We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.

In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL2(A) play a crucial role.

The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L2-space of functions; this is done in Section 2 of this paper.

Dans une note récente (1), Szekeres a donné le théorème suivant:

Sif est une fonction réelle, définie, strictement croissante et ayant la deuxième dérivée continue sur (a, b) ( — ∞ ≤ a < b ≤ ≤) alors il existe une fonction réelle ϕ, définie et strictement croissante sur (a, b), et une fonction réelle ψ définie et strictement croissante sur (ϕ(a), ϕ(b)), telles que ϕ˝,(x) ≥ 0 pour a < x < b, ψ ˝(u) < 0 pour ϕ(a) < u < ϕ(b) et f(x) = ψ(ϕ(x)) pour a < x < b.

Dans la démonstration de ce théorème, Szekeres utilise une expression où la dérivée f‘(x) apparaît au dénominateur, ce qui exige qu'on ait f'(x) ≠ 0 pour a < x < b, condition qui ne figure pas dans l'énoncé du théorème. Il est naturel alors de poser le problème si le théorème ci-dessus reste en vigueur si cette condition n'est pas remplie. Il y a lieu aussi de se demander si les autres conditions, par exemple l'existence et la continuité def“(x) sont nécessaires pour qu'une fonction strictement croissante puisse être représentée comme une superposition de deux fonctions croissantes, l'une convexe, l'autre concave.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$, where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.

2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulus

and the minimum modulus

When no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).

Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.

THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then

(1)

This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.

An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].

The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).

An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

Let Ω be a cyclic extension of degree l over the field Σ. Correcting an error which for some time had been haunting the literature, Hasse (1, p. 272) noted that for l = 2, the field Ω may contain a unit ξ such that

.

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.

C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

It is shown that the essential spectrum of a cyclic, self-dual, subnormal operator is symmetric with respect to the real axis. The study of the structure of a cyclic, irreducible, self-dual, subnormal operator is reduced to the operator Sμ with bpeμ = D. Necessary and sufficient conditions for a cyclic subnormal operator Sμ with bpeμ = D to be self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable. Finally, an approach to a general cyclic, self-dual, subnormal operator is discussed.

Let A be a C*-algebra and H a Hilbert space of large enough (infinite at least) dimension so that every πƒ, where ƒ is a factor state on A, can be unitarily represented on H. Let Fac (A, H) denote the set of all factor representations of A on H. If π is in Fac (A, H) we call its essential subspace the smallest, closed, vector subspace KoiH such that π (A ) is null on H Θ K. We define Fac∞(A, H) to be the set of elements in Fac (A, H) whose essential subspace is H.

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the set

We say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

1. Introduction. In one of his papers on limits for the characteristic roots of a Matrix, Brauer (1) has stated a theorem, which connects the roots of a given square matrix A, with those of a matrix A* derived from A by a certain process. The proof of this theorem involves a continuity argument and in a recent paper on the construction of stochastic matrices Hazel Perfect (5) has given a proof which avoids considerations of continuity.

Until recently the literature contained little on the axiomatic foundations of elliptic geometry that was non-analytical and independent of projective geometry. During the past decade this subject has come in for further study, notably by Busemann [2] and Blumenthal [1], who supplied such foundations. This paper presents another and, it is believed, simpler effort in the same general direction, proceeding by the familiar synthetic methods of elementary geometry and using only elementary topological notions and ideas concerning metric spaces. Specifically, elliptic 2-space is obtained on the basis of six axioms, most notable of which is one assuming the existence of translations. The writer wishes to express his deep appreciation to Herbert Busemann for his invaluable help.

1. Let G be a compact group, let I b e a subset of its dual object 𝚪, which, without loss of generality, will be assumed to be a countable subset. Let Di, i ∈ I , be irreducible representations of G of degree di. The Fourier series of a function F in L1(G) is denned by

where

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows:

(1)

Here π, ϕ, ϕ1 and ϕ2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes.

Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

Let I be an interval, and let for 1 ≦ j ≦ I < ∞ be abutted subintervals such that . Let τj be a linear differential expression defined on Ij. In this paper we study densely defined operators associated with

(0.1)

A U-ring is a ring in which every subring is a meta ideal. A meta ideal of a ring R is a subring I of R which lies in a chain of subrings,

with the properties:

(1) Iλ is an ideal of Iλ+1 for all λ < β;

(2) If α is a limit ordinal number, then Iα = ∪λ<αIλ.

Freidman [3] proved that every nil U-ring is a locally nilpotent ring. Since there are many locally nilpotent rings which are not U-rings, the class of locally nilpotent rings is not a very good bound for the class of nil U-rings. This paper establishes a new bound for nil U-rings based on a property of the multiplicative semigroup of the ring.