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":null,"pages":null},"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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"Translation-based completeness on compact intervals","authors":"Lukas Liehr","doi":"10.1016/j.jat.2024.106104","DOIUrl":"10.1016/j.jat.2024.106104","url":null,"abstract":"<div><div>Given a compact interval <span><math><mrow><mi>I</mi><mo>⊆</mo><mi>R</mi></mrow></math></span>, and a function <span><math><mi>f</mi></math></span> that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates <span><math><mrow><mo>{</mo><mi>f</mi><mrow><mo>(</mo><mi>⋅</mi><mo>−</mo><mi>λ</mi><mo>)</mo></mrow><mo>:</mo><mi>λ</mi><mo>∈</mo><mi>Λ</mi><mo>}</mo></mrow></math></span> are complete in <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math></span> if and only if the series of reciprocals of <span><math><mi>Λ</mi></math></span> diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299–308]. An additional characterization is obtained when <span><math><mi>Λ</mi></math></span> is an arithmetic progression, and the generator <span><math><mi>f</mi></math></span> constitutes a linear combination of translates of a function with sufficiently fast decay.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421197","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":"Monotonicity of zeros of derivatives of Bessel functions","authors":"Dimitar K. Dimitrov, Yen Chi Lun","doi":"10.1016/j.jat.2024.106102","DOIUrl":"10.1016/j.jat.2024.106102","url":null,"abstract":"<div><div>Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the <span><math><mi>n</mi></math></span>th derivative of the Bessel function of the first kind <span><math><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi>ν</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are all real when <span><math><mrow><mi>ν</mi><mo>></mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, the positive zeros of <span><math><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are increasing functions of the parameter <span><math><mi>ν</mi></math></span>, for <span><math><mrow><mi>ν</mi><mo>∈</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>. We provide two apparently distinct proofs of the conjecture.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432541","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 Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains","authors":"Feng Dai , András Kroó , Andriy Prymak","doi":"10.1016/j.jat.2024.106101","DOIUrl":"10.1016/j.jat.2024.106101","url":null,"abstract":"<div><p>We prove new Bernstein and Markov type inequalities in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spaces associated with the normal and the tangential derivatives on the boundary of a general compact <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-domain with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>2</mn></mrow></math></span>. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> norms of algebraic polynomials on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-domains with asymptotically optimal number of function samples used.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270714","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":"Lower bounds for piecewise polynomial approximations of oscillatory functions","authors":"Jeffrey Galkowski","doi":"10.1016/j.jat.2024.106100","DOIUrl":"10.1016/j.jat.2024.106100","url":null,"abstract":"<div><p>We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000881/pdfft?md5=fb33e23c82eb14bbcd3a20a9e7b11759&pid=1-s2.0-S0021904524000881-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270713","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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