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Strong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.

In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.

This paper continues the investigations in [1] and [2] and extends the results of the latter paper to functions of several complex variables. Namely let λ : R>0 —> R>0 be monotonically increasing. (If λ is non-constant, we also require that log r = 0(λ(r)). We write / = 0(g) to mean that there is a constant C > 0 such that f(r) ≦ Cg(r) except possibly on a set of intervals of finite total length.) Let 0\ denote the set of meromorphic / on Cn for which the Nevanlinna characteristic function [5, p. 174] T(f, r) satisfies T(f, r) = 0(λ(r)).

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.

In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.

Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ℂn.

In this paper we regularize the Kepler problem on S3 in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon–Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon–Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.

Let b1,…,b5 be non-zero integers and n any integer. Suppose that b1 + … + b5 ≡ n (mod 24) and (bi, bj) = 1 for 1 ≤ i < j ≤ 5. In this paper we prove that(i)if all bj are positive and , then the quadratic equation is soluble in primes pj, and(ii)if bj are not all of the same sign, then the above quadratic equation has prime solutions satisfying .

In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

An implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

An important question in the study of moment problems is to determine when a fixed point in ℝn lies in the moment cone of vectors , with μ a nonnegative measure. In associated optimization problems it is also important to be able to distinguish between the interior and boundary of the moment cone. Recent work of Dachuna-Castelle, Gamboa and Gassiat derived elegant computational characterizations for these problems, and for related questions with an upper bound on μ. Their technique involves a probabilistic interpretation and large deviations theory. In this paper a purely convex analytic approach is used, giving a more direct understanding of the underlying duality, and allowing the relaxation of their assumptions.

Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N).In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions.

In this paper we are concerned with partition functions pϒ(n) that have generating functions of the formwhere γ(n) ≧ 0. We shall obtain an asymptotic relation for pϒ(n) under suitable restrictions on ϒ (see Theorem 1.1). These restrictions are weaker than those of Brigham [2] who considered this problem previously.

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2) exists a.e. for all.

We define (as in [7]) integer sequences one for each positive integer k ≧ 2, by1.1where are the kth roots of unity and (E(k))n is replaced by after multiplying out. We note that (1.1) implies , n ≠ 0 (mod k).In [7], we considered some special properties of these number sequences, proved several congruences and conjectured several others. This paper is a continuation of the work presented in [7].In Section 2 we demonstrate the asymptotic rate of growth of the numbers by showing thatIn Section 3 we present a large number of congruences (modulo 2048), some of which are proved or can be proved by the techniques presented herein, and other congruences which appear to be true on the basis of numerical evidence.

A much-studied equation in recent years has been the second order nonlinear ordinary differential equationwhere q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroupof positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub-almost convergence.

For f ∊ L−1(0, T), we define the distribution functionwhere T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation(0.1)when the coefficient q was allowed to vary in the classWe were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

W. A. J. Luxemburg and A. C. Zaanen in [7] and W. A. J. Luxemburg in [5] have studied the order properties of the order bounded linear functionals of a given Riesz space L. In this paper we consider the vector space (L, M) of the order bounded linear transformations from a given Riesz space L into a Dedekind complete Riesz space M.

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition1.1and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.

Non-linear problems have been studied by Krasnoselski, Browder, and others; in fact Browder and independently Kirk (cf., 1; 5) have proved the following remarkable theorem: let X be a uniformly convex Banach space, U a non-expansive mapping of a bounded closed convex subset C of X into C, i.e., ||Ux — Uy|| ⩽ ||x — y|| for x, y ∊ C; then U has a fixed point in C. The aim of this paper is to give some existence theorems for non-linear functional equations in uniformly convex Banach spaces. Similar results may be found in (3 ; 6).

Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated to Lyapunov's method. Some of these results have been obtained by the author in (2). The purpose of this paper is to make use of the comparison method to obtain some refinements of Lyapunov's theory.

Disconjugacy criteria have been established for linear selfadjoint differential equations of order 2n by Sternberg [4] and Ahlbrandt [1]. Such differential equations can be written in the form1.1where it is assumed that the coefficients are real and that Pn(x) ≠ 0. We shall be interested in nontrivial solutions v(x) of (1.1), which satisfy1.2for distinct points α and β. The smallest β> α such that (1.2) is satisfied nontrivially by a solution of (1.1), is denoted by μ1(α) and called the first conjugate point of x = α with respect to (1.1). If no such conjugate point exists we write μ1(α) = ∞, and say that (1.1) is disconjugate on [α, ∞).The principal purpose of this paper is to generalize these disconjugacy criteria to the general linear nonselfadjoint differential equation of the form1.3

Methods of mechanical quadrature of functions of more than one variable apparently have received little systematic investigation and the few available results are widely scattered in the literature. In this paper a systematic approach to this problem is given and a number of formulae are derived which may prove to be useful.

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

In a previous paper (2), I discussed conditions in order that a holomorphic function f(w) be a Laplace transform of a function F(t) such that, for some a, F(t)e-at ∈ Lp(0, ∞). These conditions involved (a) the behaviour of integral transforms of the values of the function on a vertical line w = const, and (b) conditions involving the order of magnitude of the function at infinity. The present article discusses the corresponding question for Laplace transforms of distributions.

In this paper integrals are considered from the point of view of inverting differential operators. In order to do this it is necessary to introduce integrals more general than the Lebesgue integral and these integrals turn out to have other interesting properties (6, 7, 12). The integral introduced here is defined in the setting of axiomatic potential theory (2, 4). By defining it as generally as possible it not only includes the James P2-integral but inverts many of the standard second-order differential operators.

This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively.

Throughout this paper will denote an infinite dimensional, separable complex Hilbert space, and will denote the unit sphere of (i.e. ). Also will represent the algebra of all bounded linear operators on , and will represent the ideal of all compact operators on . Furthermore will denote the set of all (orthogonal) projections on and will denote the sublattice of consisting of all finite rank projections. In most of the cases (especially when limits are involved) will be regarded as a directed set with the usual order relation inherited from .

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such thatis finite.