A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field
We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [ a , b ] ⊂ ℝ, f and its analytic continuation f̅ ∞ satisfy the same properties that can be expressed in the language of rea...
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| Veröffentlicht in: | P-adic numbers, ultrametric analysis, and applications ultrametric analysis, and applications, 2018-07, Vol.10 (3), p.179-191 |
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| container_title | P-adic numbers, ultrametric analysis, and applications |
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| creator | Bottazzi, Emanuele |
| description | We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function
f
is analytic on a compact interval [
a
,
b
] ⊂ ℝ,
f
and its analytic continuation
f̅
∞
satisfy the same properties that can be expressed in the language of real closed ordered fields. If
f
is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research. |
| doi_str_mv | 10.1134/S2070046618030032 |
| format | Article |
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f
is analytic on a compact interval [
a
,
b
] ⊂ ℝ,
f
and its analytic continuation
f̅
∞
satisfy the same properties that can be expressed in the language of real closed ordered fields. If
f
is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.</description><identifier>ISSN: 2070-0466</identifier><identifier>EISSN: 2070-0474</identifier><identifier>DOI: 10.1134/S2070046618030032</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Algebra ; Analytic functions ; Mathematical analysis ; Mathematics ; Mathematics and Statistics</subject><ispartof>P-adic numbers, ultrametric analysis, and applications, 2018-07, Vol.10 (3), p.179-191</ispartof><rights>Pleiades Publishing, Ltd. 2018</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-32d1776d02febef7006fbbc07eb99daa0809cc78ad2e7bad6043c7a83111610b3</citedby><cites>FETCH-LOGICAL-c316t-32d1776d02febef7006fbbc07eb99daa0809cc78ad2e7bad6043c7a83111610b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S2070046618030032$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S2070046618030032$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Bottazzi, Emanuele</creatorcontrib><title>A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field</title><title>P-adic numbers, ultrametric analysis, and applications</title><addtitle>P-Adic Num Ultrametr Anal Appl</addtitle><description>We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function
f
is analytic on a compact interval [
a
,
b
] ⊂ ℝ,
f
and its analytic continuation
f̅
∞
satisfy the same properties that can be expressed in the language of real closed ordered fields. If
f
is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.</description><subject>Algebra</subject><subject>Analytic functions</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>2070-0466</issn><issn>2070-0474</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp1kEFLxDAQhYMouKz7A7wFPFcnSU3a41JcFQqK7p5LmiaapSZrki747-1a0YN4mmH43hveQ-icwCUhLL96piAAcs5JAQyA0SM0O5wyyEV-_LNzfooWMW4BDoworukMbZZ4HaSLRgf8GKxTdtdrbHzA6VXjyrtk3SCT9S5ib_CTlj1eDU5Nl-S_sFrvbVbZvU0Sr6zuuzN0YmQf9eJ7ztFmdbOu7rL64fa-WtaZYoSnjNGOCME7oEa32owZuGlbBUK3ZdlJCQWUSolCdlSLVnYccqaELBghhBNo2RxdTL674N8HHVOz9UNw48uGQkEFlGPQkSITpYKPMWjT7IJ9k-GjIdAcCmz-FDhq6KSJI-tedPh1_l_0CU81cLo</recordid><startdate>20180701</startdate><enddate>20180701</enddate><creator>Bottazzi, Emanuele</creator><general>Pleiades Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20180701</creationdate><title>A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field</title><author>Bottazzi, Emanuele</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-32d1776d02febef7006fbbc07eb99daa0809cc78ad2e7bad6043c7a83111610b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Algebra</topic><topic>Analytic functions</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bottazzi, Emanuele</creatorcontrib><collection>CrossRef</collection><jtitle>P-adic numbers, ultrametric analysis, and applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bottazzi, Emanuele</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field</atitle><jtitle>P-adic numbers, ultrametric analysis, and applications</jtitle><stitle>P-Adic Num Ultrametr Anal Appl</stitle><date>2018-07-01</date><risdate>2018</risdate><volume>10</volume><issue>3</issue><spage>179</spage><epage>191</epage><pages>179-191</pages><issn>2070-0466</issn><eissn>2070-0474</eissn><abstract>We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function
f
is analytic on a compact interval [
a
,
b
] ⊂ ℝ,
f
and its analytic continuation
f̅
∞
satisfy the same properties that can be expressed in the language of real closed ordered fields. If
f
is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S2070046618030032</doi><tpages>13</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 2070-0466 |
| ispartof | P-adic numbers, ultrametric analysis, and applications, 2018-07, Vol.10 (3), p.179-191 |
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| language | eng |
| recordid | cdi_proquest_journals_2082709032 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Algebra Analytic functions Mathematical analysis Mathematics Mathematics and Statistics |
| title | A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field |
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