Fourier analysis on the sphere
Thomas O.
Sherman
1-31
Abstract: A new approach to harmonic analysis on the unit sphere in ${{\mathbf{R}}^{d + 1}}$ is given, closer in form to Fourier analysis on $ {{\mathbf{R}}^d}$ than the usual development in orthonormal polynomials. Singular integrals occur in the transform formulae. The results generalize to symmetric space.
Ramsey graphs and block designs. I
T. D.
Parsons
33-44
Abstract: This paper establishes a connection between a certain class of Ramsey numbers for graphs and the class of symmetric block designs admitting a polarity. The main case considered here relates the projective planes over Galois fields to the Ramsey numbers $R({C_4},{K_{1,n}}) = f(n)$. It is shown that, for every prime power $q,f({q^2} + 1) = {q^2} + q + 2$, and $f({q^2}) = {q^2} + q + 1$.
Minimal covers and hyperdegrees
Stephen G.
Simpson
45-64
Abstract: Every hyperdegree at or above that of Kleene's $O$ is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty $\Sigma _1^1$ class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If $V = L$ or a generic extension of $L$, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If ${O^\char93 }$ exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.
Global dimension of differential operator rings. II
K. R.
Goodearl
65-85
Abstract: The aim of this paper is to find the global homological dimension of the ring of linear differential operators $R[{\theta _1}, \ldots ,{\theta _u}]$ over a differential ring $R$ with $u$ commuting derivations. When $R$ is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of $R[{\theta _1}, \ldots ,{\theta _u}]$ is the maximum of $k$ and $q + u$, where $q$ is the supremum of the ranks of all maximal ideals $ M$ of $R$ for which $R/M$ has positive characteristic, and $ k$ is the supremum of the sums $ rank(P) + diff\;dim(P)$ for all prime ideals $P$ of $R$ such that $R/P$ has characteristic zero. [The value $diff\;dim(P)$ is an invariant measuring the differentiability of $P$ in a manner defined in §3.] In case we are considering only a single derivation on $R$, this theorem leads to the result that the global dimension of $ R[\theta ]$ is the supremum of gl $dim(R)$ together with one plus the projective dimensions of the modules $R/J$, where $J$ is any primary differential ideal of $ R$. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.
Ramsey theorems for multiple copies of graphs
S. A.
Burr;
P.
Erdős;
J. H.
Spencer
87-99
Abstract: If $G$ and $H$ are graphs, define the Ramsey number $ r(G,H)$ to be the least number $p$ such that if the edges of the complete graph $ {K_p}$ are colored red and blue (say), either the red graph contains $ G$ as a subgraph or the blue graph contains $H$. Let $mG$ denote the union of $m$ disjoint copies of $G$. The following result is proved: Let $ G$ and $H$ have $k$ and $l$ points respectively and have point independence numbers of $i$ and $j$ respectively. Then $N - 1 \leqslant r(mG,nH) \leqslant N + C$, where $ N = km + ln - min(mi,mj)$ and where $C$ is an effectively computable function of $ G$ and $H$. The method used permits exact evaluation of $r(mG,nH)$ for various choices of $G$ and $H$, especially when $m = n$ or $G = H$. In particular, $r(m{K_3},n{K_3}) = 3m + 2n$ when $m \geqslant n,m \geqslant 2$.
A characterization of manifolds
Louis F.
McAuley
101-107
Abstract: The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here. Theorem. Suppose that $ f:X \Rightarrow [0,1]$, where $X$ is a compactum, and that $f$ has the properties: (1) for $0 \leqslant x < 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}$, (2) $ {f^{ - 1}}(1/2) \cong {S^n}$ with a tame (or flat) $k$-sphere ${S^k}$ shrunk to a point, (3) for $ 1/2 < x \leqslant 1,{f^{ - 1}}(x) \cong$ a compact connected $n$-manifold ${M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}$ (a spherical modification of $ {M_0}$ of type $ k$), and (4) there is a continuum $C$ in $X$ such that (letting ${C_x} = {f^{ - 1}}(x) \cap C$) (a) $0 \leqslant x < 1/2,{C_x} \cong {S^k}$, (b) $ {C_{1/2}} = \{ p\}$ a point, (c) for $1/2 < x \leqslant 1$, and (d) each of $ f\vert(X - C),f\vert{f^{ - 1}}[0,1/2)$, and $ f\vert{f^{ - 1}}(1/2,1]$ is completely regular. Then $X$ is homeomorphic to a differentiable $(n + 1)$-manifold $M$ whose boundary is the disjoint union of ${\bar M_0}$ and $ {\bar M_1}$ where ${M_i} = {\bar M_i},i = 0,1$.
Isotropic immersions and Veronese manifolds
T.
Itoh;
K.
Ogiue
109-117
Abstract: An $n$-dimensional Veronese manifold is defined as a minimal immersion of an $n$-sphere of curvature $ n/2(n + 1)$ into an $\{ n(n + 3)/2 - 1\}$-dimensional unit sphere. The purpose of this paper is to give some characterizations of a Veronese manifold in terms of isotropic immersions.
Projective limits in harmonic analysis
William A.
Greene
119-142
Abstract: A treatment of induced transformations of measures and measurable functions is presented. Given a diagram $\varphi :G \to H$ in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by $G \to {L^2}(G),{L^1}(G)$, $ M(G),W(G)$, denoting, resp., the ${L^2}$-space, ${L^1}$-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of $ {L^1}$ on ${L^2}$. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm $\leqslant 1$ is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of $ {W^ \ast }$-algebras and surjective normal $\ast$-algebra homomorphisms.
Sheaves of $H$-spaces and sheaf cohomology
James M.
Parks
143-156
Abstract: The concept of a sheaf of $H$-spaces is introduced and, using the Čech technique, a cohomology theory is defined in which the cohomology ``groups'' are $H$-spaces. The corresponding axioms of Cartan [3] for this theory are verified and other properties of the theory are investigated.
On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence
J. A.
Gosselin;
W. S.
Young
157-174
Abstract: It is known that the partial sums of Vilenkin-Fourier series of $ {L^q}$ functions $ (q > 1)$ converge a.e. In this paper we establish the ${L^2}$ result for a class of rearrangements of the Vilenkin-Fourier series, and the $ {L^q}$ result $(1 < q < 2)$ for a subclass of rearrangements. In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. Balashov. The ${L^2}$ result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.
Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions
K. Michael
Day
175-183
Abstract: Let ${T_n}(a) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let ${\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}$ be the corresponding sets of eigenvalues of ${T_n}(f)$. Define a sequence of measures ${\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}$, and $E$ a set in the $\lambda$-plane. It is shown that the weak limit $ \alpha$ of the measures ${\alpha _n}$ is unique and possesses at most two atoms, and the function $f$ which give rise to atoms are identified.
On bounded elements of linear algebraic groups
Kwan-Yuk Claire
Sit
185-198
Abstract: Let $F$ be a local field of characteristic zero and ${\text{G }}$ a connected algebraic group defined over $F$. Let $G$ be the locally compact group of $F$-rational points. One characterizes the group $B(G)$ of $g \in G$ whose conjugacy class is relatively compact. For instance, if $ {\text{G}}$ is $ F$-split or reductive without anisotropic factors then $B(G)$ is the center of $G$. If $H$ is a closed subgroup of $G$ such that $G/H$ has finite volume, then the centralizer of $ H$ in $G$ is contained in $B(G)$. If, moreover, $H$ is the centralizer of some $ x \in G$ then $ G/H$ is compact.
Interpolation and uniqueness results for entire functions
James D.
Child
199-209
Abstract: Let $K[\Omega ]$ denote the collection of entire functions of exponential type whose Borel transforms are analytic on ${\Omega ^c}$ (the complement of the simply connected domain taken relative to the sphere). Let $ f$ be in $K[\Omega ]$ and set ${L_n}(f) = {(2\pi i)^{ - 1}}\smallint \Gamma {g_n}(\lambda )F(\lambda )d\lambda (n = 0,1, \ldots )$ where $F$ is the Borel transform of $f,\Gamma \subset \Omega$ is a simple closed contour chosen so that $F$ is analytic outside and on $\Gamma$ and each ${g_n}$ is in $ H(\Omega )$ (the collection of functions analytic on $\Omega$). In what follows read 'the sequence of linear functionals $ \{ {L_n}(f)\}$' wherever the sequence of functions ' $ \{ {g_n}\}$' appears. Let $ T$ denote a continuous linear operator from $ H(\Omega )$ to $H(\Lambda )$ where $\Lambda$ is also a simply connected domain. The topologies on $ H(\Omega )$ and $H(\Lambda )$ are those of uniform convergence on compact subsets of $\Omega$ (resp. $\Lambda$). The purpose of this paper is to consider uniqueness preserving operators, i.e., operators $T$ which have the property that $K[\Lambda ]$ is a uniqueness class for $\{ T({g_n})\}$ whenever $ K[\Omega ]$ is a uniqueness class for $\{ {g_n}\}$, and to examine interpolation preserving operators, i.e., operators $ T$ which have the property that $K[\Lambda ]$ interpolates the sequence of complex numbers $ \{ {b_n}\}$ relative to $\{ T({g_n})\}$ whenever $ K[\Omega ]$ interpolates $\{ {b_n}\}$ relatives to $\{ {g_n}\}$. Once some classes of uniqueness preserving operators and some classes of interpolation preserving operators have been found, we proceed to obtain new uniqueness and interpolation results from our knowledge of these operators and from previously known uniqueness and interpolation results. Operators which multiply by analytic functions and some differential operators are considered. Composition operators are studied and the results are used to extend the interpolation results for sequences of functions of the form $ \{ {[W(\zeta )]^n}\}$ where $W$ is analytic and univalent on a simply connected domain.
Some $H\sp{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products
Max L.
Weiss
211-223
Abstract: Let $f$ be a strictly increasing continuous real function defined near ${0^ + }$ with $k,f(\theta + kf(\theta ))/f(\theta ) \to 1/$ as $\theta \to {0^ + }$. The curve in the open unit disc with corresponding representation $1 - r = f(\theta )$ is called a $K$-curve. Several analytic and geometric conditions are obtained for $K$-curves and $K$-functions. This provides a framework for some rather explicit results involving parts in the closure of $K$-curves, $ {H^\infty }$-interpolating sequences lying on $K$-curves and the behavior of their Blaschke products. In addition, a sequence of points in the disc tending upper tangentially to 1 with moduli increasing strictly to 1 and arguments decreasing strictly to 0 is proved to be interpolating if and only if the hyperbolic distance between successive points remains bounded away from zero.
The Boolean space of orderings of a field
Thomas C.
Craven
225-235
Abstract: It has been pointed out by Knebusch, Rosenberg and Ware that the set $ X$ of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings $W(a) = \{ < {\text{ in }}X\vert a < 0\}$ the Harrison subbasis of $X$. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space $X$, there exists a formally real field $ F$ such that $ X$ is homeomorphic to the space of orderings on $F$. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.
Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces
Robert
Carmignani
237-258
Abstract: For a Riemann domain $\Omega$, a connected complex manifold where $n(n = dimension)$ globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function $f$ in $\Omega$, the ``Riemann surface'' ${\Omega _f}$, a maximal holomorphic extension Riemann domain for $f$, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for $ \Omega$, a maximal holomorphic extension Riemann domain for all holomorphic functions in $\Omega$, which is shown to be the projective limit space of the ``Riemann surfaces'' ${\Omega _f}$. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of ``Riemann surfaces".
Conditions for the existence of contractions in the category of algebraic spaces
Joseph
Mazur
259-265
Abstract: Artin's conditions for the existence of a new contraction of an algebraic subspace are changed. The new conditions are more applicable in special cases, especially when the subspace has a conormal bundle.
On some classes of multivalent starlike functions
Ronald J.
Leach
267-273
Abstract: Classes of multivalent functions analogous to certain classes of univalent starlike functions are defined and studied. Estimates on coefficients and distortion are made, using a variety of techniques.
Holomorphic functions on nuclear spaces
Philip J.
Boland
275-281
Abstract: The space $\mathcal{H}(E)$ of holomorphic functions on a quasi-complete nuclear space is investigated. If $\mathcal{H}(E)$ is endowed with the compact open topology, it is shown that $\mathcal{H}(E)$ is nuclear if and only if $ E'$ (continuous dual of $ E$) is nuclear. If $ E$ is a $\mathcal{D}FN$ (dual of a Fréchet nuclear space) and $F$ is a closed subspace of $E$, then the restriction mapping $\mathcal{H}(E) \to \mathcal{H}(F)$ is a surjective strict morphism.
On differential rings of entire functions
A. H.
Cayford;
E. G.
Straus
283-293
Abstract: Consider an entire function $f$ which is a solution of the differential equation $R = {\mathbf{C}}$ and of finite exponential order in case $ R = {\mathbf{C}}[z]$. We use this result to prove a conjecture made in [2] that entire functions of order $\rho < s$, all of whose derivatives at $ s$ points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order $\leqslant 1$.
On sums over Gaussian integers
D. G.
Hazlewood
295-310
Abstract: The object of this paper is to give asymptotic estimates for some number theoretic sums over Gaussian integers. As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of Gaussian integers with only ``large'' prime factors and for the number of Gaussian integers with only ``small'' prime factors are given.
On the representation of lattices by modules
George
Hutchinson
311-351
Abstract: For a commutative ring $R$ with unit, a lattice $L$ is ``representable by $R$-modules'' if $L$ is embeddable in the lattice of submodules of some unitary left $R$-module. A procedure is given for generating an infinite first-order axiomatization of the class of all lattices representable by $ R$-modules. Each axiom is a universal Horn formula for lattices. The procedure for generating the axioms is closely related to the ring structure, and is ``effective'' in the sense that many nontrivial axioms can be obtained by moderate amounts of computation.
Product integral techniques for abstract hyperbolic partial differential equations
J. W.
Spellmann
353-365
Abstract: Explicit and implicit product integral techniques are used to represent a solution $U$ to the abstract system: ${U_{12}}(x,y) = AU(x,y);U(x,0) = p = U(0,y)$. The coefficient $A$ is a closed linear transformation defined on a dense subspace $D(A)$ of the Banach space $X$ and the point $p$ in $D(A)$ satisfies the condition that $ \vert\vert{A^i}p\vert\vert < {S^i}{(i!)^{3/2}}$ for all integers $i \geqslant 0$ and some $S > 0$. The implicit technique is developed under the additional assumption that $A$ generates a strongly continuous semigroup of bounded linear transformations on $ X$. Both methods provide representations for the ${J_0}$ Bessel function.
$a\sp*$-closures of lattice-ordered groups
Roger
Bleier;
Paul
Conrad
367-387
Abstract: A convex $ l$-subgroup of an $ l$-group $G$ is closed if it contains the join of each of its subsets that has a join in $ G$. An extension of $ G$ which preserves the lattice of closed convex $l$-subgroups of $G$ is called an $ {a^ \ast }$-extension of $G$. In this paper we consider ${a^ \ast }$-extensions and ${a^ \ast }$-closures of $G$.
The uniqueness of the one-dimensional paraboson field
Steven
Robbins
389-397
Abstract: A paraboson analog of the one-dimensional boson field is discussed and a uniqueness result similar to a result of Putnam is obtained. It is shown that the paraboson operators must be unbounded.
Compactness properties of topological groups. III
S. P.
Wang
399-418
Abstract: Compactness properties of topological groups and finiteness of Haar measure on homogeneous spaces are studied. Some concrete structure theorems are presented.
Monotone and open mappings on manifolds. I
John J.
Walsh
419-432
Abstract: Sufficient conditions are given for the existence of open mappings from a p. 1. manifold ${M^m},m \geqslant 3$, onto a polyhedron $ Q$. In addition, it is shown that a mapping $f$ from $ {M^m},m \geqslant 3$, to $ Q$ is homotopic to a monotone mapping of $M$ onto $Q$ iff ${f_ \ast }:{\pi _1}(M) \to {\pi _1}(Q)$ is onto. Finally, it is shown that a monotone mapping of ${M^m},m \geqslant 3$, onto $Q$ can be approximated by a monotone open mapping of $M$ onto $Q$.
A generalisation of supersoluble groups
R. J.
Haggarty
433-441
Abstract: A $p$-soluble group $G$ belongs to the class $F(n,p)$ whenever the ranks of the $ p$-chief factors of $ G$ divide $n$ and $G$ has order coprime to $n$. A group in $F(n,p)$ is characterised by the embedding of its maximal subgroups. Whenever ${N_1}$ and ${N_2}$ are normal subgroups of $G$, of coprime indices in $ G$, which lie in $ F(n,p)$, then $ G$ lies in $F(n,p)$ also. $F(n)$ denotes the intersection, taken over all primes $p$, of the classes $F(n,p)$. Simple groups all of whose proper subgroups lie in $F(n)$ are determined. Given an integer $ n > 2$, there exist an integer $m$ with the same prime divisors as $n$ and a soluble group $G$ such that $G$ lies in $F(m)$ but $G$ does not possess a Sylow tower. (We may take $ m = n$ provided that $ n$ is not a multiple of 1806.) Furthermore, when $n$ is odd, an example of a soluble group $ G$, all of whose proper subgroups lie in $F(n)$ but $G$ has no Sylow tower, is given.
Erratum to: ``Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics'' (Trans. Amer. Math. Soc. {\bf 186} (1973), 259--272, 1974)
Thomas G.
Kurtz
442