\begin{matrix}c \\ ◯ \end{matrix} Jockum Aniansson ISBN 91 – 7170 – 453 – 1

TRITA – MAT – 17 – MA – 1999

ISSN 1401 – 2278

ISRN KTH/MAT/R – – 99/17 – – SE KTH Högskoletryckeriet Stockholm

A D MCMXCIX

Foreword

Svenska ⟶

This thesis would never have been started without Harold Seymour Shapiro. It would never have been finished without Ari Laptev. I thank these good men with all my heart. Their help has been truly invaluable. This thesis is my tribute to their unfailing support and friendship. Only a minute fraction of all their intriguing ideas and suggestions have found its reflection in these pages.

The rate of convergence of this work has been at least quadratic.

The cardinality of the set of people I would like to thank is too large. My mathematical friends are too numerous to be enumerated.

Thank you all!

Jockum Aniansson Stockholm in September

AD MIM

Abstract

This thesis consists of two parts. The first part deals with some integral representations in n -dimensional real Euclidean space, see I. The second part deals with some integral representations in n -dimensional complex Euclidean space, see II and III. I Multi–dimensional Peano–Sard kernels and a divergence representation. We prove a divergence representation with increased smoothness and control over the support, valid for compactly supported distributions. We then construct multi– dimensional Peano–Sard kernels with optimal regularity. These kernels are used to represent remainder functionals (error functionals) in e g numerical quadrature or cubature rules. We also obtain a representation theorem for entire analytic functions of exponential order vanishing at a point. II Fischer kernels, and projectors in some spaces of analytic functions. This paper is concerned with direct sum decompositions of some classical spaces of analytic functions in n complex variables. In some specific constellations we derive concrete forms for reproducing or representing kernels for certain subspaces of functions. After a Fourier transformation this will explicitly give us the Green’s function for some characteristic and non-characteristic Cauchy problems. III Harmonic projections in Fischer–Fock space. In this article we show how to decompose any entire analytic function f(z_{1}, ⋯ z_{n}) uniquely as a sum of two entire functions f = h + g , where

△ h = △_{Z} h = ( \frac{∂}{∂z_{1}^{2}} + ⋯ + \frac{∂}{∂z_{n}^{2}} ) h( z_{1} , ⋯, z_{n}) = 0 ,

and the quotient g( z_{1} , ⋯, z_{n}) / ( z_{1}^{2} + ⋯ + z_{n}^{2} ) is entire. This is most conveniently done using orthogonal projections within the framework of Fischer–Fock space (Bargmann space) by means of the Fischer kernels we determine. Fourier transformation then gives us an explicit solution to the characteristic Cauchy problem for the wave equation.

Key words and phrases. Integral representations. Peano kernel. Sard kernel. Divergence representation. Bargmann–Segal space. Fock space. Reproducing kernel Hilbert space. Clebsch projection. Fischer pair. Characteristic Cauchy problem.

MSC 2000 Mathematics Subject Classification. Primary 41 A 80, 46 E 22. Secondary 35 C 15.

Contents

Summary  1
Peano–Sard part (paper I)  1
Fischer part (papers II and III)  8
Historical survey  8
Description of papers II and III  16
References  24

Paper I. Multi–dimensional Peano–Sard kernels and

a divergence representation  1
Preamble  1
1. Introduction  2
2. Weighted Taylor formulæ  6
3. Pseudo–differential operators  10
4. Some L^{p} mapping properties  13
5. Fourier transform of the divergence lemma  18
Appendix  19
References  21

Paper II. Fischer kernels, and projectors in some spaces

of analytic functions  1
1. Introduction  1
2. Fischer decomposition for polynomials  3
3. Space list  6
4. The Borel transformation  9
5. Martineau duality  12
6. Fischer pairs  13
7. Some examples  17
8. Fischer kernels  19
9. Constellation C  23
10. Constellation D  29
11. Fourier treatment  32
12. Fourier interpretations  35
13. Constellation E  38
14. Appendix. The Bargmann isometry  39

References  40
Illustration  42

Paper III. Harmonic projections in Fischer–Fock space  1
1. Introduction  1
2. The Clebsch projection  3
3. Fischer decomposition of exp(z_{n}) relative to z_{1}^{2} + ⋯ + z_{n}^{2}  5
4. Integral representations for Kummer’s function  8
5. Finding the kernel h  9
6. Some comments  12
7. Representation for the complementary kernel g  14
8. Splitting of e^{U·Z} , U ∈ C^{n}; Fischer kernels  14
9. Decomposition in Fischer–Fock space  16
10. Fischer decomposition of exp i(u_{1}z_{1} + ⋯ + u_{n−1}z_{n−1} − u_{n}z_{n})
relative to z_{1}^{2} + ⋯ + z_{n−1}^{2} − z_{n}^{2}  18
11. The light cone  19
12. Finding the distribution ĥ(U, ·)  20
13. Results for the wave equation  24
References  25
Illustrations  26