On metrization of unions of function spaces on different intervals

Taleb Alkurdi; Sander C. Hille; Onno Van Gaans

doi:10.1017/S1446788712000365

On metrization of unions of function spaces on different intervals

Taleb Alkurdi^*
, Sander C. Hille
, Onno Van Gaans

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Abstract This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

Original language	English
Pages (from-to)	281-297
Number of pages	17
Journal	Journal of the Australian Mathematical Society
Volume	92
Issue number	3
DOIs	https://doi.org/10.1017/S1446788712000365
State	Published - Jun 2012
Externally published	Yes

Bibliographical note

Funding Information:
Alkurdi acknowledges the Erasmus-Mundus project funding of his research. Van Gaans acknowledges the support by ‘Vidi subsidie’ (639.032.510) of the Netherlands Organisation for Scientific Research (NWO).

Keywords

Skorohod metric
concatenation map
lipeomorphism
metric space
penalty function
space of continuous functions on different intervals

ASJC Scopus subject areas

General Mathematics

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10.1017/S1446788712000365

Cite this

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title = "On metrization of unions of function spaces on different intervals",

abstract = "Abstract This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on c{\`a}dl{\`a}g functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.",

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note = "Funding Information: Alkurdi acknowledges the Erasmus-Mundus project funding of his research. Van Gaans acknowledges the support by {\textquoteleft}Vidi subsidie{\textquoteright} (639.032.510) of the Netherlands Organisation for Scientific Research (NWO).",

year = "2012",

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doi = "10.1017/S1446788712000365",

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N2 - Abstract This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

AB - Abstract This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

KW - Skorohod metric

KW - concatenation map

KW - lipeomorphism

KW - metric space

KW - penalty function

KW - space of continuous functions on different intervals

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ER -

On metrization of unions of function spaces on different intervals

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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