Pub Date : 2026-01-01Epub Date: 2025-08-19DOI: 10.1016/j.jat.2025.106219
Tamás Erdélyi
We show that if is a sequence of numbers with , and then is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.
{"title":"The sequence of partial sums of a unimodular power series is not ultraflat","authors":"Tamás Erdélyi","doi":"10.1016/j.jat.2025.106219","DOIUrl":"10.1016/j.jat.2025.106219","url":null,"abstract":"<div><div>We show that if <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> is a sequence of numbers <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>ℂ</mi></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, and <span><span><span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msup><mspace></mspace><mo>,</mo><mspace></mspace><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mspace></mspace><mo>,</mo></mrow></math></span></span></span>then <span><math><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106219"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908604","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}
Pub Date : 2025-12-01Epub Date: 2025-08-06DOI: 10.1016/j.jat.2025.106218
Pierre Bizeul , Boaz Klartag
<div><div>We study the problem of polynomial approximation in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> (<span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>), where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> (<span><math><mrow><mi>d</mi><mi>x</mi></mrow></math></span>) = <span><math><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span>. We show that for any absolutely continuous function <span><math><mi>f</mi></math></span>, <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>〈</mo><mi>f</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>〉</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>C</mi><mfenced><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></math></span> for some universal constant <span><math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math></span>, where <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> are the orthonormal polynomials associated with <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This inequality is tight in the sense that <span><math><mrow><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> on the left-hand side cannot be replaced by <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> for any sequence <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>→</mo><mi>∞</mi></mrow></math></span>. When the right-hand side is bounded, this inequality implies a logarithmic rate of approximati
{"title":"Polynomial approximation in L2 with the double-sided exponential weight via complex analysis","authors":"Pierre Bizeul , Boaz Klartag","doi":"10.1016/j.jat.2025.106218","DOIUrl":"10.1016/j.jat.2025.106218","url":null,"abstract":"<div><div>We study the problem of polynomial approximation in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>\u0000 (<span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>), where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>\u0000 (<span><math><mrow><mi>d</mi><mi>x</mi></mrow></math></span>) = <span><math><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span>. We show that for any absolutely continuous function <span><math><mi>f</mi></math></span>, <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>〈</mo><mi>f</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>〉</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>C</mi><mfenced><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></math></span> for some universal constant <span><math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math></span>, where <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> are the orthonormal polynomials associated with <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This inequality is tight in the sense that <span><math><mrow><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> on the left-hand side cannot be replaced by <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> for any sequence <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>→</mo><mi>∞</mi></mrow></math></span>. When the right-hand side is bounded, this inequality implies a logarithmic rate of approximati","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106218"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144842810","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}
Pub Date : 2025-12-01Epub Date: 2025-08-07DOI: 10.1016/j.jat.2025.106220
D. Leviatan , I. Shevchuk , V. Shevchuk
Let . For a continuous -periodic function, changing its monotonicity times in a period, and whose degree of approximation by trigonometric polynomials of degree , is , , we investigate its degree of approximation by such polynomials that, in addition, follow the changes of monotonicity. Obviously, the unconstrained degree is smaller than the constrained one, but for , there is a constant such that the constrained degree is , . On the other hand we show that, in general, this is invalid for .
{"title":"Comparing the degree of constrained and unconstrained trigonometric approximation","authors":"D. Leviatan , I. Shevchuk , V. Shevchuk","doi":"10.1016/j.jat.2025.106220","DOIUrl":"10.1016/j.jat.2025.106220","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>r</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. For a continuous <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic function, changing its monotonicity <span><math><mrow><mn>2</mn><mi>s</mi></mrow></math></span> times in a period, and whose degree of approximation by trigonometric polynomials of degree <span><math><mrow><mo><</mo><mi>n</mi></mrow></math></span>, is <span><math><mrow><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, we investigate its degree of approximation by such polynomials that, in addition, follow the changes of monotonicity. Obviously, the unconstrained degree is smaller than the constrained one, but for <span><math><mrow><mi>r</mi><mo>></mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>2</mn></mrow></math></span>, there is a constant <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> such that the constrained degree is <span><math><mrow><mo>≤</mo><mi>c</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. On the other hand we show that, in general, this is invalid for <span><math><mrow><mi>r</mi><mo>≤</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106220"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809532","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}
Pub Date : 2025-12-01Epub Date: 2025-07-08DOI: 10.1016/j.jat.2025.106212
F.D. Kovac , F.E. Levis , L. Zabala
In this article, we investigate the best approximation operator from a finite-dimensional linear space defined on Lorentz Gamma spaces for . We extend the best approximation operator from to the larger space and establish several key properties of these operators.
{"title":"Best approximations and their extensions in Lorentz Gamma spaces","authors":"F.D. Kovac , F.E. Levis , L. Zabala","doi":"10.1016/j.jat.2025.106212","DOIUrl":"10.1016/j.jat.2025.106212","url":null,"abstract":"<div><div>In this article, we investigate the best approximation operator from a finite-dimensional linear space defined on Lorentz Gamma spaces <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>p</mi></mrow></msub></math></span> for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>. We extend the best approximation operator from <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>p</mi></mrow></msub></math></span> to the larger space <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>w</mi><mo>,</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and establish several key properties of these operators.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106212"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587795","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}
Pub Date : 2025-12-01Epub Date: 2025-07-23DOI: 10.1016/j.jat.2025.106214
Alicja Dota , Leszek Skrzypczak
Let be a subspace of the Besov space that consists of block-radial (multi-radial) functions. We study the asymptotic behaviour of approximation numbers of compact embeddings . Moreover, we find a sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces .
{"title":"Embeddings of block-radial functions — approximation properties and nuclearity","authors":"Alicja Dota , Leszek Skrzypczak","doi":"10.1016/j.jat.2025.106214","DOIUrl":"10.1016/j.jat.2025.106214","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>γ</mi></mrow></msub><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> be a subspace of the Besov space <span><math><mrow><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> that consists of block-radial (multi-radial) functions. We study the asymptotic behaviour of approximation numbers of compact embeddings <span><math><mrow><mi>id</mi><mo>:</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>γ</mi></mrow></msub><msubsup><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow><mo>→</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>γ</mi></mrow></msub><msubsup><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Moreover, we find a sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>γ</mi></mrow></msub><msubsup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106214"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144711188","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}
Pub Date : 2025-12-01Epub Date: 2025-05-24DOI: 10.1016/j.jat.2025.106201
Hans-Peter Blatt
<div><div>Let <span><math><mi>E</mi></math></span> be compact and connected with <span><math><mrow><mi>cap</mi><mspace></mspace><mi>E</mi><mo>></mo><mn>0</mn></mrow></math></span> and connected complement <span><math><mrow><mi>Ω</mi><mo>=</mo><mover><mrow><mi>ℂ</mi></mrow><mo>¯</mo></mover><mo>∖</mo><mi>E</mi></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> be the Green’s function of <span><math><mi>Ω</mi></math></span> with pole at infinity and let <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>Ω</mi><mo>:</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo><</mo><mo>log</mo><mi>σ</mi><mo>}</mo></mrow><mo>∪</mo><mi>E</mi><mo>,</mo><mspace></mspace><mn>1</mn><mo><</mo><mi>σ</mi><mo><</mo><mi>∞</mi><mo>,</mo></mrow></math></span> be the Green domains with boundaries <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>. Let <span><math><mi>f</mi></math></span> be holomorphic on <span><math><mi>E</mi></math></span> and let <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></math></span> denote the maximal parameter of holomorphy of <span><math><mi>f</mi></math></span> and let <span><math><msub><mrow><mfenced><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a sequence of polynomials converging maximally to <span><math><mi>f</mi></math></span> on <span><math><mi>E</mi></math></span>. If <span><math><mi>σ</mi></math></span>, <span><math><mrow><mn>1</mn><mo><</mo><mi>σ</mi><mo><</mo><mi>ρ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo><</mo><mi>∞</mi></mrow></math></span>, is fixed and if <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of interpolation points of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to <span><math><mi>f</mi></math></span> in <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span> with normalized counting measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, then there exists a subset <span><math><mrow><mi>Λ</mi><mo>⊂</mo><mi>N</mi></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mi>n</mi><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mspace></mspace><mtext>as</mtext><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Λ</mi><mo>,</mo><mi>n</mi><mo>→</mo><mi>∞</mi><mo>,</mo></mrow></math></s
设E是紧致的，并且与capE>；0和连通补Ω=¯∈E相连，设gΩ（z,∞）是Ω的极点在无穷远处的Green函数，设Eσ∈Ω：gΩ（z,∞）<logσ}∪E,1<σ<；∞是有边界的Green域Γσ。让f E和上全纯让ρ(f)表示最大的正则参数f对所测试,让∈N是一个多项式序列收敛最大f E .如果σ,1 & lt;σ& lt;ρ(f) & lt;∞,是固定的,如果mn(σ)表示pn的数量的插值点与规范化计数测量μf Eσσ,N,那么存在一个子集Λ⊂N, mn(σ)= N + o (N) asn∈Λ,N→∞,μσ,N | E +μσ,N |Ω⟶∗μEasn∈Λ,N→∞,在μσ,N =μσ,N | E +μσ,N |Ω,μσ，n|Ê表示μσ的平衡测度，n|E在E的边界上，μE是E的平衡测度，并且存在一个收敛于σ的序列σnn∈Λ，使得闭合曲线γn=(f−pn)（Γσn）不经过0点，圈数Indγn(0)满足Indγn(0)=mn(σn)=n+o(n)asn∈Λ，n→∞。
{"title":"Intrinsic interpolation, near-circularity and maximal convergence","authors":"Hans-Peter Blatt","doi":"10.1016/j.jat.2025.106201","DOIUrl":"10.1016/j.jat.2025.106201","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi></math></span> be compact and connected with <span><math><mrow><mi>cap</mi><mspace></mspace><mi>E</mi><mo>></mo><mn>0</mn></mrow></math></span> and connected complement <span><math><mrow><mi>Ω</mi><mo>=</mo><mover><mrow><mi>ℂ</mi></mrow><mo>¯</mo></mover><mo>∖</mo><mi>E</mi></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> be the Green’s function of <span><math><mi>Ω</mi></math></span> with pole at infinity and let <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>Ω</mi><mo>:</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo><</mo><mo>log</mo><mi>σ</mi><mo>}</mo></mrow><mo>∪</mo><mi>E</mi><mo>,</mo><mspace></mspace><mn>1</mn><mo><</mo><mi>σ</mi><mo><</mo><mi>∞</mi><mo>,</mo></mrow></math></span> be the Green domains with boundaries <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>. Let <span><math><mi>f</mi></math></span> be holomorphic on <span><math><mi>E</mi></math></span> and let <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></math></span> denote the maximal parameter of holomorphy of <span><math><mi>f</mi></math></span> and let <span><math><msub><mrow><mfenced><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a sequence of polynomials converging maximally to <span><math><mi>f</mi></math></span> on <span><math><mi>E</mi></math></span>. If <span><math><mi>σ</mi></math></span>, <span><math><mrow><mn>1</mn><mo><</mo><mi>σ</mi><mo><</mo><mi>ρ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo><</mo><mi>∞</mi></mrow></math></span>, is fixed and if <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of interpolation points of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to <span><math><mi>f</mi></math></span> in <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span> with normalized counting measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, then there exists a subset <span><math><mrow><mi>Λ</mi><mo>⊂</mo><mi>N</mi></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mi>n</mi><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mspace></mspace><mtext>as</mtext><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Λ</mi><mo>,</mo><mi>n</mi><mo>→</mo><mi>∞</mi><mo>,</mo></mrow></math></s","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106201"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144271754","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}
Pub Date : 2025-12-01Epub Date: 2025-07-28DOI: 10.1016/j.jat.2025.106215
Dansheng Yu
The construction and approximation properties of exact neural network interpolation are the important and challenging topics on approximation by neural networks. Most research on exact neural network interpolation has focused on establishing existence, with very few specifically constructed interpolation neural networks proposed. The main purpose of the present paper is to provide a method for directly constructing exact neural network interpolation operators, which has the advantages that all the components in the neural network operators are explicitly known, such as the coefficients, the weights and the thresholds. By employing some important methods in approximation theory, such as the equivalence between the functional and the modulus of continuity of the function, Berens–Lorentz Lemma, and two useful estimates of the derivatives of the operators, we establish both the direct and the converse results of approximation by the new interpolation operators, and thus obtain an equivalence characterization theorem. We also introduce a type of neural network interpolation operators with four layers and a type of max-product neural network operators, rigorously analyzing their approximation properties.
{"title":"Construction and approximation properties of exact neural network interpolation operators activated by entire functions","authors":"Dansheng Yu","doi":"10.1016/j.jat.2025.106215","DOIUrl":"10.1016/j.jat.2025.106215","url":null,"abstract":"<div><div>The construction and approximation properties of exact neural network interpolation are the important and challenging topics on approximation by neural networks. Most research on exact neural network interpolation has focused on establishing existence, with very few specifically constructed interpolation neural networks proposed. The main purpose of the present paper is to provide a method for directly constructing exact neural network interpolation operators, which has the advantages that all the components in the neural network operators are explicitly known, such as the coefficients, the weights and the thresholds. By employing some important methods in approximation theory, such as the equivalence between the <span><math><mrow><mi>K</mi><mo>−</mo></mrow></math></span>functional and the modulus of continuity of the function, Berens–Lorentz Lemma, and two useful estimates of the derivatives of the operators, we establish both the direct and the converse results of approximation by the new interpolation operators, and thus obtain an equivalence characterization theorem. We also introduce a type of neural network interpolation operators with four layers and a type of max-product neural network operators, rigorously analyzing their approximation properties.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106215"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721942","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}
Pub Date : 2025-11-01Epub Date: 2025-05-16DOI: 10.1016/j.jat.2025.106200
Anton Baranov , Ilgiz Kayumov , Rachid Zarouf
We derive new integral estimates of the derivatives of mean -valent functions in the unit disc. Our results develop and complement estimates obtained by E. P. Dolzhenko and A. A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.
给出了单位圆盘中n价平均函数导数的新的积分估计。我们的结果发展和补充了E. P. Dolzhenko和A. A. Pekarskii的估计，以及作者最近得到的不等式。作为应用，我们改进了一些由于Dolzhenko的有理逼近的逆定理。
{"title":"Bernstein-type inequalities for mean n-valent functions","authors":"Anton Baranov , Ilgiz Kayumov , Rachid Zarouf","doi":"10.1016/j.jat.2025.106200","DOIUrl":"10.1016/j.jat.2025.106200","url":null,"abstract":"<div><div>We derive new integral estimates of the derivatives of mean <span><math><mi>n</mi></math></span>-valent functions in the unit disc. Our results develop and complement estimates obtained by E.<!--> <!-->P. Dolzhenko and A.<!--> <!-->A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106200"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116319","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}
Pub Date : 2025-11-01Epub Date: 2025-05-09DOI: 10.1016/j.jat.2025.106191
Tomasz Beberok
The study of interpolation nodes and their associated Lebesgue constants is a cornerstone of numerical analysis, directly influencing the stability and accuracy of polynomial approximations. In this paper, we examine the Morrow–Patterson points, a specific set of interpolation nodes introduced to construct cubature formulas with the minimal number of points in a square for a fixed degree . We prove that their Lebesgue constant has minimal rate of growth of at least .
{"title":"A lower bound for the Lebesgue constant of the Morrow–Patterson points","authors":"Tomasz Beberok","doi":"10.1016/j.jat.2025.106191","DOIUrl":"10.1016/j.jat.2025.106191","url":null,"abstract":"<div><div>The study of interpolation nodes and their associated Lebesgue constants is a cornerstone of numerical analysis, directly influencing the stability and accuracy of polynomial approximations. In this paper, we examine the Morrow–Patterson points, a specific set of interpolation nodes introduced to construct cubature formulas with the minimal number of points in a square for a fixed degree <span><math><mi>n</mi></math></span>. We prove that their Lebesgue constant has minimal rate of growth of at least <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106191"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935524","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}
Pub Date : 2025-11-01Epub Date: 2025-05-09DOI: 10.1016/j.jat.2025.106190
T.M. Nikiforova
Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar–Rakhmanov–Saff theorem, too.
{"title":"Minimax and maximin problems for sums of translates on the real axis","authors":"T.M. Nikiforova","doi":"10.1016/j.jat.2025.106190","DOIUrl":"10.1016/j.jat.2025.106190","url":null,"abstract":"<div><div>Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar–Rakhmanov–Saff theorem, too.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106190"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936458","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}
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