Microlocal Analysis and Complex Fourier Analysis

PREFACE

CONTENTS

The following sections are included:

• Introduction

• An equation with simple turning points
  • Verification by the steepest descent method
  • Verification by using the connection formula

• A degenerate equation with simple turning points
  • Verification by the exact steepest descent method
  • Discussion by adding a small perturbation

• An equation with double turning points
  • Verification by the exact steepest descent method
  • Discussion by adding a small perturbation
  • Verification by using the connection formula

• Conclusions

• Acknowledgments

• References

The existence and blowing up of solutions are discussed for the parabolic equation

We discuss the key role the theory of residues plays in complexity problems in Algebra and constructive procedures in Algebraic Geometry.

We extend the results of [4] to include integrals with a more general measure. Our goal is to utilize a distribution based on an arbitrary radial measure and including a moment factor, i.e.,

The following sections are included:

• Introduction

• Notations and Preliminaries

• Uniqueness Theorems

• Example of Nonuniqueness

• Reflection Principles

• References

We consider the Bergman kernel for the N_p-ball which is defined by the N_p-norm and is related to the Lie ball. Then we study the Bergman transformation for analytic functionals on the N_p-ball and study a relation with the Fourier-Borel transformation.

We generalize in two ways the method of construction of Fourier hyperfunctions supported at infinity proposed by Morimoto-Yoshino. Namely, on one hand, we generalize the growth order to most general admissible ones. On the other hand, we construct the example in general quasi-analytic ultradistributions.

The following sections are included:

• Introduction

• Characteristic features of the problem

• Study of some basic examples

• References

In this paper we concerned with properties of the closed linear hull of exponentials in the space of entire functions in Cⁿ. Some applications to convolution equations are considered.

The following sections are included:

• Introduction

• Generalized functions as initial values of solutions of the heat equation

• Heat kernel method and hyperfunctions
  • Positive definite hyperfunctions
  • Hyperfunctions of L^P growth
  • Periodic hyperfunctions
  • Almost periodic hyperfunctions
  • Mehler Kernel Approach

• Acknowledgments

• References

Salomon Bochner (1899–1982) and Torsten Carleman (1892–1949) presented generalizations of the Fourier transform of functions defined on the real axis. While Bochner's idea was to define the Fourier transform as a (formal) derivative of high order of a function, Carleman, in his lectures in 1935, defined his Fourier transform as a pair of holomorphic functions and thus foreshadowed the definition of hyperfunctions. Jesper Lützen, in his book on the prehistory of the theory of distributions, stated two problems in connection with Carleman's generalization of the Fourier transform. In the article these problems are discussed and solved.

The following sections are included:

• Introduction — Brief review of exact steepest descent method and several problems of it

• The effect of a new Stokes curve

• The effect of a singular point of simple pole type

• The effect of a constant characteristic root

• Concluding remarks

• Acknowledgments

• References

Mathematical folklores are convincing, especially when told by great mathematicians, but they are not necessarily based on the historical facts. “Fourier formulated his expansion theorem but failed to give a proof.” “Heaviside invented operational calculus.” “The δ-function was introduced by Dirac.” “The theory of distributions clarifies all mysteries in operational calculus.” Reading Fourier's book and Heaviside's papers, we give negative evidence to the above. A knowledge of hyperfunctions and ultradistributions makes it easy to understand these works.

This note discusses briefly some geometric aspects of large deviation for random walks on a crystal lattice. The motivation arises from our attempts to apply ideas in the Bloch-Floquet theory, developed originally for periodic Schrödinger operators on the Euclidean space, to the transition operators. Indeed, if L denotes the transition operator associated with a periodic transition probability on a crystal lattice, then I – L is regarded as a discrete analogue of the Laplacian which is equivariant under the action of a lattice group. As was established in our previous work, unitary characters of a lattice group play a significant role in the proof of the central limit theorem. What we shall observe in this note is that real characters show up in the study of large deviations. We also see that the Gromov-Hausdorff limit of metric spaces homothetic to a crystal lattice has something to do with the large deviation property of random walks. The detailed accounts of this on-going research will be published elsewhere.

There are two definitions of Boehmians on the sphere; the first one relied on zonal delta sequences, while the second on central delta sequences. In this note we introduce semi-groups of zonal delta sequences to distinguish several spaces of Boehmians on the sphere and investigate their relations. We also show the second method reduces to the first one.

A new Lax pair for the sixth Painlevé equation P_VI is constructed in the framework of the loop algebra . The Bäcklund transformations for P_VI are interpreted as gauge transformations of the corresponding linear problem.

A generalization of the Laurent expansion of holomorphic functions of one variable to the case of linear reductive complex Lie groups is discussed.

An open subset D ⊂ ℝⁿ is called a Laplace domain (resp. complex Laplace domain), if there exists a series where Q_j is a homogeneous harmonic polynomial of n variables of degree j such that D = {a ∈ ℝⁿ; the series is convergent in a neighborhood of a in ℝⁿ} (resp. D = {a ∈ ℂⁿ; the series is convergent in a neighborhood of a in ℂⁿ}). On the plane ℝ² the only Laplace domains are disks with centre 0. For all n > 2 Laplace domains exhibit grater variety, even non-convex ones are among them. So far we do not know a geometrical characterization of Laplace domains for n > 2. We show that a balanced domain D in ℂⁿ is a complex Laplace domain if it is a domain of harmonic holomorphicity. A convex balanced domain is a complex Laplace domain iff whenever the normal vector v exists at a point a of ∂D, it is a null vector, i.e. .

This paper considers the following nonlinear first order partial differential equation ∂u/∂t = F(t, x, u, ∂u/∂x) in the complex domain. The main purpose is to examine whether or not the equation possesses solutions which admit singularities only on the hypersurface S = {t = 0}. This will be done either by examining the possibility of analytic continuation of solutions or by actually constructing solutions that possess singularities only on S.

The following sections are included:

• Introduction

• The Fourier-Borel transform of an algebraic local cohomology class

• Noetherian differential operators

• Algorithm

• Cauchy problem

• References

The following sections are included:

• Introduction

• Reproducing kernels of the space of harmonic polynomials in case.

• Reproducing kernels of the space of harmonic polynomials in case.

• Proof of Lemma 1.2.

• References

Generalized functions and hyperfunctions have been characterized in terms of their expansions in Fourier series and other series of orthogonal functions, such as Legendre, Jacobi, Laguerre, Hermite, and Faber. In this paper we give a characterization of a class of analytic hyperfunctions on a two-dimensional complex manifold in terms their expansions in series of products of Lame functions.