Journal of Approximation Theory最新文献

{"title":"On simultaneous density order from shift invariant subspaces in Sobolev spaces","authors":"Ch. Boukeffous ,&nbsp;A. San Antolín","doi":"10.1016/j.jat.2025.106147","DOIUrl":"10.1016/j.jat.2025.106147","url":null,"abstract":"<div><div>The notion of simultaneous approximation order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> from shift-invariant subspaces in Sobolev spaces was introduced in the paper by Zhao (1995). Moreover, a characterization of those principal shift-invariant subspaces that provide simultaneous approximation order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> was proved there. In this note, we prove another characterization using dilated by some adequate expansive linear maps of a shift-invariant subspace. In addition, we introduce the notion of simultaneous density order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> and give necessary and sufficient conditions on a shift-invariant subspace to have a simultaneous density desired. To give our conditions, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of the generators of a shift-invariant subspace. For this, we will use the classical notion of approximate continuity.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106147"},"PeriodicalIF":0.9,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379119","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":"Iterated entropy derivatives and binary entropy inequalities","authors":"Tanay Wakhare","doi":"10.1016/j.jat.2025.106143","DOIUrl":"10.1016/j.jat.2025.106143","url":null,"abstract":"<div><div>We embark on a systematic study of the <span><math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-th derivative of <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>r</mi></mrow></msup><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≔</mo><mo>−</mo><mi>x</mi><mo>log</mo><mi>x</mi><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is the binary entropy and <span><math><mrow><mi>k</mi><mo>≥</mo><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></math></span> are integers. Our motivation is the conjectural entropy inequality <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>&lt;</mo><mn>1</mn></mrow></math></span> is given by a functional equation. The <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span> case was the key technical tool driving recent breakthroughs on the union-closed sets conjecture. We express <span><math><mrow><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>r</mi></mrow></msup><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> as a rational function, an infinite series, and a sum over generalized Stirling numbers. This allows us to reduce the proof of the entropy inequality for real <span><math><mi>k</mi></math></span> to showing that an associated polynomial has only two real roots in the interval <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, which also allows us to prove the inequality for fractional exponents such as <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math></span>. The proof suggests a new framework for proving tight inequalities for the sum of polynomials times the logarithms of polynomials, which converts the inequality into a statement about the real roots of a simpler associated polynomial.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106143"},"PeriodicalIF":0.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154843","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":"Optimal heat kernel bounds and asymptotics on Damek–Ricci spaces","authors":"Tommaso Bruno ,&nbsp;Federico Santagati","doi":"10.1016/j.jat.2025.106144","DOIUrl":"10.1016/j.jat.2025.106144","url":null,"abstract":"<div><div>We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace–Beltrami operator on Damek–Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106144"},"PeriodicalIF":0.9,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154844","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":"Bounds for extreme zeros of Meixner–Pollaczek polynomials","authors":"A.S. Jooste ,&nbsp;K. Jordaan","doi":"10.1016/j.jat.2024.106142","DOIUrl":"10.1016/j.jat.2024.106142","url":null,"abstract":"<div><div>In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>k</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner–Pollaczek polynomial. When <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is orthogonal with respect to a weight <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub></math></span> is orthogonal with respect to the weight <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, we show that <span><math><mrow><mi>k</mi><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></mrow></math></span> is a necessary and sufficient condition for existence of a connection formula involving a polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> of degree <span><math><mrow><mo>(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>, such that the <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> zeros of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub></mrow></math></span> and the <span><math><mi>n</mi></math></span> zeros of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> interlace. We analyse the new inner bounds for the extreme zeros of Meixner–Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106142"},"PeriodicalIF":0.9,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154842","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":"Twenty-five years of greedy bases","authors":"Fernando Albiac ,&nbsp;José L. Ansorena ,&nbsp;Vladimir Temlyakov","doi":"10.1016/j.jat.2024.106141","DOIUrl":"10.1016/j.jat.2024.106141","url":null,"abstract":"<div><div>Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106141"},"PeriodicalIF":0.9,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154841","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":"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}