Criteria for absolute convegence of Fourier series of functions of bounded variation
Ingemar
Wik
1-24
Abstract: The usual criteria for establishing that a function of bounded variation or an absolutely continuous function has an absolutely convergent Fourier series are given in terms of the modulus of continuity, the integrated modulus of continuity or conditions on the derivative. The relations between these criteria are investigated. A class of functions is constructed to provide counterexamples which show to what extent the existing theorems are best possible. In the case of absolutely continuous functions a few new criteria are given involving the variation of the given function. A couple of necessary and sufficient conditions are given for a class of absolutely continuous functions to have absolutely convergent Fourier series.
Sequences of divided powers in irreducible, cocommutative Hopf algebras
Kenneth
Newman
25-34
Abstract: In Hopf algebras with one grouplike element, M. E. Sweedler showed that over perfect fields, sequences of divided powers in cocommutative, irreducible Hopf algebras can be extended if certain ``coheight'' conditions are met. Here, we show that with a suitable generalization of ``coheight", Sweedler's theorem is true over nonperfect fields. (We also point out, that in one case Sweedler's theorem was false, and additional conditions must be assumed.) In the same paper, Sweedler gave a structure theorem for irreducible, cocommutative Hopf algebras over perfect fields. We generalize this theorem in both the perfect and nonperfect cases. Specifically, in the nonperfect case, while a cocommutative, irreducible Hopf algebra does not, in general, satisfy the structure theorem, the sub-Hopf algebra, generated by all sequences of divided powers, does. Some additional properties of this sub-Hopf algebra are also given, including a universal property.
On the summation formula of Voronoi
C.
Nasim
35-45
Abstract: A formula involving sums of the form $ \Sigma d(n)f(n)$ and $\Sigma d(n)g(n)$ is derived, where $ d(n)$ is the number of divisors of $n$, and $f(x),g(x)$ are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi's summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between $ f(x)$ and $g(x)$ is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class $ {L^2}(0,\infty )$. We use $ {L^2}$-theory of Mellin and Fourier-Watson transformations.
Diffeomorphic invariants of immersed circles
Roger F.
Verhey
47-63
Abstract: The intersection sequences of a normal immersion form a complete invariant for diffeomorphically equivalent normal immersions. Numerical invariants and inequalities on numerical invariants are obtained using intersection sequences.
Conformality and isometry of Riemannian manifolds to spheres
Chuan-chih
Hsiung;
Louis W.
Stern
65-73
Abstract: Suppose that a compact Riemannian manifold ${M^n}$ of dimension $n > 2$ admits an infinitesimal nonisometric conformal transformation $\upsilon$. Some curvature conditions are given for ${M^n}$ to be conformal or isometric to an $ n$-sphere under the initial assumption that $ {L_\upsilon }R = 0$, where ${L_\upsilon }$ is the operator of the infinitesimal transformation $\upsilon$ and $R$ is the scalar curvature of ${M^n}$. For some special cases, these conditions were given by Yano [10] and Hsiung [2].
Inverse $H$-semigroups and $t$-semisimple inverse $H$-semigroups
Mary Joel
Jordan
75-84
Abstract: An $H$-semigroup is a semigroup such that both its right and left congruences are two-sided. A semigroup is $t$-semisimple provided the intersection of all its maximal modular congruences is the identity relation. We prove that a semigroup is an inverse $H$-semigroup if and only if it is a semilattice of disjoint Hamiltonian groups. Using the set $E$ of idempotents of $S$ as the semilattice, we show that an inverse $ H$-semigroup $ S$ is $t$-semisimple if and only if for each pair of groups $ {G_e},{G_f}$, in the semilattice, with $f \geqq e$ in $E$, the homomorphism ${\varphi _{f,e}}$ on ${G_f}$, into ${G_e}$, defined by $a{\varphi _{f,e}} = ae$, is a monomorphism; and for each $e$ in $E$, for each $a \ne e$ in ${G_e}$, there exists a subsemigroup ${T_p}$ of $S$ such that $ a \notin {T_p}$ and, for each $f$ in $E$, $ {T_p} \cap {G_f} = {H_f}$, where ${H_f} = {G_f}$ or ${H_f}$ is a maximal subgroup of prime index $p$ in ${G_f}$.
$F$-minimal sets
Nelson G.
Markley
85-100
Abstract: An $F$-minimal set is the simplest proximal extension of an equicontinuous minimal set. It has one interesting proximal cell and all the points in this proximal cell are uniformly asymptotic. The Sturmian minimal sets are the best known examples of $F$-minimal sets. Our analysis of them is in terms of their maximal equicontinuous factors. Algebraically speaking $F$-minimal sets are obtained by taking an invariant *-closed algebra of almost periodic functions and adjoining some suitable functions to it. Our point of view is to obtain these functions from the maximal equicontinuous factor. In §3 we consider a subclass of $F$-minimal sets which generalize the classical Sturmian minimal sets, and in §4 we examine the class of minimal sets obtained by taking the minimal right ideal of an $F$-minimal set and factoring by a closed invariant equivalence relation which is smaller than the proximal relation.
Homology invariants of cyclic coverings with application to links
Y.
Shinohara;
D. W.
Sumners
101-121
Abstract: The main purpose of this paper is to study the homology of cyclic covering spaces of a codimension two link. The integral (rational) homology groups of an infinite cyclic cover of a finite complex can be considered as finitely generated modules over the integral (rational) group ring of the integers. We first describe the properties of the invariants of these modules for certain finite complexes related to the complementary space of links. We apply this result to the homology invariants of the infinite cyclic cover of a higher dimensional link. Further, we show that the homology invariants of the infinite cyclic cover detect geometric splittability of a link. Finally, we study the homology of finite unbranched and branched cyclic covers of a link.
The sign of Lommel's function
J.
Steinig
123-129
Abstract: Lommel's function ${s_{\mu ,\nu }}(x)$ is a particular solution of the differential equation ${s_{\mu ,\nu }}(x) > 0$ for $x > 0$, if $\mu = \tfrac{1}{2}$ and $\vert\nu \vert < \tfrac{1}{2}$, or if $\mu > \tfrac{1}{2}$ and $\vert\nu \vert \leqq \mu $. This includes earlier results of R. G. Cooke's. The sign of ${s_{\mu ,\nu }}(x)$ for other values of $ \mu$ and $\nu$ is also discussed.
Using additive functionals to embed preassigned distributions in symmetric stable processes
Itrel
Monroe
131-146
Abstract: Following Skorokhod, several authors in recent years have proposed methods to define a stopping time $T$ for Brownian motion $({X_t},{\mathcal{F}_t})$ such that ${X_T}$ will have some preassigned distribution. In this paper a method utilizing additive functionals is explored. It is applicable not only to Brownian motion but all symmetric stable processes of index $\alpha > 1$. Using this method one is able to obtain any distribution having a finite $\alpha - 1$ absolute moment. There is also a discussion of the problem of approximating symmetric stable processes with random walks.
A theorem of completeness for families of compact analytic spaces
John J.
Wavrik
147-155
Abstract: A sufficient condition is given for a family of compact analytic spaces to be complete. This condition generalizes to analytic spaces the Theorem of Completeness of Kodaira and Spencer [6]. It contains, as a special case, the rigidity theorem proved by Schuster in [11].
Approximation in the mean by analytic functions
Lars Inge
Hedberg
157-171
Abstract: Let $E$ be a compact set in the plane, let ${L^p}(E)$ have its usual meaning, and let $ L_a^p(E)$ be the subspace of functions analytic in the interior of $E$. The problem studied in this paper is whether or not rational functions with poles off $ E$ are dense in $ L_a^p(E)$ (or in $ {L^p}(E)$ in the case when $ E$ has no interior). For $1 \leqq p \leqq 2$ the problem has been settled by Bers and Havin. By a method which applies for $1 \leqq p < \infty$ we give new results for $p > 2$ which improve earlier results by Sinanjan. The results are given in terms of capacities.
Conjugacy separability of certain Fuchsian groups
P. F.
Stebe
173-188
Abstract: Let $G$ be a group. An element $g$ is c.d. in $G$ if and only if given any element $h$ of $G$, either it is conjugate to $h$ or there is a homomorphism $ \xi$ from $G$ onto a finite group such that $ \xi (g)$ is not conjugate to $\xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. Let $F$ be a Fuchsian group, i.e. let $F$ be presented as $\displaystyle F = ({S_1}, \ldots ,{S_n},{a_1}, \ldots ,{a_{2r}},{b_1}, \ldots ,... ..._n}{a_1} \ldots {a_{2r}}a_1^{ - 1} \ldots a_{2r}^{ - 1}{b_1} \ldots {b_t} = 1).$ In this paper, we show that every element of infinite order in $F$ is c.d. and if $t \ne 0$ or $r \ne 0$, $F$ is c.s.
Measure algebras and functions of bounded variation on idempotent semigroups
Stephen E.
Newman
189-205
Abstract: Our main result establishes an isomorphism between all functions on an idempotent semigroup $S$ with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of $S$, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on $S$ and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case $S$ is the unit interval and the ``product'' of two numbers in $S$ is their maximum.
Discrete sufficient sets for some spaces of entire functions
B. A.
Taylor
207-214
Abstract: Let $E$ denote the space of all entire functions $f$ of exponential type (i.e. $\vert f(z)\vert = O(\exp (B\vert z\vert))$) for some $B > 0$). Let $ \mathcal{K}$ denote the space of all positive continuous functions $ k$ on the complex plane $ C$ with $\exp (B\vert z\vert) = O(k(z))$ for each $B > 0$. For $k \in \mathcal{K}$ and $ S \subset C$, let $ \vert\vert f\vert{\vert _{k,s}} = \sup \{ \vert f(z)\vert/k(z):z \in S\}$. We prove that the two families of seminorms $ {\{ \vert\vert\vert{\vert _{k,C}}\} _{k \in \mathcal{K}}}$ and $ {\{ \vert\vert\vert{\vert _{k,s}}\} _{k \in \mathcal{K}}}$, where $\displaystyle S = \{ n + im: - \infty < n,m < + \infty \}$ , determine the same topology on $E$.
Noncommutative Jordan division algebras
Kevin
McCrimmon
215-224
Abstract: The structure theory for noncommutative Jordan algebras with chain conditions leads to the following simple algebras: (I) division algebras, (II) forms of nodal algebras, (III) algebras of generic degree two, (IV) commutative Jordan matrix algebras, (V) quasi-associative algebras. The chain condition is always satisfied in a division algebra, hence does not serve as a finiteness restriction. Consequently, the general structure of noncommutative Jordan division algebras, even commutative Jordan division algebras, is unknown. In this paper we will classify those non-commutative Jordan division algebras which are forms of algebras of types (II)-(V); this includes in particular all the finite-dimensional ones.
Extreme points in a class of polynomials having univalent sequential limits
T. J.
Suffridge
225-237
Abstract: This paper concerns a class $ {\mathcal{P}_n}$ (defined below) of polynomials of degree less than or equal to $ n$ having the properties: each polynomial which is univalent in the unit disk and of degree $n$ or less is in $ {\mathcal{P}_n}$ and if $ \{ {P_{{n_k}}}\} _{k = 1}^\infty$ is a sequence of polynomials such that $ {P_{{n_k}}} \in {\mathcal{P}_{{n_k}}}$ and ${\lim _{k \to \infty }}{P_{{n_k}}} = f$ (uniformly on compact subsets of the unit disk) then $ f$ is univalent. The approach is to study the extreme points in ${\mathcal{P}_n}$ ( $P \in {\mathcal{P}_n}$ is extreme if $P$ is not a proper convex combination of two distinct elements of ${\mathcal{P}_n}$). Theorem 3 shows that if $ P \in {\mathcal{P}_n}$ is extreme then $P(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n},{a_n} = 1/n$, are dense in the class $S$ of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve $P({e^{i\theta }})$, $0 \leqq \theta \leqq 2\pi $, turns at a constant rate (between cusps) as $\theta$ varies.
The local spectral behavior of completely subnormal operators
K. F.
Clancey;
C. R.
Putnam
239-244
Abstract: For any compact set $ X$, let $C(X)$ denote the continuous functions on $ X$ and $R(X)$ the functions on $X$ which are uniformly approximable by rational functions with poles off $X$. Let $A$ denote a subnormal operator having no reducing space on which it is normal. It is shown that a necessary and sufficient condition that $X$ be the spectrum of such an operator $ A$ is that $ R(X \cap \overline D ) \ne C(X \cap \overline D )$ whenever $D$ is an open disk intersecting $ X$ in a nonempty set.
Symmetric Massey products and a Hirsch formula in homology
Stanley O.
Kochman
245-260
Abstract: A Hirsch formula is proved for the singular chains of a second loop space and is applied to show that the symmetric Massey produce $ {\langle x\rangle ^p}$ is defined for $x$ an odd dimensional $\bmod p$ homology class of a second loop space with $ p$ an odd prime. ${\langle x\rangle ^p}$ is then interpreted in terms of the Dyer-Lashof and Browder operations.
Branched structures on Riemann surfaces
Richard
Mandelbaum
261-275
Abstract: Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures. We define the notion of a branched analytic cover on a Riemann surface $M$ (of genus $g$) and specialize this to the case of branched projective and affine structures. Establishing a correspondence between branched projective and affine structures on $M$ and the classical projective and affine connections on $M$ we show that if a certain linear homogeneous differential equation involving the connection has only meromorphic solutions on $M$ then the connection corresponds to a branched structure on $M$. Utilizing this fact we then determine classes of positive divisors on $M$ such that for each divisor $\mathfrak{D}$ in the appropriate class the branched structures having $ \mathfrak{D}$ as their branch locus divisor form a nonempty affine variety. Finally we apply some of these results to study the structures on a fixed Riemann surface of genus 2.
The $L\sp{1}$- and $C\sp{\ast} $-algebras of $[FIA]\sp{-}\sb{B}$ groups, and their representations
Richard D.
Mosak
277-310
Abstract: Let $G$ be a locally compact group, and $ B$ a subgroup of the (topologized) group $ \operatorname{Aut} (G)$ of topological automorphisms of $G$; $G$ is an $ [FIA]_B^ -$ group if $ B$ has compact closure in $ \operatorname{Aut} (G)$. Abelian and compact groups are $[FIA]_B^ -$ groups, with $B = I(G)$; the purpose of this paper is to generalize certain theorems about the group algebras and representations of these familiar groups to the case of general $ [FIA]_B^ -$ groups. One defines the set $ {\mathfrak{X}_B}$ of $ B$-characters to consist of the nonzero extreme points of the set of continuous positive-definite $B$-invariant functions $\phi$ on $G$ with $ \phi (1) \leqq 1$. ${\mathfrak{X}_B}$ is naturally identified with the set of pure states on the subalgebra of $B$-invariant elements of ${C^\ast}(G)$. When this subalgebra is commutative, this identification yields generalizations of known duality results connecting the topology of $G$ with that of $\hat G$. When $B = I(G),{\mathfrak{X}_B}$ can be identified with the structure spaces of ${C^\ast}(G)$ and ${L^1}(G)$, and one obtains thereby information about representations of $G$ and ideals in ${L^1}(G)$. When $G$ is an $ [FIA]_B^ -$ group, one has under favorable conditions a simple integral formula and a functional equation for the $ B$-characters. $ {L^1}(G)$ and ${C^\ast}(G)$ are ``semisimple'' in a certain sense (in the two cases $B = (1)$ and $B = I(G)$ this ``semisimplicity'' reduces to weak and strong semisimplicity, respectively). Finally, the $B$-characters have certain separation properties, on the level of the group and the group algebras, which extend to ${[SIN]_B}$ groups (groups which contain a fundamental system of compact $B$-invariant neighborhoods of the identity). When $B = I(G)$ these properties generalize known results about separation of conjugacy classes by characters in compact groups; for example, when $B = (1)$ they reduce to a form of the Gelfand-Raikov theorem about ``sufficiently many'' irreducible unitary representations.
A characterization of odd order extensions of the finite projective symplectic groups ${\rm PSp}(4,\,q)$
Morton E.
Harris
311-327
Abstract: In a recent paper, W. J. Wong characterized the finite projective symplectic groups $ {\text{PSp}}(4,q)$ where $ q$ is a power of an odd prime integer by the structure of the centralizer of an involution in the center of a Sylow $2$-subgroup of ${\text{PSp}}(4,q)$. In the present paper, finite groups which contain an involution in the center of a Sylow $2$-subgroup whose centralizer has a more general structure than in the ${\text{PSp}}(4,q)$ case are classified by showing them to be odd ordered extensions of ${\text{PSp}}(4,q)$.
Factoring functions on Cartesian products
N.
Noble;
Milton
Ulmer
329-339
Abstract: A function on a product space is said to depend on countably many coordinates if it can be written as a function defined on some countable subproduct composed with the projection onto that subproduct. It is shown, for $X$ a completely regular Hausdorff space having uncountably many nontrivial factors, that each continuous real-valued function on $ X$ depends on countably many coordinates if and only if $X$ is pseudo- $ {\aleph _1}$-compact. It is also shown that a product space is pseudo- ${\aleph _1}$-compact if and only if each of its finite subproducts is. (This fact derives from a more general theorem which also shows, for example, that a product satisfies the countable chain condition if and only if each of its finite subproducts does.) All of these results are generalized in various ways.
Regular modules
J.
Zelmanowitz
341-355
Abstract: In analogy to the elementwise definition of von Neumann regular rings an $ R$-module $M$ is called regular if given any element $m \in M$ there exists $f \in {\operatorname{Hom} _R}(M,R)$ with $ (mf)m = m$. Other equivalent definitions are possible, and the basic properties of regular modules are developed. These are applied to yield several characterizations of regular self-injective rings. The endomorphism ring $ E(M)$ of a regular module $ _RM$ is examined. It is in general a semiprime ring with a regular center. An immediate consequence of this is the recently observed fact that the endomorphism ring of an ideal of a commutative regular ring is again a commutative regular ring. Certain distinguished subrings of $E(M)$ are also studied. For example, the ideal of $ E(M)$ consisting of the endomorphisms with finite-dimensional range is a regular ring, and is simple when the socle of $ _RM$ is homogeneous. Finally, the self-injectivity of $E(M)$ is shown to depend on the quasi-injectivity of $ _RM$.
Some invariant $\sigma $-algebras for measure-preserving transformations
Peter
Walters
357-368
Abstract: For an invertible measure-preserving transformation $ T$ of a Lebesgue measure space $ (X,\mathcal{B},m)$ and a sequence $N$ of integers, a $T$-invariant partition ${\alpha _N}(T)$ of $ (X,\mathcal{B},m)$ is defined. The relationship of these partitions to spectral properties of $T$ and entropy theory is discussed and the behaviour of the partitions $ {\alpha _N}(T)$ under group extensions is investigated. Several examples are discussed.
A method of symmetrization and applications
W. E.
Kirwan
369-377
Abstract: In this paper we define a method of symmetrization for plane domains that includes as special cases methods of symmetrization considered by Szegö and by Marcus. We prove that under this method of symmetrization the mapping radius of a fixed point is not decreased. This fact is used to obtain some results concerning covering properties of Bieberbach-Eilenberg functions.
Quadratic extensions of linearly compact fields
Ron
Brown;
Hoyt D.
Warner
379-399
Abstract: A group valuation is constructed on the norm factor group of a quadratic extension of a linearly compact field, and the norm factor group is explicitly computed as a valued group. Generalizations and applications of this structure theory are made to cyclic extensions of prime degree, to square (and $p$th power) factor groups, to generalized quaternion algebras, and to quadratic extensions of arbitrary fields.
Convergence, uniqueness, and summability of multiple trigonometric series
J. Marshall
Ash;
Grant V.
Welland
401-436
Abstract: In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let $E$ be a subset of positive (Lebesgue) measure of the $k$ dimensional torus. The principal result is that the convergence of a trigonometric series on $ E$ forces the boundedness of the partial sums almost everywhere on $E$ where the system of partial sums is the one associated with the system of all rectangles situated symmetrically about the origin in the lattice plane with sides parallel to the axes. If $E$ has a countable complement, then the partial sums are bounded at every point of $ E$. This result implies a uniqueness theorem for double trigonometric series, namely, that if a double trigonometric series converges unrestrictedly rectangularly to zero everywhere, then all the coefficients are zero. Although uniqueness is still conjectural for dimensions greater than two, we obtain partial results and indicate possible lines of attack for this problem. We carry out an extensive comparison of various modes of convergence (e.g., square, triangular, spherical, etc.). A number of examples of pathological double trigonometric series are displayed, both to accomplish this comparison and to indicate the ``best possible'' nature of some of the results on the growth of partial sums. We obtain some compatibility relationships for summability methods and finally we present a result involving the $ (C,\alpha ,0)$ summability of multiple Fourier series.
Essential spectrum for a Hilbert space operator
Richard
Bouldin
437-445
Abstract: Various notions of essential spectrum have been defined for densely defined closed operators on a Banach space. This paper shows that the theory for those notions of essential spectrum simplifies if the underlying space is a Hilbert space and the operator is reduced by its finite-dimensional eigenspaces. In that situation this paper classifies each essential spectrum in terms of the usual language for the spectrum of a Hilbert space operator. As an application this paper deduces the main results of several recent papers dealing with generalizations of the Weyl theorem.
On a problem of Tur\'an about polynomials with curved majorants
Q. I.
Rahman
447-455
Abstract: Let $\phi (x) \geqq 0$ for $- 1 \leqq x \leqq 1$. For a fixed ${x_0}$ in $[ - 1,1]$ what can be said for $ \vert{p_n}(x)\vert \leqq \phi (x)$ for $ - 1 \leqq x \leqq 1$? The case $\phi (x) = 1$ was considered by A. A. Markov and S. N. Bernstein. We investigate the problem when $ \phi (x) = {(1 - {x^2})^{1/2}}$. We also study the case $\phi (x) = \vert x\vert$ and the subclass consisting of polynomials typically real in $\vert z\vert < 1$.
Fully nuclear and completely nuclear operators with applications to $\mathcal{L}_1-$ and $\mathcal{L}_\infty$-spaces
C. P.
Stegall;
J. R.
Retherford
457-492
Abstract: This paper is devoted to a study of the conjecture of A. Grothendieck that if $E$ and $F$ are Banach spaces and all operators from $ E$ to $F$ are nuclear, then $E$ or $F$ must be finite dimensional. Two partial solutions are given to this conjecture (Chapters II and IV). In these chapters, operators we call fully nuclear and completely nuclear are introduced and studied. The principal result of these two chapters is that if $\mathcal{L}(E,F) = \operatorname{FN} (E,F)$ or $\mathcal{L}(E,F) = \operatorname{CN} (E,F)$ (and $E$ is isomorphic to a conjugate space or $ E'$ contains a reflexive subspace in the latter case) then one of $E$, $F$ is finite dimensional. Two new properties of Banach spaces are introduced in Chapter I. We call these properties ``sufficiently Euclidean'' and ``the two-series property". Chapter I provides the machinery for all the subsequent chapters. The principal part of the paper (Chapters II and V) is devoted to internal characterizations of the $ {\mathcal{L}_\infty }$ - and $ {\mathcal{L}_1}$-spaces of Lindenstrauss and Pełlczyhski. These characterizations are in terms of the behavior of various classes of operators from or into these spaces. As a by-product an apparently new characterization of Hilbert spaces is obtained. Finally, Chapter VI is a summary of the known characterizations of ${\mathcal{L}_1}$ - and ${\mathcal{L}_\infty }$ -spaces.
The class group of Dedekind domains
C. R.
Leedham-Green
493-500
Abstract: A new proof is given of Claborn's theorem, namely that every abelian group is the class group of a Dedekind domain. A variation of the proof shows that the Dedekind domain can be constructed to be a quadratic extension of a principal ideal ring; a Dedekind domain is also constructed that is unrelated in a certain sense to any principal ideal ring.