# Problems in Analysis: A Symposium in Honor of Salomon Bochner (PMS-31)

ROBERT C. GUNNING GENERAL EDITOR
Pages: 366
https://www.jstor.org/stable/j.ctt130hk8g

1. Front Matter
(pp. i-vi)
2. Foreword
(pp. vii-viii)

A symposium on problems in analysis in honor of Salomon Bochner was held in Fine Hall, Princeton University, April 1–3, 1969, to celebrate his seventieth birthday, which took place on August 20, 1969. The symposium was sponsored by Princeton University and the United States Air Force Office of Scientific Research; the organizing committee consisted of W. Feller, R. C. Gunning, G. A. Hunt, D. Montgomery, R. G. Pohrer, and W. R. Trott.

This volume contains some of the papers delivered by the invited speakers at the symposium, together with a number of papers contributed by former students of Professor...

(pp. ix-xii)
4. ### PART I: LECTURES AT THE SYMPOSIUM

• On the Group of Automorphisms of a Symplectic Manifold
(pp. 1-26)
EUGENIO CALABI

Let X be a connected, differential manifold of 2ndimensions. Asymplectic structureon X is the geometrical structure induced by a differentiable exterior 2-formωdefined on X, satisfying the following conditions:

(i) The formωis closed:= 0;

(ii) It is everywhere of maximal rank; this means that the 2n-formωn(nth exterior power ofω) is everywhere different from zero, or equivalently, the skew-symmetric (2n) × (2n) matrix of coefficients ofω, in terms of a basis for the cotangent space, is everywhere nonsingular.

A classical theorem, ordinarily attributed to Darboux, states that a 2n-dimensional...

• On the Minimal Immersions of the Two-sphere in a Space of Constant Curvature
(pp. 27-40)
SHIING-SHEN CHERN

This study arose from reading Calabi’s interesting paper [2] on the minimal, immersions of the two-sphereS² into anN-dimensional sphereSN. Using an idea of H. Hopf, Calabi observed the strong implications of the fact that the submanifold is a two-sphere. His treatment made essential use of the global coordinates onSN. We shall give a different approach by supposing only that the ambient space is a Riemannian manifold of constant sectional curvature. Some general remarks on the higher osculating spaces are intended to make the development more natural and to prepare the way for future applications to the...

• Intersections of Cantor Sets and Transversality of Semigroups
(pp. 41-60)
HARRY FURSTENBERG

LetXbe a compact metric space endowed with a notion of dimension for subsets ofX(analogous to Hausdorff dimension for subsets of a manifold). Two closed subsetsA, BXwill be calledtransverseif

(1)\mathrm{dim}\; A\cap B\leqq\left\{\begin{matrix} \mathrm{dim}\; A+\mathrm{dim}\; B-\mathrm{dim}\; X\; \mathrm{when\; this\; is}\geqq0 \\ 0\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \mathrm{otherwise} \end{matrix}\right.

LetS1andS2be two semigroups (or groups) acting onX(i.e., eachS1is a semigroup of continuous transformations ofXinto itself).S1andS2will be calledtransverseif wheneverAis a closedS1-invariant set andBa closedS2-invariant set,AandBare transverse. Finally, two transformationsT1andT2ofXX...

• Kählersche Mannigfaltigkeiten mit hyper-q-konvexem Rand
(pp. 61-80)
HANS GRAUERT and OSWALD RIEMENSCHNEIDER

1953 zeigte K. Kodaira [3], daß die KohomologiegruppenHs(X, F) fürs<nverschwinden, wennXeinen-dimensionale kompakte kählersche Mannigfaltigkeit undFein negatives komplex-analytisches Geradenbündel aufXist. Hierbei bezeichnetFwie üblich die Garbe der Keime von lokalen holomorphen Schnitten inF. Kodaira verwandte beim Beweis dieses Ergebnisses neben der harmonischen Analysis wesentlich eine Ungleichung, die zuerst von S. Bochner angegeben wurde (man vgl. [2]). Bis heute ist es nicht gelungen, diesen Satz für vollständige projektiv-algebraische Mannigfaltigkeiten auf algebraischem Wege herzuleiten.

Man gelangte jedoch zu mehreren Verschärfungen und Verallgemeinerungen. Von Y. Akizuki und S. Nakano wurde bewiesen [1],...

• Iteration of Analytic Functions of Several Variables
(pp. 81-92)
SAMUEL KARLIN and JAMES McGREGOR

Professor Salomon Bochner has contributed notably in many areas of mathematics including the theory of functions of several complex variables and the theory of stochastic processes.

In line with these interests, this paper reports results on iteration of holomorphic functions of several complex variables motivated by investigations pertaining to multi-type branching Markoff processes. Apart from its intrinsic importance and independent interest, iteration of holomorphic mappings plays a fundamental role in celestial mechanics, population genetics, numerical analysis, and other areas.

Consider a vector-valued mapping

(1)f(z)=(f_{1}(z),f_{2}(z),\cdots ,f_{p}(z)),

z=(z_{1},z_{2},\cdots ,z_{p})\in Z_{p}

whereZpdenotes the space ofp-tuples of complex numbers. Suppose the mapping...

• A Class of Positive-Difinite Functions
(pp. 93-110)
J. F. C. KINGMAN

Among the many distinguished results associated with the name of Salomon Bochner, one of the more widely known is his characterization of the positive-definite functions, those continuous functionsfof a real variable for which any choice of the valuest_{1},t_{2},\cdots ,t_{n}makes

(1)(f(t_{\alpha }-t_{\beta });\alpha ,\beta =1,2,\cdots ,n)

a non-negative-definite Hermitian matrix. These are exactly the functions expressible in the form

(2)f(t)=\int e^{i\omega t}\phi (d\omega),

for totally finite (positive) measuresϕon the Borel subsets of the real line. In the case of a real-valued functionf(necessarily even), (2) takes the form

(3)f(t)=\int \mathrm{cos}\; \omega t\; \phi (d\omega),

andϕmay then be regarded as concentrated on...

• Local Noncommutative Analysis
(pp. 111-130)
IRVING SEGAL

Despite the inroads of linear functional analysis, much of analysis still deals ultimately not with vectors in a linear space, but with functions defined on a suitably structured point set, generally endowed with properties reminiscent of physical space. It seems as if some notion of physical space is quite possibly a primordial concept in the human mind and inevitably colors all our perceptions of nature and formulations of natural law. The mathematical emphasis on equations which arelocalin geometrical space (or transforms of such equations) and the complementary physical idea that the primary forces in nature are exerted through...

5. ### PART II: PAPERS ON PROBLEMS IN ANALYSIS

• Linearization of the Product of Orthogonal Polynomials
(pp. 131-138)

In [3] Bochner showed that there is a convolution structure associated with ultraspherical polynomials which generalizes the classicalL¹ convolution algebra of even functions on the circle. Bochner uses the addition formula to obtain the essential positivity result. In [18] Weinberger shows that this positivity property follows from a maximum principle for a class of hyperbolic equations. Hirschman [8] has dualized this convolution structure and has proven the required positivity result by means of a formula of Dougall which linearizes the product of two ultraspherical polynomials. We will prove a theorem which gives most of Hirschman’s results as well as...

• Eisenstein Series on Tube Domains
(pp. 139-156)
WALTER L. BAILY JR.

We wish to prove here that under certain conditions the Eisenstein series for an arithmetic group acting on a tube domain [8], [9] in Cmgenerate the field of automorphic functions for that group. The conditions are that the domain be equivalent to a symmetric bounded domain having a O-dimensional rational boundary component (with respect to the arithmetic group) [3, Section 3] and that the arithmetic group be maximal discrete in the (possibly not connected) Lie group of all holomorphic automorphisms of the domain. More precisely, under the same conditions, we prove that certain types of polynomials in certain linear...

• Laplace–Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physic
(pp. 157-174)
JOHN L. BARNES

For clarity this paper begins with the very simple examples shown in Figs. 1a, lb, and 2. These illustrate the idea that a simple function can contain choice information (see [2]) in two ways. In these figures the elements chosen are shaded. To construct messages they are placed in positions located in intervals on the scale of abscissas calledcellsand on the scale of ordinates calledstates. It is assumed that they are chosen independently and that the binary locations are chosen with equal statistical frequency. In Fig. 1a the information is contained in the height, i.e., ordinate of...

• An Integral Equation Related to the Schroedinger Equation with an Application to Integration in Function Space
(pp. 175-194)
R. H. CAMERON and D. A. STORVICK

In a previous paper [l]² the authors proved the existence of a solution of the equation³

\Gamma (t,\xi,q)=(\frac{q}{2\pi it})^{1/2\: (\xi)}\int_{-\infty }^{\infty }\psi (u)e^{iq(\xi -u)^{2/}2t}du

1.1+(\frac{q}{2\pi i})^{1/2}\int_{o}^{t}\frac{1}{(t-s)^{1/2}}^{(\xi )}\int_{-\infty }^{\infty }\theta (s,u,q)e^{iq(\xi -u)^{2}2(t-s)}\; du\; ds

for almost every realqunder the following conditions. It was assumed that\theta (t,u)is continuous almost everywhere in the stripR:0≦tt0, —∞<u<∞, |θ(t, u)| ≦Mfor (t, u) inRand that\psi (u)\in L_{2}(-\infty ,\infty ).

Moreover it was shown⁴ that for eacht\in (0,t_{0}]and for almost every reqlq, the solution satisfies the inequality

(1.2)\int_{-\infty }^{\infty }|\Gamma (t,\xi ,q)|^{2}d\xi \leqq||\psi ||^{2}e^{2Mt_{0}}.

It is the purpose of thise paper to obtain a corresponding uniqueness theorem and then to strengthen both the existence and the uniqueness...

• A Lower Bound for the Smallest Eigenvalue of the Laplacian
(pp. 195-200)
JEFF CHEEGER

Various authors have studied the geometrical and topological significance of the spectrum of the Laplacian Δ², on a Riemannian manifold. (The excellent survey article of Berger [2] contains background, references, and open problems.) The purpose of this note is to give a lower bound for the smallest eigenvalue λ > 0 of Δ² applied to functions. The bound is in terms of a certain global geometric invariant, essentially the constant in the isoperimetric inequality. The technique works for compact manifolds of arbitrary dimension with or without boundary.

The author wishes to thank J. Simons for helpful conversations and in particular for...

• The Integral Equation Method in Scattering Theory
(pp. 201-228)
C. L. DOLPH

The subject of this essay seems singularly appropriate to this occasion for several reasons. Much of the material on which it depends stems from the Berlin and Munich schools where Salomon Bochner spent many of his early mathematical years. The foundations of the theory are perhaps best expressed via the Bochner integral (see Wilcox [1] and Dolph [2]). Finally, the subject has reached some sort of culmination for the direct problems with the as yet unpublished work of Shenk and Thoe [3], while the inverse problem has achieved a significant new impetus from the recently published work of Lax and...

• Group Algebra Bundles
(pp. 229-238)
BERNARD R. GELBAUM

LetGbe a topological space and letAbe a Banach space. OnG, the σ-ring generated by the compact sets ofG, letμbe a measure (such thatμ(Κ) < ∞ for all compactK). LetEbe a Banach algebra bundle (an equivalence class of Banach algebra coordinate bundles) with total spaceE, base spaceG, fibreA, projectionp: E->G. Let one coordinate bundle\tilde{B}inEbe specified by an open coveringŨofGand for eachUŨa coordinate map (fibre-preserving homeomorphism)\varphi _{u}:U\times A\rightarrow p^{-1}(U). Furthermore, let the group of the bundle...

• Quadratic Periods of Hyperelliptic Abelian Integrals
(pp. 239-248)
R. C. GUNNING

Consider a compact Riemann surfaceMof genusg> 0, represented in the familiar manner as the quotient space of its universal covering surface\tilde{M}by the covering translation group Γ. For the present purposes the only thing one needs to know about the surface\tilde{M}is that it is a simply connected noncompact Riemann surface. The group Γ is properly discontinuous and has no fixed points; and the quotient space\tilde{M}/\Gammais analytically equivalent to the Riemann surfaceM. The image of a pointz\in \tilde{M}under an automorphism T ∈ Γ will be denoted byTz. Some fixed but...

• The Existence of Complementary Series
(pp. 249-260)
A. W. KNAPP and E. M. STEIN

LetGbe a semisimple Lie group. The principal series forGconsists of unitary representations induced from finite-dimensional unitary representations of a certain subgroup ofG. These representations are not all mutually inequivalent, and their study begins with a study of the operators that give the various equivalences—the so-called intertwining operators.

ForG=SL(2,R), these operators are classical transformations. The principal series can be viewed conveniently as representations onL² of the line orL² of the circle. In the first case, the operators are given formally by scalar multiples of

(1.1a)f(x)\rightarrow \int_{-\infty }^{\infty }f(x-y)|y|^{-1+it}dy

and

(1.1b)f(x)\rightarrow \int_{-\infty }^{\infty }f(x-y)(\mathrm{sign}\; y)|y|^{-1+it}dy....

• Some Recent Developments in the Theory of Singular Perturbations
(pp. 261-272)
P. A. LAGERSTROM

In the nineteenth century celestial mechanics played an essential role in developing perturbation methods and asymptotic theory. This work culminated in Poincaré’s great treatise,Les Méthodes Nouvelles de la Mechanique Céleste. In the twentieth century fluid mechanics has played a somewhat similar role. The present paper attempts to draw mathematicians’ attention to some important ideas in the theory of singular perturbations developed recently by workers in applied fields. The ideas will be illustrated by simple model equations; the references to fluid mechanics are merely historical and are not needed for the mathematical discussion which starts in Section 2.

We first...

• Sequential Convergence in Lattice Groups
(pp. 273-290)

We introduce here the concept ofRiesz convergence, any kind of sequential convergence in a lattice group subject to the four conditions given below. Riesz convergence includes as special cases order convergence and relative uniform convergence in lattice groups as well as sequential convergence in any locally o-convex topological vector lattice (see [4]). Each Riesz convergenceCinduces a number of related Riesz convergences which are eitner stronger or weaker thanCand which in special cases coincide withC. We shall be particularly concerned with Riesz convergences induced by seminorms.

All functions and scalars are assumed to be finite,...

• A Group-theoretic Lattice-point Problem
(pp. 291-296)
BURTON RANDOL

LetG=SL(2,R),\Gamma =SL(2,Z), and letPbe the set of primitive integral lattice-points inR². ForgGandr> 0, letNr(g) be the number of points in the setg(P) which intersect the disk |x| <r. Then, inasmuch as ; (P) =Pfor anyy eΓ, it is evident thatNr(g) can be regarded as a function on the quotient spaceG/Γ, which has a normalized Haar measuredg. Moreover, as a result of Siegel, the mean, or integral, ofNr(g) overG/Fis simply (ζ(2))-1πr² (see [4], Formula 25).

Consider now the following question. What can...

• The Riemann Surface of Klein with 168 Automorphisms
(pp. 297-308)
HARRY E. RAUCH and J. LEWITTES

In the following we consider Klein’s compact Riemann surface of genus three admitting a simple group of 168 automorphisms (conformal self-homeomorphisms) and adduce certain of its properties which to our knowledge and, indeed, surprise do not seem to be in the literature. Our main results are the explicit exhibition (i) of a canonical integral homology basis in dimension one (Section 3), (ii) of the action on this basis of Klein’s group of automorphisms in the form of integral matrices representing a set of generators of the group inSp(3, Z), although actually we give more (Section 4), and (iii) of...

• Envelopes of Holomorphy of Domains in Complex Lie Groups
(pp. 309-318)
O. S. ROTHAUS

There are two well known results in the literature concerning envelopes of holomorphy that we want to single out here, namely the ones describing the completion of Reinhardt domains and tube domains. The statement for the latter type we owe to Salomon Bochner.

There is a feature common to both the results which is worth noting. Roughly speaking it may be described as follows: a domain in a complex Lie group which is stable under a real form of the group may be completed by forming certain “averages” in the complex group.

In this note we shall give one reasonably...

• Automorphisms of Commutative Banach Algebras
(pp. 319-324)
STEPHEN SCHEINBERG

This presentation consists of a few observations related to joint work with Herbert Kamowitz. In [2] we showed that ifTis an automorphism of a semisimple commutative Banach algebra and ifTnI(alln), then the spectrum ofT,σ(T), must contain the unit circle. Examples were given to show that this containment can be proper. Section 1 that follows contains more complicated examples of suchσ(T) plus the theorem thatσ(T) is necessarily connected. In [1] we studied derivations and automorphisms of a particular radical algebra. Section 2 concerns another class of radical algebras, those of...

• Historical Notes on Analyticity as a Concept in Functional Analysis
(pp. 325-344)
ANGUS E. TAYLOR

This is an essay—one of a projected series—on certain aspects of the history of functional analysis. The emphasis of this essay is on the way in which the classical theory of analytic functions of a complex variable was extended and generalized and came to play a significant role in functional analysis. One can perceive two lines of development: (i) the extension of the classical theory to cases in which the function of a complex variable has its values in a function space or in an abstract space, and (ii) the development of a theory of analytic functions from...

• A-Almost Automorphic Functions
(pp. 345-351)
WILLIAM A. VEECH

Twill denote a locally compact,σ-compact, Abelian group, andC=C(T) will be the algebra of bounded, complex-valued, uniformly continuous functions onT. Bochner definesfCto bealmost automorphicif from every sequence {βm} =βTmay be extracted a subsequence\left \{ \alpha _{n} \right \}=\alpha \left ( \alpha _{n}=\beta _{m_{n}} \right )such that (a) the limitT_{\alpha }f(t)=\lim_{n}f(t+\alpha_{n})exists for eacht, and (b) the equation

(1)T_{-\alpha }T_{\alpha }f(t)=\lim_{n}T_{\alpha }f(t-\alpha _{n})

=f(t)

holds identically.

The limits in (a) and (b) are not required to be uniform (but they are automatically locally uniform), and, in fact, becauseTisσ-compact, (a) places no restriction at all onf, so...