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Sufficient conditions for convergence of riemann sums for function space defined by the k-modulus of continuity

October 2021
Real Analysis Exchange 46(1)

DOI:10.14321/realanalexch.46.1.0037

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Kathryn Porter

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Families of bloch functions on higher dimensional complex spaces

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A holomorphic function f for which (1+|z|f(z)| is bounded on the complex disk D{zεC.|z|<} is classically called Bloch. We introduce the notion of uniformly Bloch families on complex spaces, which notion includes that of Bloch function as defined and studied by authors in various settings, show that several known characterizations and properties of Bloch functions have generalizations to uniformly ... [Show full abstract] Bloch families on higher dimensional complex spaces and establish some new properties of Bloch functions. For example we show that if X is a complex space. AC and C-A does not contain arbitrarily large disks, then FH(X)is uniformly Bloch iff for some . As is the case for spaces of Bloch functions defined on the unit disk and on homogeneous bounded domains in the space of Bloch functions (mod constants) defined on a hyperbolic space is a complex Banach space. We also show that members of uniformly Bloch families on complex manifolds satisfy extension theorems of big Picard type if the nature of the singularies is that of normal crossings; moreover, the function which maps a member of a uniformly Bloch family to its extension is a topological imbedding. The family of continuous extensions of the members of a uniformly Bloch family from the punctured disk D * or more generally from ( D *)n is shown to be uniformly Bloch.

Chapter

Function spaces in topological categories

November 2006

Oswald Wyler

We do not attempt a comprehensive survey; this paper concentrates on some recent developments in which the author was involved. We shall discuss fine proper function spaces, closed reflective subcategories of cartesian closed topological categories, continuous uniform convergence for uniform convergence spaces, and spaces of closed sets of locally compact spaces. The title is somewhat inaccurate: ... [Show full abstract] we work with initially structured categories as defined in [22], not with the topological categories of [14] and [23]. The llst of references is far from comprehensive, but it should be adequate.

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Second dual projection characterizations of three classes of L0-closed, convex, bounded sets in L1

June 2008 · Journal of Mathematical Analysis and Applications

Let τλ be the topology of convergence locally in measure on L1=L1(λ) and P be the Yosida–Hewitt projection from onto L1. We characterize convex, τλ-compact subsets C of L1 as precisely those for which P is a compactness preserving map from with the weak∗-topology to C with the τλ-topology. We further show that a convex, τλ-closed, L1-norm bounded subset C of L1 is a Schur set if and only if is ... [Show full abstract] sequentially continuous. Finally, we discover which τλ-closed, bounded, convex subsets C of L1 are such that is continuous. We call such sets C good. They turn out to be precisely the pluriweak-to-measure-continuity sets, in the sense defined below.

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Asymptotic Suprema of Averaged Random Functions

December 1991 · The Annals of Statistics

Asad Zaman

Suppose

X_i

are i.i.d. random variables taking values in

\mathscr{X}, \Theta

is a parameter space and

y: \mathscr{X} \times \Theta \rightarrow \mathbf{R}

is a map. Define the averages

S_n(y, \theta) = (1/n)\sum^n_{i = 1}y(X_i, \theta)

and the truncated expectations

T_m(y, \theta) = \mathbf{E} \max(y(X_1, \theta), - m)

. Under the hypothesis of global dominance [i.e., $\mathbf{E} ... [Show full abstract] \sup_\Theta y(X_1, \theta) < \infty

] and some regularity conditions, the main result of the paper characterizes the asymptotic suprema of

S_n

as follows. For any subset

G of \Theta

, with probability 1,

\lim_{n \rightarrow \infty} \sup_{\theta \in G} S_N(y, \theta) = \lim_{m \rightarrow \infty} \sup_{\theta \in G} T_m(y, \theta).

This has immediate application to consistency of

M

-estimators. In particular, under global dominance, maxima of

S_n

must converge to the same limit as the maxima of

T_m(y, \theta)

almost surely. We also obtain necessary and sufficient conditions for consistency resembling Huber's in the case of local dominance [i.e., each

\theta \in \Theta

has a neighborhood

N(\theta)

such that

\mathbf{E} \sup_{\psi \in N(\theta)}y(X_1, \psi) < \infty\rbrack.

In this case there must exist a function

b(\theta) \geq 1

such that

y/b

is globally dominated and maxima of

T_m(y/b,\theta)$ converge.

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Uniformizable and realcompact bornological universes

September 2009 · Applied General Topology

Tom Vroegrijk

Bornological universes were introduced some time ago by Hu and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. One of Hu's results gives us a necessary and sufficient condition for which a bornological universe is metrizable. In this article we will extend this result and give a characterization of uniformizable bornological ... [Show full abstract] universes. Furthermore, a construction on bornological universes that the author used to find the bornological dual of function spaces endowed with the bounded-open topology will be used to define realcompactness for bornological universes. We will also give various characterizations of this new concept.

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Last Updated: 22 Oct 2024

Sufficient conditions for convergence of riemann sums for function space defined by the k-modulus of continuity

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