A Note on Derivative of Sine Series with Square Root

Q3 Mathematics

Abstract and Applied Analysis Pub Date : 2021-11-08 DOI:10.1155/2021/7035776

Sergiusz Kęska

{"title":"A Note on Derivative of Sine Series with Square Root","authors":"Sergiusz Kęska","doi":"10.1155/2021/7035776","DOIUrl":null,"url":null,"abstract":"Chaundy and Jolliffe proved that if \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 is a nonnegative, nonincreasing real sequence, then series \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 converges uniformly if and only if \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 . The purpose of this paper is to show that if \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 is nonincreasing and \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 , then the series \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 can be differentiated term-by-term on \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 for \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 . However, \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 may not exist.","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}

引用次数: 0

Abstract

Chaundy and Jolliffe proved that if

\{a_{n}\}

is a nonnegative, nonincreasing real sequence, then series

\sum 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) = \sum a_{n} \sin (\sqrt{n} x)

can be differentiated term-by-term on

[c, d]

for

c, d > 0

. However,

f^{'} (0)

may not exist.

查看原文本刊更多论文

关于平方根正弦级数导数的一个注记

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。

求助全文

约1分钟内获得全文求助全文

来源期刊

Abstract and Applied Analysis 数学-数学

CiteScore

2.30

自引率

0.00%

发文量

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.