{"title":"周期函数的 Comonotone 近似值","authors":"D. Leviatan , M.V. Shchehlov , I.O. Shevchuk","doi":"10.1016/j.jat.2024.106015","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> be the space of continuous <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic functions <span><math><mi>f</mi></math></span>, endowed with the uniform norm <span><math><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></msub><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, and denote by <span><math><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, the <span><math><mi>m</mi></math></span>th modulus of smoothness of <span><math><mi>f</mi></math></span>. Denote by <span><math><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></math></span>, the subspace of <span><math><mi>r</mi></math></span><span> times continuously differentiable functions </span><span><math><mrow><mi>f</mi><mo>∈</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow></math></span>, and let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span>, be the set of trigonometric polynomials </span><span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree <span><math><mrow><mo><</mo><mi>n</mi></mrow></math></span>. If <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, has <span><math><mrow><mn>2</mn><mi>s</mi></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span>, extremal points in </span><span><math><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></math></span>, denote by <span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>inf</mo></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>≥</mo><mn>0</mn></mrow></munder><mo>‖</mo><mi>f</mi><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo><mo>,</mo></mrow></math></span> the error of its best comonotone approximation. We prove, that if <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, then for either <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math></span>, or <span><math><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn><mi>s</mi></mrow></math></span>, or <span><math><mrow><mi>m</mi><mo>∈</mo><mi>N</mi></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>></mo><mn>2</mn><mi>s</mi></mrow></math></span>, <span><span><span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>c</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msup><mo>,</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></math></span></span></span>where the constant <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow></math></span> depends only on <span><math><mi>m</mi></math></span>, <span><math><mi>r</mi></math></span> and <span><math><mi>s</mi></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Comonotone approximation of periodic functions\",\"authors\":\"D. Leviatan , M.V. Shchehlov , I.O. Shevchuk\",\"doi\":\"10.1016/j.jat.2024.106015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> be the space of continuous <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic functions <span><math><mi>f</mi></math></span>, endowed with the uniform norm <span><math><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></msub><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, and denote by <span><math><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, the <span><math><mi>m</mi></math></span>th modulus of smoothness of <span><math><mi>f</mi></math></span>. Denote by <span><math><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></math></span>, the subspace of <span><math><mi>r</mi></math></span><span> times continuously differentiable functions </span><span><math><mrow><mi>f</mi><mo>∈</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow></math></span>, and let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span>, be the set of trigonometric polynomials </span><span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree <span><math><mrow><mo><</mo><mi>n</mi></mrow></math></span>. If <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, has <span><math><mrow><mn>2</mn><mi>s</mi></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span>, extremal points in </span><span><math><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></math></span>, denote by <span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>inf</mo></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>≥</mo><mn>0</mn></mrow></munder><mo>‖</mo><mi>f</mi><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo><mo>,</mo></mrow></math></span> the error of its best comonotone approximation. We prove, that if <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, then for either <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math></span>, or <span><math><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn><mi>s</mi></mrow></math></span>, or <span><math><mrow><mi>m</mi><mo>∈</mo><mi>N</mi></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>></mo><mn>2</mn><mi>s</mi></mrow></math></span>, <span><span><span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>c</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msup><mo>,</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></math></span></span></span>where the constant <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow></math></span> depends only on <span><math><mi>m</mi></math></span>, <span><math><mi>r</mi></math></span> and <span><math><mi>s</mi></math></span>.</p></div>\",\"PeriodicalId\":54878,\"journal\":{\"name\":\"Journal of Approximation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Approximation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021904524000017\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Approximation Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021904524000017","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let be the space of continuous -periodic functions , endowed with the uniform norm , and denote by , the th modulus of smoothness of . Denote by , the subspace of times continuously differentiable functions , and let , be the set of trigonometric polynomials of degree . If , has , , extremal points in , denote by the error of its best comonotone approximation. We prove, that if , then for either , or and , or and , where the constant depends only on , and .
期刊介绍:
The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others:
• Classical approximation
• Abstract approximation
• Constructive approximation
• Degree of approximation
• Fourier expansions
• Interpolation of operators
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• Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth)
• Gabor (Weyl-Heisenberg) expansions and sampling theory.