Harmonic analysis for functors on categories of Banach spaces of distributions
Thomas
Donaldson
1-82
Abstract: This paper develops a theory of harmonic analysis (Fourier series, approximation, convolution, and singular integrals) for a general class of Banach function or distribution spaces. Continuity of singular convolution operators and convergence of trigonometric series is shown with respect to the norms of all the spaces in this class; the maximal class supporting a theory of the type developed in this paper is characterized (for other classes other theories exist). Theorems are formulated in category language throughout. However only elementary category theory is needed, and for most results the notions of functor and natural mapping are sufficient.
Homology of the classical groups over the Dyer-Lashof algebra
Stanley O.
Kochman
83-136
Abstract: The action of the Dyer-Lashof algebra is computed on the homology of the infinite classical groups (including Spin), their classifying spaces, their homogeneous spaces, Im J, B Im J and BBSO. Some applications are given while applications by other authors appear elsewhere.
The calculation of penetration indices for exceptional wild arcs
James M.
McPherson
137-149
Abstract: A new class of wild arcs is defined, the class of ``exceptional'' arcs, which is a subclass of the class of arcs whose only wildpoint is an endpoint. This paper then uses geometric techniques to calculate the penetration indices of these exceptional arcs.
On coefficient means of certain subclasses of univalent functions
F.
Holland;
J. B.
Twomey
151-163
Abstract: Let $\mathcal{R}$ denote the class of regular functions whose derivatives have positive real part in the unit disc $\gamma$ and let $ \mathcal{S}$ denote the class of functions starlike in $\gamma$. In this paper we investigate the rates of growth of the means ${s_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n\vert{a_k}{\vert^\lambda }(0 < \lambda \leq 1)$ and ${t_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n{k^\lambda }\vert{a_k}{\vert^\lambda }\;(\lambda > 0)$ as $n \to + \infty$ for bounded $f(z) = \Sigma _1^\infty {a_k}{z^k} \in \mathcal{R} \cup \mathcal{S}$. It is proved, for example, that the estimate ${t_n}(\lambda ) = o(1){(\log n)^{ - \alpha (\lambda )}}(n \to + \infty )$, where $\alpha (\lambda ) = \lambda /2$ for $0 < \lambda < 2$ and $\alpha (\lambda ) = 1$ for $\lambda \geq 2$, holds for such functions f, and that it is best possible for each fixed $\lambda > 0$ within the class $\mathcal{R}$ and for each fixed $\lambda \geq 2$ within the class $\mathcal{S}$. It is also shown that the inequality ${s_n}(1) = o(1){n^{ - 1}}{(\log n)^{1/2}}$, which holds for all bounded univalent functions, cannot be improved for bounded $ f \in \mathcal{R}$. The behavior of $ {t_n}(\lambda )$ as $n \to + \infty$ when ${a_k} \geq 0(k \geq 1)$ and $\lambda \geq 1$ is also examined.
Multipliers for certain convolution measure algebras
Charles Dwight
Lahr
165-181
Abstract: Let ($A,\ast$) be a commutative semisimple convolution measure algebra with structure semigroup $ \Gamma$, and let S denote a commutative locally compact topological semigroup. Under the assumption that A possesses a weak bounded approximate identity, it is shown that there is a topological embedding of the multiplier algebra $ \mathcal{M}(A)$ of A in $M(\Gamma )$. This representation leads to a proof of the commutative case of Wendel's theorem for $A = {L_1}(G)$, where G is a commutative locally compact topological group. It is also proved that if ${l_1}(S)$ has a weak bounded approximate identity of norm one, then $ \mathcal{M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$, where $\Omega (S)$ is the multiplier semigroup of S. Likewise, if S is cancellative, then $ \mathcal{M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$. An example is provided of a semigroup S for which ${l_1}(\Omega (S))$ is isomorphic to a proper subset of $ \mathcal{M}({l_1}(S))$.
Stability properties of a class of attractors
Jorge
Lewowicz
183-198
Abstract: Let A be an attractor of an analytical dynamical system defined in ${R^n} \times R$. The class of attractors considered in this paper consists of those A which remain stable as invariant subsets of the complex extension of the flow to ${C^n} \times R$. If A is a critical point or a closed orbit, these are the elementary or generic attractors. It is shown that such an A is always a submanifold of ${R^n}$ and that there exists a Lie group acting on A and containing the given flow as a one parameter dense subgroup; as a consequence, some necessary and sufficient conditions for an analytical dynamical system to have an attracting generic periodic motion are given. It is also shown that for any flow $ {C^1}$-close to the given one, there is a unique retraction of a neighbourhood of A onto a submanifold of $ {R^n}$ homeomorphic to A that commutes with the flow.
Representations of Jordan triples
Ottmar
Loos
199-211
Abstract: Some standard results on representations of quadratic Jordan algebras are extended to Jordan triples. It is shown that the universal envelope of a finite-dimensional Jordan triple is finite-dimensional, and that it is nilpotent if the Jordan triple is radical. A permanence principle and a duality principle are proved which are useful in deriving identities.
On recurrent random walks on semigroups
T. C.
Sun;
A.
Mukherjea;
N. A.
Tserpes
213-227
Abstract: Let $\mu$ be a regular Borel probability measure on a locally compact semigroup S and consider the right (resp. left) random walk on $D = \overline {{\text{U}}{F^n}} ,F = {\text{Supp}}\;\mu$, with transition function ${P^n}(x,B) \equiv {\mu ^n}({x^{ - 1}}B)\;({\text{resp}}.\;{\mu ^n}(B{x^{ - 1}}))$. These Markov chains can be represented as $ {Z_n} = {X_1}{X_2} \cdots {X_n}\;({\text{resp}}.\;{S_n} = {X_n}{X_{n - 1}} \cdots {X_1}),\;{X_i}$'s independent $\mu $-distributed with values in S defined on an infinite-sequence space $ (\Pi _1^\infty {S_i},P),{S_i} = S$ for all i. Let $ {R_r}\;({\text{resp}}.\;{R_t}) = \{ x \in D;{P_x}({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all neighborhoods $ {N_x}$ of x} and ${R'_r}({R'_t}) = \{ x \in D;P({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all ${N_x}$ of x}. Let S be completely simple ( $ = E \times G \times F$, usual Rees product) in the results (1), (2), (3), (4), (5) below: (1) $x \in {R_r}\;iff\;\Sigma \;{\mu ^n}({x^{ - 1}}{N_x}) = \infty $ for all neighborhoods $ {N_x}$ of $ x\;iff\;\Sigma \;{\mu ^n}({N_x}) = \infty$ for all ${N_x}$ of x. (2) Either ${R_r} = {R_t} = \emptyset$ or ${R_r} = {R_t} = D =$ also completely simple. (3) If the group factor G is compact, then there are recurrent values and we have ${R_r} = {R_t} = D =$ completely simple. (4) $R' = R = K = $ the kernel of S. These results extend previously known results of Chung and Fuchs and Loynes.
On strictly cyclic algebras, $\mathcal{P}$-algebras and reflexive operators
Domingo A.
Herrero;
Alan
Lambert
229-235
Abstract: An operator algebra $\mathfrak{A} \subset \mathcal{L}(\mathcal{X})$ (the algebra of all operators in a Banach space $\mathcal{X}$ over the complex field C) is called a ``strictly cyclic algebra'' (s.c.a.) if there exists a vector $ {x_0} \in \mathcal{X}$ such that $ \mathfrak{A}({x_0}) = \{ A{x_0}:A \in \mathfrak{A}\} = \mathcal{X};{x_0}$ is called a ``strictly cyclic vector'' for $\mathfrak{A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak{A}$ (i.e., if $A \in \mathfrak{A}$ and $ A{x_0} = 0$, then $ A = 0$), then $\mathfrak{A}$ is called a ``separated s.c.a." $\mathfrak{A}$ is a $ \mathcal{P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal{X}$, there exists $ {x_0} \in \mathcal{X}$ such that $\left\Vert {A{x_j}} \right\Vert \leq \left\Vert {A{x_0}} \right\Vert$, for all $A \in \mathfrak{A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak{A}$ is an s.c.a., then $\mathfrak{A}$ is a $ \mathcal{P}$-algebra and the strong and the uniform operator topology coincide on $ \mathfrak{A}$; these results are specialized for the case when $\mathfrak{A}$ and
Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification
David
Ruelie
237-251
Abstract: We consider a compact set $\Omega$ with a homeomorphism (or more generally a $ {{\mathbf{Z}}^\nu }$ action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on $ \mathcal{C}(\Omega )$--the real continuous functions on $\Omega$. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated.
The law of the iterated logarithm for Brownian motion in a Banach space
J.
Kuelbs;
R.
Lepage
253-264
Abstract: Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result.
Extreme points for some classes of univalent functions
W.
Hengartner;
G.
Schober
265-270
Abstract: Monotonicity properties are given for extreme points in classes of normalized analytic and univalent mappings of an arbitrary domain. For the familiar class of normalized univalent mappings of the unit disk, extreme points f are shown to have the remarkable property that $f/z$ is univalent.
Triangular representations of splitting rings
K. R.
Goodearl
271-285
Abstract: The term ``splitting ring'' refers to a nonsingular ring R such that for any right R-module M, the singular submodule of M is a direct summand of M. If R has zero socle, then R is shown to be isomorphic to a formal triangular matrix ring $\left( {\begin{array}{*{20}{c}} A & 0 B & C \end{array} } \right)$, where A is a semiprime ring, C is a left and right artinian ring, and $_C{B_A}$ is a bimodule. Also, necessary and sufficient conditions are found for such a formal triangular matrix ring to be a splitting ring.
Multiplicative structure of generalized Koszul complexes
Eugene H.
Gover
287-307
Abstract: A multiplicative structure is defined for the generalized Koszul complexes $K({ \wedge ^p}f)$ associated with the exterior powers of a map $f:{R^m} \to {R^n}$ where R is a commutative ring and $m \geq n$. With this structure $K({ \wedge ^n}f)$ becomes a differential graded R-algebra over which each $K({ \wedge ^p}f),1 \leq p \leq n$, is a DG right $K({ \wedge ^n}f)$-module. For $f = 0$ and $n > 1$, the multiplication and all higher order Massey operations of $K({ \wedge ^n}f)$ are shown to be trivial. When R is noetherian local, $K({ \wedge ^n}f)$ is used to define a class of local rings which includes the local complete intersections. The rings obtained for $n > 1$ are Cohen-Macaulay but not Gorenstein. Their Betti numbers and Poincaré series are computed but these do not characterize the rings.
Isomorphisms of the lattice of inner ideals of certain quadratic Jordan algebras
Jerome M.
Katz
309-329
Abstract: The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the associative theory. In particular, the simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals play a role analogous to that of the simple artinian algebras in the associative theory. In this paper, we investigate the automorphism group of the lattice of inner ideals of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. For the case $ \mathfrak{H}(\mathfrak{A}{,^ \ast })$ where $(\mathfrak{A}{,^ \ast })$ is a simple artinian algebra with hermitian involution, we show that the automorphism group of the lattice of inner ideals is isomorphic to the group of semilinear automorphisms of $\mathfrak{A}$. For the case $ \mathfrak{H}({\mathfrak{Q}_n}{,^ \ast })$ where $ \mathfrak{Q}$ is a split quaternion algebra, we obtain only a partial result. For the cases $J = \mathfrak{H}({\mathfrak{O}_3})$ and $J = {\text{Jord}}(Q,1)$ for $\mathfrak{O}$ an octonion algebra, $(Q,1)$ a nondegenerate quadratic form with base point of Witt index at least three and J finite dimensional, it is shown that every automorphism of the lattice of inner ideals is induced by a norm semisimilarity. Finally, we determine conditions under which two algebras of the type under consideration can have isomorphic lattices of inner ideals.
Hermitian operators and one-parameter groups of isometries in Hardy spaces
Earl
Berkson;
Horacio
Porta
331-344
Abstract: Call an operator A with domain and range in a complex Banach space X hermitian if and only if iA generates a strongly continuous one-parameter group of isometries on X. Hermitian operators in the Hardy spaces of the disc $({H^p},1 \leq p \leq \infty )$ are investigated, and the following results, in particular, are obtained. For $1 \leq p \leq \infty ,p \ne 2$, the bounded hermitian operators on ${H^p}$ are precisely the trivial ones--i.e., the real scalar multiples of the identity operator. Furthermore, as pointed out to the authors by L. A. Rubel, there are no unbounded hermitian operators in ${H^\infty }$. To each unbounded hermitian operator in the space ${H^p},1 \leq p < \infty ,p \ne 2$, there corresponds a uniquely determined one-parameter group of conformal maps of the open unit disc onto itself. Such unbounded operators are classified into three mutually exclusive types, an operator's type depending on the nature of the common fixed points of the associated group of conformal maps. The hermitian operators falling into the classification termed ``type (i)'' have compact resolvent function and one-dimensional eigenmanifolds which collectively span a dense linear manifold in $ {H^p}$.
Almost everywhere convergence of Vilenkin-Fourier series
John
Gosselin
345-370
Abstract: It is shown that the partial sums of Vilenkin-Fourier series of functions in $ {L^q}(G),q < 1$, converge almost everywhere, where G is a zero-dimensional, compact abelian group which satisfies the second axiom of countability and for which the dual group X has a certain bounded subgroup structure. This result includes, as special cases, the Walsh-Paley group $ {2^w}$, local rings of integers, and countable products of cyclic groups for which the orders are uniformly bounded.
The strong law of large numbers when the mean is undefined
K. Bruce
Erickson
371-381
Abstract: Let ${S_n} = {X_1} + \cdots + {X_n}$ where $\{ {X_n}\}$ are i.i.d. random variables with $EX_1^ \pm = \infty$. An integral test is given for each of the three possible alternatives $\lim ({S_n}/n) = + \infty $ a.s.; $\lim ({S_n}/n) = - \infty $ a.s.; $\lim \sup ({S_n}/n) = + \infty$ and $\lim \inf ({S_n}/n) = - \infty$ a.s. Some applications are noted.
Neocontinuous Mikusi\'nski operators
Carl C.
Hughes;
Raimond A.
Struble
383-400
Abstract: A class of Mikusiński-type operators in several variables, called neocontinuous operators, is studied. These particular operators are closely affiliated with Schwartz distributions on ${R^k}$ and share certain continuity properties with them. This affiliation is first of all revealed through a common algebraic view of operators and distributions as homomorphic mappings and a new representation theory, and is then characterized in terms of continuity properties of the mappings. The traditional procedures of the operational calculus apply to the class of neocontinuous operators. Moreover, the somewhat vague association of operational and distributional solutions of partial differential equations is replaced by the decisive representation concept, thus illustrating the appropriateness of the study of neocontinuous operators.
Perturbations of nonlinear differential equations
R. E.
Fennell;
T. G.
Proctor
401-411
Abstract: Scalar and vector comparison techniques are used to study the comparative asymptotic behavior of the systems (1)
Convex hulls and extreme points of families of starlike and convex mappings
L.
Brickman;
D. J.
Hallenbeck;
T. H.
Macgregor;
D. R.
Wilken
413-428
Abstract: The closed convex hull and extreme points are obtained for the starlike functions of order $\alpha$ and for the convex functions of order $ \alpha$. More generally, this is determined for functions which are also k-fold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order $ \alpha$. Also, the lower bound on $\operatorname{Re} \{ f(z)/z\}$ is found for each $z\;(\vert z\vert < 1)$ where f varies over the convex functions of order $ \alpha$ and $\alpha \geq 0$.
Cauchy problem and analytic continuation for systems of first order elliptic equations with analytic coefficients
Chung Ling
Yu
429-443
Abstract: Let a, b, c, d, f, g be analytic functions of two real variables x, y in the $z = x + iy$ plane. Consider the elliptic equation (M) $\partial u/\partial x - \partial v/\partial y = au + bv + f,\partial u/\partial y + \partial v/\partial x = cu + dv + g$. The following areas will be investigated: (1) the integral respresentations for solutions of (M) to the boundary $\partial G$ of a simply connected domain G; (2) reflection principles for (M) under nonlinear analytic boundary conditions; (3) the sufficient conditions for the nonexistence and analytic continuation for the solutions of the Cauchy problem for (M).
Square integrable representations of nilpotent groups
Calvin C.
Moore;
Joseph A.
Wolf
445-462
Abstract: We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if $\Gamma$ is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in ${L_2}(N/\Gamma )$, and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.
Appell polynomials and differential equations of infinite order
J. D.
Buckholtz
463-476
Abstract: Let $ \Phi (z) = \Sigma _0^\infty {\beta _j}{z^j}$ have radius of convergence $r\;(0 < r < \infty )$ and no singularities other than poles on the circle $\vert z\vert = r$. A complete solution is obtained for the infinite order differential equation $( \ast )\;\Sigma _0^\infty {\beta _j}{u^{(j)}}(z) = g(z)$. It is shown that $(\ast)$ possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by $\Phi$. The solutions of $( \ast )$ are expressed in terms of the coefficients which appear in the Appell polynomial expansions of g. An alternate method of solution is obtained, in which the problem of solving $( \ast )$ is reduced to the problem of finding a solution, within a certain space of entire functions, of a finite order linear differential equation with constant coefficients. Additionally, differential operator techniques are used to study Appell polynomial expansions.
Mechanical systems with symmetry on homogeneous spaces
Ernesto A.
Lacomba
477-491
Abstract: The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets
A geometrical characterization of Banach spaces with the Radon-Nikodym property
Hugh B.
Maynard
493-500
Abstract: A characterization of Banach spaces having the Radon-Nikodym property is obtained in terms of a convexity requirement on all bounded subsets. In addition a Radon-Nikodym theorem, utilizing this convexity property, is given for the Bochner integral and it is easily shown that this theorem is equivalent to the Phillips-Metivier Radon-Nikodym theorem as well as all the standard Radon-Nikodym theorems for the Bochner integral.
Weakly almost periodic functionals on the Fourier algebra
Charles F.
Dunkl;
Donald E.
Ramirez
501-514
Abstract: The theory of weakly almost periodic functional on the Fourier algebra is herein developed. It is the extension of the theory of weakly almost periodic functions on locally compact abelian groups to the duals of compact groups. The complete direct product of a countable collection of nontrivial compact groups furnishes an important example for some of the constructions.