfunctions.

Theory of fourier hyperfunctions and its applications to quantum field theory

A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.

Theory of Vector-Valued Hyperfunctions

Theory of Vector-Valued Hyperfunctions

Equations with constant coefficients invariant under a group of linear transformations

If P P is a linear differential operator on R n {{\mathbf {R}}^n} with constant coefficients, which is invariant under a group G G of linear transformations, it is not true in general that the equation P u = f Pu = f always has a G G -invariant solution u u for a G G -invariant f f . We elucidate here the particular case of a “big” group G G , and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.

Pseudo-differential operators in the theory of hyperfunctions

Pseudo-differential operators in the theory of hyperfunctions

Pseudo-differential Operators in the Theory of Hyperfunctions

Pseudo-differential Operators in the Theory of Hyperfunctions

The Theory of Hyperfunctions on Totally Real Subsets of a Complex Manifold with Applications to Extension Problems

Introduction. Suppose K is a compact subset of a Stein manifold. The set K is said to be holomorphically convex if K is homeomorphic to the spectrum of the topological algebra of analytic functions on K (Definition 2. 1). Because of the results proved in iHarvey-Wells [7], the proof of Sato's duality theorem [15] given in Martineau [10] applies to compact holomorphically convex sets (see Theorem 2. 2).