scholarly journals Strict and pointwise convergence of BV functions in metric spaces

2017 ◽  
Vol 455 (2) ◽  
pp. 1005-1021 ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Pointwise Convergence ◽  
Metric Spaces ◽  
Bv Functions

scholarly journals Metric spaces with nice closed balls and distance functions for closed sets

1987 ◽  
Vol 35 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Closed Ball ◽  
Distance Functions ◽  
Entire Space ◽  
Closed Sets ◽  
Bounded Sets

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Boundary Values ◽  
Poincare Inequality ◽  
Compactly Supported ◽  
Bv Functions

AbstractIn the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of{\mathrm{BV}}functions with zero boundary values. In particular, we show that the class is the closure of compactly supported{\mathrm{BV}}functions in the{\mathrm{BV}}norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and{\mathrm{BV}}analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

The set of filter cluster functions

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3057-3071
Author(s):  
Hüseyin Albayrak ◽  
Serpil Pehlivan
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Limit Function ◽  
Cluster Function

In this work, we are concerned with the concepts of F-?-convergence, F-pointwise convergence and F-uniform convergence for sequences of functions on metric spaces, where F is a filter on N. We define the concepts of F-limit function, F-cluster function and limit function respectively for each of these three types of convergence, and obtain some results about the sets of F-cluster and F-limit functions for sequences of functions on metric spaces. We use the concept of F-exhaustiveness to characterize the relations between these points.

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