scholarly journals Local uniform convergence of the Riesz means of Laplace and Dirac expansions

1997 ◽  
Vol 6 (4) ◽  
pp. 653-696 ◽  
Author(s):  
Miklòs Horváth
Keyword(s):  
Uniform Convergence ◽  
Riesz Means ◽  
Local Uniform Convergence

scholarly journals LIMITS OF MULTIPOLE PLURICOMPLEX GREEN FUNCTIONS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250065 ◽  
Author(s):  
JÓN I. MAGNÚSSON ◽  
ALEXANDER RASHKOVSKII ◽  
RAGNAR SIGURDSSON ◽  
PASCAL J. THOMAS
Keyword(s):  
Green Function ◽  
Uniform Convergence ◽  
Complete Intersection ◽  
Green Functions ◽  
Ideal Formula ◽  
Vanishing Ideal ◽  
Pluricomplex Green Function ◽  
Samuel Multiplicity ◽  
Hyperconvex Domain ◽  
Local Uniform Convergence

Let Ω be a bounded hyperconvex domain in ℂn, 0 ∈ Ω, and Sε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each Sε we associate its vanishing ideal [Formula: see text] and pluricomplex Green function [Formula: see text]. Suppose that, as ε tends to 0, [Formula: see text] converges to [Formula: see text] (local uniform convergence), and that (Gε)ε converges to G, locally uniformly away from 0; then [Formula: see text]. If the Hilbert–Samuel multiplicity of [Formula: see text] is strictly larger than its length (codimension, equal to N here), then (Gε)ε cannot converge to [Formula: see text]. Conversely, if [Formula: see text] is a complete intersection ideal, then (Gε)ε converges to [Formula: see text]. We work out the case of three poles.

WEAK CONTINUITY OF THE COMPLEX -HESSIAN OPERATORS WITH RESPECT TO LOCAL UNIFORM CONVERGENCE

2013 ◽  
Vol 89 (2) ◽  
pp. 227-233
Author(s):  
NEIL S. TRUDINGER ◽  
WEI ZHANG
Keyword(s):  
Uniform Convergence ◽  
Monotonicity Formula ◽  
Alternative Proof ◽  
Weak Continuity ◽  
Plurisubharmonic Functions ◽  
Local Uniform Convergence

AbstractIn this paper, we study the properties of $k$-plurisubharmonic functions defined on domains in ${ \mathbb{C} }^{n} $. By the monotonicity formula, we give an alternative proof of the weak continuity of complex $k$-Hessian operators with respect to local uniform convergence.

scholarly journals On the class gamma and related classes of functions

2013 ◽  
Vol 93 (107) ◽  
pp. 1-18 ◽  
Author(s):  
Edward Omey
Keyword(s):  
Uniform Convergence ◽  
Auxiliary Function ◽  
Higher Order ◽  
Representation Theorems ◽  
Order Relations ◽  
Measurable Functions ◽  
Local Uniform Convergence

The gamma class ??(g) consists of positive and measurable functions that satisfy f(x + yg(x))/f(x) ? exp(?y). In most cases the auxiliary function g is Beurling varying and self-neglecting, i.e., g(x)/x ? 0 and g??0(g). Taking h = log f, we find that h?E??(g, 1), where E??(g, a) is the class of positive and measurable functions that satisfy (f(x + yg(x))? f(x))/a(x) ? ?y. In this paper we discuss local uniform convergence for functions in the classes ??(g) and E??(g, a). From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class ??(g) and related classes. Two applications are given.

The Schwartz property and nuclearity of spaces of smooth and holomorphic functions in infinite dimensions

1990 ◽  
Vol 107 (2) ◽  
pp. 377-385
Author(s):  
Sten Bjon
Keyword(s):  
Uniform Convergence ◽  
Open Subset ◽  
Convex Space ◽  
Holomorphic Functions ◽  
Locally Convex Space ◽  
Infinite Dimensions ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Locally Convex ◽  
Local Uniform Convergence

In [8] it was shown that a locally convex space E is a Schwartz space if and only if the convergence algebras Hc(U) and He(U) of holomorphic functions on an open subset of E coincide, i.e. if and only if continuous convergence c (see [1]) and the associated equable convergence structure e (= local uniform convergence, see [2, 13]) coincide.

On the Convergence of the Density of the Power Normalized Maximum1

1998 ◽  
Vol 48 (1-2) ◽  
pp. 13-20 ◽  
Author(s):  
N. R. Mohan ◽  
U. R. Subramanya
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Laws ◽  
Stable Law ◽  
Necessary And Sufficient ◽  
Local Uniform Convergence

Necessary and sufficient conditions for local uniform convergence of the density of the power normalized maximum to the corresponding density of a p-max stable law is derived for each of the six p-max stable laws. An unified criterian is also obtained.

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