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{"title":"关于平方根正弦级数导数的一个注记","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":"{\"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}
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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.