Journal of Approximation Theory最新文献

{"title":"Orthogonal polynomials in weighted Bergman spaces","authors":"Erwin Miña-Díaz","doi":"10.1016/j.jat.2023.105972","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105972","url":null,"abstract":"<div><p>Let <span><math><mi>w</mi></math></span> be a weight on the unit disk <span><math><mi>D</mi></math></span> having the form <span><span><span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mo>|</mo><mi>v</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><munderover><mrow><mo>∏</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></munderover><msup><mrow><mfenced><mrow><mfrac><mrow><mi>z</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mn>1</mn><mo>−</mo><mi>z</mi><msub><mrow><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mspace></mspace><mo>,</mo><mspace></mspace><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>&gt;</mo><mo>−</mo><mn>2</mn><mo>,</mo><mspace></mspace><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mo>&lt;</mo><mn>1</mn><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>v</mi></math></span> is analytic and free of zeros in <span><math><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></math></span>, and let <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> be the sequence of polynomials (<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>) orthonormal over <span><math><mi>D</mi></math></span> with respect to <span><math><mi>w</mi></math></span>. We give an integral representation for <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span> from which it is in principle possible to derive its asymptotic behavior as </span><span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span> at every point <span><math><mi>z</mi></math></span> of the complex plane, the asymptotic analysis of the integral being primarily dependent on the nature of the first singularities encountered by the function <span><math><mrow><mi>v</mi><msup><mrow><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>z</mi><msub><mrow><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"296 ","pages":"Article 105972"},"PeriodicalIF":0.9,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191859","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":"Analytical study of the pantograph equation using Jacobi theta functions","authors":"Changgui Zhang","doi":"10.1016/j.jat.2023.105974","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105974","url":null,"abstract":"<div><p>The aim of this paper is to use the analytic theory of linear <span><math><mi>q</mi></math></span>-difference equations for the study of the functional-differential equation <span><math><mrow><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mi>y</mi><mrow><mo>(</mo><mi>q</mi><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mi>y</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span> are two non-zero real or complex numbers. When <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>y</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span><span>, the associated Cauchy problem admits a unique power series solution, </span><span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><mfrac><mrow><msub><mrow><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>/</mo><mi>b</mi><mo>;</mo><mi>q</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mspace></mspace><msup><mrow><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, that converges in the whole complex <span><math><mi>x</mi></math></span><span>-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination<span><span> of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the </span>asymptotic behavior of the solutions over the real axis.</span></span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"296 ","pages":"Article 105974"},"PeriodicalIF":0.9,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191860","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":"Counterexamples in isometric theory of symmetric and greedy bases","authors":"Fernando Albiac ,&nbsp;José L. Ansorena ,&nbsp;Óscar Blasco ,&nbsp;Hùng Việt Chu ,&nbsp;Timur Oikhberg","doi":"10.1016/j.jat.2023.105970","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105970","url":null,"abstract":"<div><p>We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"297 ","pages":"Article 105970"},"PeriodicalIF":0.9,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50180339","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":"The upper bound for the Lebesgue constant for Lagrange interpolation in equally spaced points of the triangle","authors":"Natalia Baidakova","doi":"10.1016/j.jat.2023.105969","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105969","url":null,"abstract":"<div><p><span>An upper bound for the Lebesgue constant, i.e., the supremum norm of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to </span><span><math><mi>n</mi></math></span> is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to <span><math><mi>n</mi></math></span> for an arbitrary <span><math><mi>d</mi></math></span>-dimensional simplex was established by the author. The upper bound proved in this article refines this result for <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"296 ","pages":"Article 105969"},"PeriodicalIF":0.9,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191861","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":"Quarklet characterizations for Triebel–Lizorkin spaces","authors":"Marc Hovemann ,&nbsp;Stephan Dahlke","doi":"10.1016/j.jat.2023.105968","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105968","url":null,"abstract":"<div><p>In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span><span> can be characterized in terms of quarklets. So for functions from Triebel–Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed by means of biorthogonal compactly supported Cohen–Daubechies–Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel–Lizorkin–Morrey spaces </span><span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"295 ","pages":"Article 105968"},"PeriodicalIF":0.9,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50183871","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 sections of ℓp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators","authors":"Aicke Hinrichs ,&nbsp;Joscha Prochno ,&nbsp;Mathias Sonnleitner","doi":"10.1016/j.jat.2023.105919","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105919","url":null,"abstract":"<div><p>We study the circumradius of a random section of an <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-ellipsoid, <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math></span> and compressed sensing if <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math></span>. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value <span><math><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></math></span> or not. If <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></math></span>, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021] for the case <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"293 ","pages":"Article 105919"},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191929","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":"Telescoping continued fractions for the error term in Stirling’s formula","authors":"Gaurav Bhatnagar ,&nbsp;Krishnan Rajkumar","doi":"10.1016/j.jat.2023.105943","DOIUrl":"10.1016/j.jat.2023.105943","url":null,"abstract":"<div><p>In this paper, we introduce telescoping continued fractions to find lower bounds for the error term <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in Stirling’s approximation <span><math><mrow><mi>n</mi><mo>!</mo><mo>=</mo><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msup><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo>.</mo></mrow></math></span><span> This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.</span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"293 ","pages":"Article 105943"},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42705947","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 discretization error of integral norms for function classes with small smoothness","authors":"V.N. Temlyakov","doi":"10.1016/j.jat.2023.105913","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105913","url":null,"abstract":"<div><p><span>We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of </span>multivariate functions with mixed smoothness.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"293 ","pages":"Article 105913"},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191924","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":"Sharp estimates for Jacobi heat kernels in conic domains","authors":"Dawid Hanrahan ,&nbsp;Dariusz Kosz","doi":"10.1016/j.jat.2023.105921","DOIUrl":"10.1016/j.jat.2023.105921","url":null,"abstract":"<div><p>We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone <span><math><msup><mrow><mi>V</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> and its surface <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span><span>. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.</span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"293 ","pages":"Article 105921"},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42715263","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":"Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials","authors":"Diego Dominici ,&nbsp;Juan José Moreno-Balcázar","doi":"10.1016/j.jat.2023.105918","DOIUrl":"10.1016/j.jat.2023.105918","url":null,"abstract":"<div><p><span><span>In this paper we tackle the asymptotic behavior of a family of </span>orthogonal polynomials with respect to a nonstandard inner product involving the forward operator </span><span><math><mi>Δ</mi></math></span>. Concretely, we treat the generalized Charlier weights in the framework of <span><math><mi>Δ</mi></math></span><span><span>-Sobolev orthogonality. We obtain an </span>asymptotic expansion<span> for these orthogonal polynomials where the falling factorial polynomials play an important role.</span></span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"293 ","pages":"Article 105918"},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46496151","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}