关于平方根正弦级数导数的一个注记

Q3 Mathematics
Sergiusz Kęska
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The purpose of this paper is to show that if <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mfenced open=\"{\" close=\"}\">\n <mrow>\n <mi>n</mi>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> is nonincreasing and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>n</mi>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>⟶</mo>\n <mn>0</mn>\n </math>\n </jats:inline-formula>, then the series <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <mi>f</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <mo>∑</mo>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mi mathvariant=\"normal\">sin</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <msqrt>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msqrt>\n <mi>x</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> can be differentiated term-by-term on <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mfenced open=\"[\" close=\"]\">\n <mrow>\n <mi>c</mi>\n <mo>,</mo>\n <mi>d</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> for <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi>c</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>></mo>\n <mn>0</mn>\n </math>\n </jats:inline-formula>. However, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <msup>\n <mrow>\n <mi>f</mi>\n </mrow>\n <mrow>\n <mo>′</mo>\n </mrow>\n </msup>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mn>0</mn>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> may not exist.</jats:p>","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Note on Derivative of Sine Series with Square Root\",\"authors\":\"Sergiusz Kęska\",\"doi\":\"10.1155/2021/7035776\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>Chaundy and Jolliffe proved that if <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M1\\\">\\n <mfenced open=\\\"{\\\" close=\\\"}\\\">\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> is a nonnegative, nonincreasing real sequence, then series <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M2\\\">\\n <mo>∑</mo>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mi mathvariant=\\\"normal\\\">sin</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>n</mi>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> converges uniformly if and only if <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\">\\n <mi>n</mi>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>⟶</mo>\\n <mn>0</mn>\\n </math>\\n </jats:inline-formula>. The purpose of this paper is to show that if <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M4\\\">\\n <mfenced open=\\\"{\\\" close=\\\"}\\\">\\n <mrow>\\n <mi>n</mi>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> is nonincreasing and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M5\\\">\\n <mi>n</mi>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>⟶</mo>\\n <mn>0</mn>\\n </math>\\n </jats:inline-formula>, then the series <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M6\\\">\\n <mi>f</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n <mo>=</mo>\\n <mo>∑</mo>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mi mathvariant=\\\"normal\\\">sin</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <msqrt>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msqrt>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> can be differentiated term-by-term on <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M7\\\">\\n <mfenced open=\\\"[\\\" close=\\\"]\\\">\\n <mrow>\\n <mi>c</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> for <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M8\\\">\\n <mi>c</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </math>\\n </jats:inline-formula>. However, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M9\\\">\\n <msup>\\n <mrow>\\n <mi>f</mi>\\n </mrow>\\n <mrow>\\n <mo>′</mo>\\n </mrow>\\n </msup>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mn>0</mn>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> may not exist.</jats:p>\",\"PeriodicalId\":7061,\"journal\":{\"name\":\"Abstract and Applied Analysis\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Abstract and Applied Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1155/2021/7035776\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abstract and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2021/7035776","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0

摘要

Chaundy和Jolliffe证明了如果是非负的、不增加的实数序列,则级数∑a n sin nx一致收敛当且仅当n⟶ 0。本文的目的是证明如果n是非递增的并且n⟶ 0,则级数f x=∑an sin n x可以在c上逐项微分,d>0。然而f′0可能不存在。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Note on Derivative of Sine Series with Square Root
Chaundy and Jolliffe proved that if a n is a nonnegative, nonincreasing real sequence, then series a n sin n x converges uniformly if and only if n a n 0 . The purpose of this paper is to show that if n a n is nonincreasing and n a n 0 , then the series f x = a n sin n x can be differentiated term-by-term on c , d for c , d > 0 . However, f 0 may not exist.
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
36
审稿时长
3.5 months
期刊介绍: Abstract and Applied Analysis is a mathematical journal devoted exclusively to the publication of high-quality research papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimization theory, and control theory. Abstract and Applied Analysis supports the publication of original material involving the complete solution of significant problems in the above disciplines. Abstract and Applied Analysis also encourages the publication of timely and thorough survey articles on current trends in the theory and applications of analysis.
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