{"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, 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}
M. Dressler , S. Foucart , M. Joldes , E. de Klerk , J.B. Lasserre , Y. Xu
{"title":"Optimization-aided construction of multivariate Chebyshev polynomials","authors":"M. Dressler , S. Foucart , M. Joldes , E. de Klerk , J.B. Lasserre , 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}
{"title":"Gaussian quadrature formulae are strongly asymptotically optimal for a class of infinitely differentiable functions","authors":"Guiqiao Xu","doi":"10.1016/j.jat.2024.106117","DOIUrl":"10.1016/j.jat.2024.106117","url":null,"abstract":"<div><div>This paper investigates the optimal quadrature formulae of a class <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> of infinitely differentiable functions on <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>. We obtain the strong equivalences of the optimal worst case errors of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> for standard information and Hermite data. We proved that the Gaussian quadrature formulae are strongly asymptotically optimal.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106117"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660806","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":"In search of a higher Bochner theorem","authors":"Emil Horozov , Boris Shapiro , Miloš Tater","doi":"10.1016/j.jat.2024.106114","DOIUrl":"10.1016/j.jat.2024.106114","url":null,"abstract":"<div><div>We initiate the study of a natural generalisation of the classical Bochner–Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106114"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530115","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":"Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials","authors":"Sergey M. Zagorodnyuk","doi":"10.1016/j.jat.2024.106115","DOIUrl":"10.1016/j.jat.2024.106115","url":null,"abstract":"<div><div>In this paper we study the generalized Bessel polynomials <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> (in the notation of Krall and Frink). Let <span><math><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>b</mi><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. In this case we present the following positive continuous weights <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> on the unit circle for <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span>: <span><math><mrow><mn>2</mn><mi>π</mi><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>b</mi><mi>u</mi><mo>cos</mo><mi>θ</mi></mrow></msup><mo>cos</mo><mrow><mo>(</mo><mi>b</mi><mi>u</mi><mo>sin</mo><mi>θ</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>d</mi><mi>u</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>]</mo></mrow></mrow></math></span>. Namely, we have <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mi>d</mi><mi>θ</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>n</mi><mo>,</mo><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>.</mo></mrow></math></span> Notice that this orthogon","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106115"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552898","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":"Barycentric rational interpolation method for solving 3 dimensional convection–diffusion equation","authors":"Jin Li, Yongling Cheng","doi":"10.1016/j.jat.2024.106106","DOIUrl":"10.1016/j.jat.2024.106106","url":null,"abstract":"<div><div>Barycentric rational interpolation collocation method (BRICM) is presented to solve 3-dimensional convection–diffusion (CD) equation. The unknown value is approximated by barycentric rational interpolation basis, the discrete CD equation is written into the matrix equation. At last, the stability and convergence rate of BRIM for CD equation are proven and a numerical example is illustrated in our results.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106106"},"PeriodicalIF":0.9,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424155","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 the representability of a continuous multivariate function by sums of ridge functions","authors":"Rashid A. Aliev , Fidan M. Isgandarli","doi":"10.1016/j.jat.2024.106105","DOIUrl":"10.1016/j.jat.2024.106105","url":null,"abstract":"<div><div>In this paper, new conditions are found for the representability of a continuous multivariate function as a sum of ridge functions. Using these conditions, we give a new proof for the earlier theorem solving the problem, posed by A.Pinkus in his monograph “Ridge Functions”, up to a multivariate polynomial. That is, we show that if a continuous multivariate function has a representation as a sum of arbitrarily behaved ridge functions, then it can be represented as a sum of continuous ridge functions and some multivariate polynomial.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106105"},"PeriodicalIF":0.9,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416752","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 sharp heat kernel estimates in the context of Fourier–Dini expansions","authors":"Bartosz Langowski , Adam Nowak","doi":"10.1016/j.jat.2024.106103","DOIUrl":"10.1016/j.jat.2024.106103","url":null,"abstract":"<div><div>We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106103"},"PeriodicalIF":0.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424156","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}
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