Journal of Approximation Theory最新文献

{"title":"Improved Stein inequalities for the Fourier transform","authors":"Erlan D. Nursultanov ,&nbsp;Durvudkhan Suragan","doi":"10.1016/j.jat.2024.106126","DOIUrl":"10.1016/j.jat.2024.106126","url":null,"abstract":"<div><div>In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>2</mn></mrow></math></span> to <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo>&lt;</mo><mi>∞</mi></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106126"},"PeriodicalIF":0.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722538","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":"Spherical basis functions in Hardy spaces with localization constraints","authors":"C. Gerhards,&nbsp;X. Huang","doi":"10.1016/j.jat.2024.106124","DOIUrl":"10.1016/j.jat.2024.106124","url":null,"abstract":"<div><div>Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mo>−</mo></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span>, respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106124"},"PeriodicalIF":0.9,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722537","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":"Yosida approximations of the cumulative distribution function and applications in survival analysis","authors":"Miroslav Bačák","doi":"10.1016/j.jat.2024.106123","DOIUrl":"10.1016/j.jat.2024.106123","url":null,"abstract":"<div><div>The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.</div><div>When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.</div><div>After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106123"},"PeriodicalIF":0.9,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706208","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":"Classification of 2-orthogonal polynomials with Brenke type generating functions","authors":"Hamza Chaggara ,&nbsp;Abdelhamid Gahami","doi":"10.1016/j.jat.2024.106125","DOIUrl":"10.1016/j.jat.2024.106125","url":null,"abstract":"<div><div>The Brenke type generating functions are the polynomial generating functions of the form <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><msup><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mi>A</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are two formal power series subject to the conditions <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mi>⋅</mi><msup><mrow><mi>B</mi></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>≠</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></mrow></math></span>. In this work, we shall determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sequences, that is to say, polynomials with Brenke type generating function and satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of <span><math><mi>d</mi></math></span>-orthogonality.</div><div>The classification is based on the discussion of a three-order difference equation induced by the four-term recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara (1968) who gave all pairs <span><math><mrow><mo>(</mo><mi>A</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mi>B</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></math></span> for which <span><math><msub><mrow><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></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is an orthogonal polynomial sequence. In some cases, we give the expression of the moments associated to the two-dimensional functional of orthogonality.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106125"},"PeriodicalIF":0.9,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722535","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":"Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)","authors":"Tamás Erdélyi","doi":"10.1016/j.jat.2024.106122","DOIUrl":"10.1016/j.jat.2024.106122","url":null,"abstract":"<div><div>We prove that there is an absolute constant <span><math><mrow><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> such that for every <span><span><span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo>]</mo></mrow><mspace></mspace><mo>,</mo><mspace></mspace><mn>1</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>exp</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>9</mn></mrow></mfrac></mrow></mfenced><mo>,</mo></mrow></math></span></span></span>there are <span><span><span><math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span></span></span>such that the polynomial <span><math><mi>P</mi></math></span> of the form <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>b</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></math></span> has at least <span><math><mrow><mi>c</mi><msup><mrow><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mo>log</mo><mrow><mo>(</mo><mn>4</mn><mi>M</mi><mo>)</mo></mrow></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> distinct sign changes in <span><math><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>a</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>a</mi><mo>:</mo><mo>=</mo><msup><mrow><mfenced><mrow><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mn>4</mn><mi>M</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>≤</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span>. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106122"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706248","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":"Random sampling and polynomial-free interpolation by Generalized MultiQuadrics","authors":"A. Sommariva,&nbsp;M. Vianello","doi":"10.1016/j.jat.2024.106119","DOIUrl":"10.1016/j.jat.2024.106119","url":null,"abstract":"<div><div>We prove that interpolation matrices for Generalized MultiQuadrics (GMQ) of order greater than one are almost surely nonsingular without polynomial addition, in any dimension and with any continuous random distribution of sampling points. We also include a new class of generalized MultiQuadrics recently proposed by Buhmann and Ortmann.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106119"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705738","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":"Strictly positive definite functions on spheres","authors":"Tianshi Lu","doi":"10.1016/j.jat.2024.106120","DOIUrl":"10.1016/j.jat.2024.106120","url":null,"abstract":"<div><div>In this paper, we proved that a positive definite radial function on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with support in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></mrow></math></span> is strictly positive definite on the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and real projective space <span><math><msup><mrow><mi>RP</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for odd <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. We also proved that the truncated power function <span><math><msubsup><mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mo>+</mo></mrow><mrow><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msubsup></math></span> is strictly positive definite on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>RP</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>t</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106120"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662967","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":"Sampling theorems with derivatives in shift-invariant spaces generated by periodic exponential B-splines","authors":"Karlheinz Gröchenig,&nbsp;Irina Shafkulovska","doi":"10.1016/j.jat.2024.106118","DOIUrl":"10.1016/j.jat.2024.106118","url":null,"abstract":"<div><div>We derive sufficient conditions for sampling with derivatives in shift-invariant spaces generated by a periodic exponential B-spline. The sufficient conditions are expressed with a new notion of measuring the gap between consecutive samples. These conditions are near optimal, and, in particular, they imply the existence of sampling sets with lower Beurling density arbitrarily close to the necessary density.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106118"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722536","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":"Generalized Bell polynomials","authors":"Antonio J. Durán","doi":"10.1016/j.jat.2024.106121","DOIUrl":"10.1016/j.jat.2024.106121","url":null,"abstract":"<div><div>In this paper, generalized Bell polynomials <span><math><msub><mrow><mrow><mo>(</mo><msubsup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></math></span> associated to a sequence of real numbers <span><math><mrow><mi>ϕ</mi><mo>=</mo><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></mrow></math></span> are introduced. Bell polynomials correspond to <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>i</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. We prove that when <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>i</mi><mo>≥</mo><mn>1</mn></mrow></math></span>: (a) the zeros of the generalized Bell polynomial <span><math><msubsup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math></span> are simple, real and non positive; (b) the zeros of <span><math><msubsup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math></span> interlace the zeros of <span><math><msubsup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math></span>; (c) the zeros are decreasing functions of the parameters <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106121"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706209","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":"Optimization-aided construction of multivariate Chebyshev polynomials","authors":"M. Dressler ,&nbsp;S. Foucart ,&nbsp;M. Joldes ,&nbsp;E. de Klerk ,&nbsp;J.B. Lasserre ,&nbsp;Y. Xu","doi":"10.1016/j.jat.2024.106116","DOIUrl":"10.1016/j.jat.2024.106116","url":null,"abstract":"<div><div>This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial <span><math><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> on the euclidean ball and for the monomial <span><math><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> on the simplex.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106116"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552896","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}