Journal of Approximation Theory最新文献

{"title":"Localization for random CMV matrices","authors":"Xiaowen Zhu","doi":"10.1016/j.jat.2023.106008","DOIUrl":"10.1016/j.jat.2023.106008","url":null,"abstract":"<div><p>We prove Anderson localization (AL) and dynamical localization in expectation (EDL, also known as strong dynamical localization) for random CMV matrices for arbitrary distribution of i.i.d. Verblunsky coefficients.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106008"},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Comonotone approximation of periodic functions","authors":"D. Leviatan ,&nbsp;M.V. Shchehlov ,&nbsp;I.O. Shevchuk","doi":"10.1016/j.jat.2024.106015","DOIUrl":"10.1016/j.jat.2024.106015","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>&lt;</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<","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"299 ","pages":"Article 106015"},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial approximation on disjoint segments and amplification of approximation","authors":"Yu. Malykhin ,&nbsp;K. Ryutin","doi":"10.1016/j.jat.2023.106010","DOIUrl":"10.1016/j.jat.2023.106010","url":null,"abstract":"<div><p><span>We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments (see </span><span>(1)</span><span><span>). This problem has important applications in several areas of numerical analysis, complexity theory, </span>quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations of higher degree </span><span><math><mi>M</mi></math></span><span> and better accuracy from the approximations of degree </span><span><math><mi>m</mi></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106010"},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints","authors":"German Dzyubenko ,&nbsp;Kirill A. Kopotun","doi":"10.1016/j.jat.2023.106012","DOIUrl":"10.1016/j.jat.2023.106012","url":null,"abstract":"<div><p>Given <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, a nonnegative function <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span>, <span><math><mrow><mi>r</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, an arbitrary finite collection of points <span><math><mrow><msub><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub><mo>⊂</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span><span>, and a corresponding collection of nonnegative integers </span><span><math><msub><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub></math></span> with <span><math><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>r</mi></mrow></math></span>, <span><math><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></math></span>, is it true that, for sufficiently large <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there exists a polynomial <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi></math></span> such that</p><p>(i) <span><math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><mi>c</mi><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</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><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>;</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≔</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the classical <span><math><mi>k</mi></math></span>th modulus of smoothness.</p><p>(ii) <span><math><mrow><msup><mrow><mi>P</mi></mrow><mrow><mrow","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"299 ","pages":"Article 106012"},"PeriodicalIF":0.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?","authors":"Congpei An ,&nbsp;Hao-Ning Wu","doi":"10.1016/j.jat.2023.106013","DOIUrl":"10.1016/j.jat.2023.106013","url":null,"abstract":"<div><p><span>Singular and oscillatory functions play a crucial role in various applications, and their approximation is crucial for solving applied mathematics problems efficiently. Hyperinterpolation is a discrete projection method approximating functions with the </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span> orthogonal projection coefficients obtained by numerical integration. However, this approach may be inefficient for approximating singular and oscillatory functions, requiring a large number of integration points to achieve satisfactory accuracy. To address this issue, we propose a new approximation scheme in this paper, called efficient hyperinterpolation, which leverages the product-integration methods to attain the desired accuracy with fewer numerical integration points than the original scheme. We provide theorems that explain the superiority of efficient hyperinterpolation over the original scheme in approximating such functions belonging to </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and continuous function spaces, respectively, and demonstrate through numerical experiments on the interval and the sphere that our approach outperforms the original method in terms of accuracy when using a limited number of integration points.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"299 ","pages":"Article 106013"},"PeriodicalIF":0.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139092666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On some identities for confluent hypergeometric functions and Bessel functions","authors":"Yoshitaka Okuyama","doi":"10.1016/j.jat.2023.106014","DOIUrl":"10.1016/j.jat.2023.106014","url":null,"abstract":"<div><p>Mathematical functions<span>, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer’s confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results.</span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106014"},"PeriodicalIF":0.9,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Onesided Korovkin approximation","authors":"Michele Campiti","doi":"10.1016/j.jat.2023.106011","DOIUrl":"10.1016/j.jat.2023.106011","url":null,"abstract":"<div><p>In this paper we study in detail some characterizations of Korovkin closures and we also introduce the notions of onesided upper and lower Korovkin closures. We provide some complete characterizations of these new closures which separate the roles of approximating functions in a Korovkin system. We also present some new characterizations of the classical Korovkin closure in spaces of integrable functions. Again we can introduce and characterize the upper and lower Korovkin closures. Finally, we provide some examples which justify the interest in these new closures.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106011"},"PeriodicalIF":0.9,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904523001491/pdfft?md5=c52ebe4199164b358a170c0ce8a2ccd0&pid=1-s2.0-S0021904523001491-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Protein Carbonyl, Lipid Peroxidation, Glutathione and Enzymatic Antioxidant Status in Male Wistar Brain Sub-regions After Dietary Copper Deficiency.","authors":"Ankita Rajendra Kurup, Neena Nair","doi":"10.1007/s12291-022-01093-1","DOIUrl":"10.1007/s12291-022-01093-1","url":null,"abstract":"<p><p>Copper a quintessential transitional metal is required for development and function of normal brain and its deficiency has been associated with impairments in brain function. The present study investigates the effects of dietary copper deficiency on brain sub-regions of male Wistar rats for 2-, 4- and 6-week. Pre-pubertal rats were divided into four groups: negative control (NC), copper control (CC), pairfed (PF) and copper deficient (CD). In brain sub regions total protein concentration, glutathione concentration and Cu-Zn SOD activity were down regulated after 2-, 4- and 6 weeks compared to controls and PF groups. Significant increase in brain sub regions was observed in protein carbonyl and lipid peroxidation concentration as well as total SOD, Mn SOD and catalase activities after 2-, 4- and 6 weeks of dietary copper deficiency. Experimental evidences indicate that impaired copper homeostasis has the potential to generate reactive oxygen species enhancing the susceptibility to oxidative stress by inducing up- and down-regulation of non-enzymatic and enzymatic profile studied in brain sub regions causing loss of their normal function which can consequently lead to deterioration of cell structure and death if copper deficiency is prolonged.</p>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"2 1","pages":"73-82"},"PeriodicalIF":0.9,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10784247/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74702717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Chebyshev unions of planes, and their approximative and geometric properties","authors":"A.R. Alimov ,&nbsp;I.G. Tsar’kov","doi":"10.1016/j.jat.2023.106009","DOIUrl":"10.1016/j.jat.2023.106009","url":null,"abstract":"<div><p><span>We study approximative and geometric properties of Chebyshev sets composed of at most countably many planes (i.e., closed affine subspaces). We will assume that the union of planes is irreducible, i.e., no plane in this union contains another plane from the union. We show, in particular, that if a Chebyshev subset </span><span><math><mi>M</mi></math></span><span> of a Banach space </span><span><math><mi>X</mi></math></span> consists of at least two planes, then it is not <span><math><mi>B</mi></math></span>-connected (i.e., its intersection with some closed ball is disconnected) and is not <span><math><mover><mrow><mi>B</mi></mrow><mrow><mo>̊</mo></mrow></mover></math></span>-complete. We also verify that, in reflexive <span><math><mrow><mo>(</mo><mi>CLUR</mi><mo>)</mo></mrow></math></span>-spaces (and, in particularly, in complete uniformly convex spaces), a set composed of countably many planes is not a Chebyshev set. For finite unions, we show that any finite union of planes (involving at least two planes) is not a Chebyshev set for any norm on the space. Several applications of our results in the spaces <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> are also given.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106009"},"PeriodicalIF":0.9,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An asymptotic development of the Poisson integral for Laguerre polynomial expansions","authors":"Ulrich Abel","doi":"10.1016/j.jat.2023.106007","DOIUrl":"https://doi.org/10.1016/j.jat.2023.106007","url":null,"abstract":"<div><p><span>The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span><span> spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions </span><span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mfenced><mrow><mn>0</mn><mo>,</mo><mo>+</mo><mi>∞</mi></mrow></mfenced></mrow></math></span> with <span><math><mrow><mn>4</mn><mo>/</mo><mn>3</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>4</mn></mrow></math></span>. In this paper we deal with the Poisson integral <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>f</mi></mrow></math></span>\u0000<span><math><mfenced><mrow><mn>0</mn><mo>&lt;</mo><mi>r</mi><mo>&lt;</mo><mn>1</mn></mrow></mfenced></math></span><span> which arises by applying Abel’s summation method to the Laguerre expansion of the function </span><span><math><mi>f</mi></math></span><span>. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence<span>, the convergence by norm, and that the Poisson integral is a contraction mapping in </span></span><span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mfenced><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow></mfenced></mrow></math></span>. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit <span><math><mrow><msup><mrow><mfenced><mrow><mn>1</mn><mo>−</mo><mi>r</mi></mrow></mfenced></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced><mrow><mfenced><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>f</mi></mrow></mfenced><mfenced><mrow><mi>x</mi></mrow></mfenced><mo>−</mo><mi>f</mi><mfenced><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow></math></span> as <span><math><mrow><mi>r</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow></math></span>, provided that <span><math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mfenced><mrow><mi>x</mi></mrow></mfenced></mrow></math></span> exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of <span><math><mrow><mfenced><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>f</mi></mrow></mfenced><mfenced><mrow><mi>x</mi></mrow></mfenced></mrow></math></span> as <span><math><mrow><mi>r</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"298 ","pages":"Article 106007"},"PeriodicalIF":0.9,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138501441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}