Journal of Approximation Theory最新文献

{"title":"Best constants in a class of Landau type inequalities","authors":"JinYan Miao, Silvestru Sever Dragomir","doi":"10.1016/j.jat.2026.106285","DOIUrl":"10.1016/j.jat.2026.106285","url":null,"abstract":"<div><div>We first prove the best constant <span><math><mi>C</mi></math></span> in a Landau type inequality <span><span><span><math><mrow><msubsup><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mn>5</mn></mrow></msubsup><mo>≤</mo><mi>C</mi><msubsup><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mn>3</mn></mrow></msubsup><msub><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow></msub><msub><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow></msub><mo>,</mo></mrow></math></span></span></span>and with similar approach, we prove the best constants in a more general form <span><span><span><math><mrow><msubsup><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mn>2</mn><mo>+</mo><mi>η</mi></mrow></msubsup><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>η</mi></mrow></msub><msubsup><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>η</mi></mrow></msubsup><msubsup><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>η</mi></mrow></msubsup><msubsup><mrow><mo>‖</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow><mrow><mi>η</mi></mrow></msubsup></mrow></math></span></span></span>for all <span><math><mrow><mn>0</mn><mo>≤</mo><mi>η</mi><mo>≤</mo><mn>1</mn></mrow></math></span>. Here we have the direct expression for each of the best constants <span><span><span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>η</mi></mrow><mrow><mi>η</mi></mrow></msubsup><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>+</mo><mi>η</mi></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>−</mo><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>/</mo><mn>3</mn><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>η</mi></mrow></msup></mrow></mfrac><mo>,</mo></mrow></math></span></span></span>where <span><span><span><math><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>3</mn><mo>+</mo><mn>3</mn><mi>η</mi><mo>−</mo><msqrt><mrow><mo>−</mo><mn>3</mn><msup><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>6</mn><mi>η</mi><mo>+</mo><mn>9</mn></mrow></msqrt></mrow><mrow><mn>8</mn><mo>+</mo><mn>4</mn><mi>η</mi></mrow></mfrac><mo>.</mo></mrow></math></span></span></span>A Landau type inequality and another special cas","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"317 ","pages":"Article 106285"},"PeriodicalIF":0.6,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039782","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":"Anisotropic approximation on space–time domains","authors":"Pedro Morin ,&nbsp;Cornelia Schneider ,&nbsp;Nick Schneider","doi":"10.1016/j.jat.2025.106282","DOIUrl":"10.1016/j.jat.2025.106282","url":null,"abstract":"<div><div>We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spatial moduli of smoothness and their properties. In particular, we prove Jackson- and Whitney-type inequalities on Lipschitz cylinders, i.e., space–time domains <span><math><mrow><mi>I</mi><mo>×</mo><mi>D</mi></mrow></math></span> with a finite interval <span><math><mi>I</mi></math></span> and a bounded Lipschitz domain <span><math><mrow><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. As an application, we prove a direct estimate result for adaptive space–time finite element approximation in the discontinuous setting.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"317 ","pages":"Article 106282"},"PeriodicalIF":0.6,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145969435","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":"Smoothness and time–frequency analysis in Sobolev–Besicovitch spaces of almost periodic functions","authors":"Juan Miguel Medina ,&nbsp;Hernán D. Centeno ,&nbsp;Raúl F. Florentín ,&nbsp;Mónica Miralles","doi":"10.1016/j.jat.2025.106255","DOIUrl":"10.1016/j.jat.2025.106255","url":null,"abstract":"<div><div>Here, smoothness analysis of almost periodic functions is studied. Analogously to the case of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, the smoothness of the class of Besicovitch almost periodic functions is measured in a classic form by controlling, in some sense, the increments <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and in a dual form by the decay of its Fourier–Bohr transform or by its approximation properties. The same problem is also treated considering the time–frequency representation given by the Gabor transform. Some results are given as equivalence of norms between appropriate function spaces.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106255"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600527","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":"Edge detection with polynomial frames on the sphere","authors":"Frederic Schoppert","doi":"10.1016/j.jat.2025.106279","DOIUrl":"10.1016/j.jat.2025.106279","url":null,"abstract":"<div><div>In a recent article, we have shown that a variety of localized polynomial frames, including isotropic as well as directional spherical systems, are suitable for detecting jump discontinuities that lie along circles on the sphere. More precisely, such edges can be identified in terms of their position and orientation by the asymptotic decay of the frame coefficients in an arbitrary small neighborhood. In this paper, we will extend these results to discontinuities which lie along general smooth curves. In particular, we prove upper and lower estimates for the frame coefficients when the analysis function is concentrated in the vicinity of such a singularity. The estimates are given in an asymptotic sense, with respect to some dilation parameter, and they hold uniformly in a neighborhood of the smooth curve segment under consideration.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106279"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883524","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 image of the limit q-Durrmeyer operator","authors":"Sofiya Ostrovska,&nbsp;Mehmet Turan","doi":"10.1016/j.jat.2025.106280","DOIUrl":"10.1016/j.jat.2025.106280","url":null,"abstract":"<div><div>The focus of this work is on the properties of the <span><math><mi>q</mi></math></span>-Durrmeyer operators <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>,</mo></mrow></math></span>\u0000 <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi><mo>,</mo></mrow></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub></math></span> introduced, for <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo></mrow></math></span> by V. Gupta and H. Wang. First, it is shown that, for each <span><math><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo></mrow></math></span> the sequence <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>q</mi></mrow></msub><mi>f</mi><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converges to <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub><mi>f</mi></mrow></math></span> uniformly on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> with a rate not slower than <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>f</mi></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, which refines the previously available result by V. Gupta and H. Wang, and implies the possibility of an analytic continuation for <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub><mi>f</mi></mrow></math></span> into a neighbourhood of <span><math><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>.</mo></mrow></math></span> Further investigation shows that <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub><mi>f</mi></mrow></math></span> admits an analytic continuation as an entire function regardless of <span><math><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>. Finally, the growth estimates for these functions are received and applied to describe the point spectrum of <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>.</mo></mrow></math></span> The paper also addresses the significant differences between the properties of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub></math></span> and the previously well-known limit <span><math><mi>q</mi></math></span>-Bernstein operator <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106280"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884113","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 on the Haraux function","authors":"Patrick L. Combettes,&nbsp;Julien N. Mayrand","doi":"10.1016/j.jat.2026.106284","DOIUrl":"10.1016/j.jat.2026.106284","url":null,"abstract":"<div><div>The Haraux function is an important tool in monotone operator theory and its applications. One of its salient properties for a maximally monotone operator is to be valued in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mi>∞</mi><mo>]</mo></mrow></math></span> and to vanish only on the graph of the operator. Sharper lower bounds for this function have been proposed in specific cases. We derive lower bounds in the general context of set-valued operators in reflexive real Banach spaces. These bounds are new, even for maximally monotone operators acting on Euclidean spaces, a scenario in which we show that they can be better than existing ones. As a by-product, we obtain lower bounds on the Fenchel–Young function in variational analysis. Several examples are given and applications to composite monotone inclusions are discussed.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106284"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077058","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 electrostatic model for the roots of discrete classical orthogonal polynomials","authors":"Joaquín Sánchez-Lara","doi":"10.1016/j.jat.2025.106256","DOIUrl":"10.1016/j.jat.2025.106256","url":null,"abstract":"<div><div>An electrostatic model is presented to describe the behaviour of the roots of classical discrete orthogonal polynomials. Indeed, this model applies more generally to the roots of polynomial solutions of second-order linear difference equations <span><math><mrow><mi>A</mi><msub><mrow><mi>Δ</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mo>∇</mo></mrow><mrow><mi>h</mi></mrow></msub><mi>y</mi><mo>+</mo><mi>B</mi><msub><mrow><mi>Δ</mi></mrow><mrow><mi>h</mi></mrow></msub><mi>y</mi><mo>+</mo><mi>C</mi><mi>y</mi><mo>=</mo><mn>0</mn><mspace></mspace></mrow></math></span> where <span><math><mi>A</mi></math></span>, <span><math><mi>B</mi></math></span> and <span><math><mi>C</mi></math></span> are polynomials and <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>h</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><mo>∇</mo></mrow><mrow><mi>h</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>h</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. The existence of a unique distribution of points which minimizes the energy of the system is guaranteed under some assumptions on <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span>. Furthermore, interlacing properties and the monotonicity of the roots depending on the parameters which appear in the difference equation are obtained from this electrostatic model.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106256"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694484","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":"Weighted estimates for dyadic maximal operators","authors":"Ferenc Weisz ,&nbsp;Guangheng Xie ,&nbsp;Dachun Yang","doi":"10.1016/j.jat.2025.106278","DOIUrl":"10.1016/j.jat.2025.106278","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span> be the dyadic maximal operator. In this article, the authors find some sufficient conditions on the indices <span><math><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></math></span> to guarantee the boundedness of the operators <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span> on weighted Lebesgue spaces. These conditions in special weight case turn out to be necessary and hence are sharp in some sense. Moreover, the authors show that a weight <span><math><mi>w</mi></math></span> belongs to the Muckenhoupt weight class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, if and only if the operator <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span> is bounded on weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spaces. Furthermore, the authors establish the weighted vector-valued dyadic maximal inequality and the weighted weak-type <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> estimate for the operators <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span>. As applications, the authors obtain the convergence of Fejér means of Walsh–Fourier series of martingales in both pointwise and Musielak–Orlicz spaces. These main results in Musielak–Orlicz space case remedy a missing necessary assumption of Theorems 3.2 and 3.4 in Weisz et al. (2023).</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106278"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839928","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 of finite orders and multivariate polynomial interpolation","authors":"Shelby Kilmer,&nbsp;Xingping Sun,&nbsp;Matthew Wright","doi":"10.1016/j.jat.2025.106283","DOIUrl":"10.1016/j.jat.2025.106283","url":null,"abstract":"<div><div>We study strictly positive definite functions of finite orders on real inner product spaces. Via exploring new connections to theories of Chebyshev ranks and Schur polynomials, we obtain quantifiable conditions for such functions. To assist field scientists in selecting an ideal multivariate polynomial interpolant, we propose and study the notions of “interpolating rank” and “minimal seminorm interpolation.” We quantify both Chebyshev ranks and interpolating ranks in terms of the number of common zeros of a certain finite collection of Schur polynomials. As a byproduct, we develop a mechanism to construct positive-rank Chebyshev systems of which examples are scarce in the literature.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"315 ","pages":"Article 106283"},"PeriodicalIF":0.6,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925083","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":"Runge-type approximation theorem for Banach-valued H∞ functions on a polydisk","authors":"Alexander Brudnyi","doi":"10.1016/j.jat.2025.106221","DOIUrl":"10.1016/j.jat.2025.106221","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denote the open unit polydisk, and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be a Cartesian product of planar compacta. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̂&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be an open neighborhood of the closure &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the maximal ideal space of the algebra &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; of bounded holomorphic functions on &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Given a complex Banach space &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, denote by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; the Banach space of bounded &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-valued holomorphic functions on an open set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We prove that any &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̂&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, can be uniformly approximated on &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; by functions of the form &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a finite product of interpolating Blaschke products satisfying &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;inf&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, if &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is contained in a compact holomorphically convex subset of &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̂&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;, then such approximations can be achieved without denominators: that is, &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; can be approximated uniformly on &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106221"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683674","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}