Pub Date : 2024-07-05DOI: 10.1016/j.jat.2024.106071
David H. Bailey
{"title":"Peter Borwein: A philosopher’s mathematician","authors":"David H. Bailey","doi":"10.1016/j.jat.2024.106071","DOIUrl":"10.1016/j.jat.2024.106071","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141690540","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 : 2024-07-05DOI: 10.1016/j.jat.2024.106072
Stephen Choi
{"title":"In memory of Peter Borwein","authors":"Stephen Choi","doi":"10.1016/j.jat.2024.106072","DOIUrl":"10.1016/j.jat.2024.106072","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000583/pdfft?md5=cbc6f03a9f2eb4bee6ad9cfdd54f3ec1&pid=1-s2.0-S0021904524000583-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141695786","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}
Pub Date : 2024-07-05DOI: 10.1016/j.jat.2024.106075
Boris Shekhtman
{"title":"In memory of Peter Borwein","authors":"Boris Shekhtman","doi":"10.1016/j.jat.2024.106075","DOIUrl":"10.1016/j.jat.2024.106075","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141708036","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 : 2024-07-05DOI: 10.1016/j.jat.2024.106073
Doron S. Lubinsky
{"title":"Remembering Peter Benjamin Borwein (May 10, 1953–August 23, 2020)","authors":"Doron S. Lubinsky","doi":"10.1016/j.jat.2024.106073","DOIUrl":"10.1016/j.jat.2024.106073","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141688829","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 : 2024-07-05DOI: 10.1016/j.jat.2024.106074
Michael J. Mossinghoff
Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.
{"title":"In Memoriam: Peter Benjamin Borwein (1953–2020)","authors":"Michael J. Mossinghoff","doi":"10.1016/j.jat.2024.106074","DOIUrl":"10.1016/j.jat.2024.106074","url":null,"abstract":"<div><p>Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal<span><span> problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the </span>Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.</span></p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142040780","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 : 2024-06-26DOI: 10.1016/j.jat.2024.106067
F.E. Levis , C.V. Ridolfi , L. Zabala
We correct an error in the statement of Levis et al. (2023, Theorem 4.5).
我们纠正了莱维斯等人（2023，定理 4.5）的陈述中的一个错误。
{"title":"Corrigendum to “Strong uniqueness and alternation theorems for relative Chebyshev centers” [J. Approx. Theory, 293 (2023) 105917]","authors":"F.E. Levis , C.V. Ridolfi , L. Zabala","doi":"10.1016/j.jat.2024.106067","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106067","url":null,"abstract":"<div><p>We correct an error in the statement of Levis et al. (2023, Theorem 4.5).</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000534/pdfft?md5=351282467b6da49e539feca5eabfeab4&pid=1-s2.0-S0021904524000534-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594108","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}
Pub Date : 2024-06-12DOI: 10.1016/j.jat.2024.106056
Robert J. Kunsch , Erich Novak , Marcin Wnuk
We prove lower bounds for the randomized approximation of the embedding based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix . These lower bounds reflect the increasing difficulty of the problem for , namely, a term in the complexity . This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity only exhibits a -dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order .
{"title":"Randomized approximation of summable sequences — adaptive and non-adaptive","authors":"Robert J. Kunsch , Erich Novak , Marcin Wnuk","doi":"10.1016/j.jat.2024.106056","DOIUrl":"10.1016/j.jat.2024.106056","url":null,"abstract":"<div><p>We prove lower bounds for the randomized approximation of the embedding <span><math><mrow><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>↪</mo><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></math></span> based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix <span><math><mrow><mi>N</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></mrow></math></span>. These lower bounds reflect the increasing difficulty of the problem for <span><math><mrow><mi>m</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, namely, a term <span><math><msqrt><mrow><mo>log</mo><mi>m</mi></mrow></msqrt></math></span> in the complexity <span><math><mi>n</mi></math></span>. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity <span><math><mi>n</mi></math></span> only exhibits a <span><math><mrow><mo>(</mo><mo>log</mo><mo>log</mo><mi>m</mi><mo>)</mo></mrow></math></span>-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order <span><math><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568583","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 : 2024-06-08DOI: 10.1016/j.jat.2024.106057
Ni Li , Jin Yang
Motivated by some properties of the geometric measures for compact convex sets in the Brunn–Minkowski theory, such as the properties of the volume, the -capacity and the torsional rigidity for compact convex sets, we introduce a more general geometric invariant, called the compatible functional . Inspired also by the Minkowski problem associated with the volume, the -capacity and the torsional rigidity for compact convex sets, we pose the Minkowski problem associated with the compatible functional and prove the existence of the solutions to this problem for . We will show that the volume, the -capacity and the torsional rigidity for compact convex sets are the compatible functionals. Thus, as an application, we provide the solution to the Minkowski problem for arbitrary measure associated with -capacity .
{"title":"The Lp Minkowski problem associated with the compatible functional F","authors":"Ni Li , Jin Yang","doi":"10.1016/j.jat.2024.106057","DOIUrl":"10.1016/j.jat.2024.106057","url":null,"abstract":"<div><p>Motivated by some properties of the geometric measures for compact convex sets in the Brunn–Minkowski theory, such as the properties of the volume, the <span><math><mi>p</mi></math></span>-capacity <span><math><mrow><mo>(</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>n</mi><mo>)</mo></mrow></math></span> and the torsional rigidity for compact convex sets, we introduce a more general geometric invariant, called the compatible functional <span><math><mi>F</mi></math></span>. Inspired also by the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> Minkowski problem associated with the volume, the <span><math><mi>p</mi></math></span>-capacity and the torsional rigidity for compact convex sets, we pose the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> Minkowski problem associated with the compatible functional <span><math><mi>F</mi></math></span> and prove the existence of the solutions to this problem for <span><math><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></math></span>. We will show that the volume, the <span><math><mi>p</mi></math></span>-capacity <span><math><mrow><mo>(</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mo>)</mo></mrow></math></span> and the torsional rigidity for compact convex sets are the compatible functionals. Thus, as an application, we provide the solution to the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> Minkowski problem <span><math><mrow><mo>(</mo><mn>0</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn><mo>)</mo></mrow></math></span> for arbitrary measure associated with <span><math><mi>p</mi></math></span>-capacity <span><math><mrow><mo>(</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141410757","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 : 2024-05-28DOI: 10.1016/j.jat.2024.106055
Filip Tronarp , Toni Karvonen
We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of . This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.
{"title":"Orthonormal expansions for translation-invariant kernels","authors":"Filip Tronarp , Toni Karvonen","doi":"10.1016/j.jat.2024.106055","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106055","url":null,"abstract":"<div><p>We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of <span><math><mrow><msub><mrow><mi>ℒ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000418/pdfft?md5=e69b571338e575238b6d57e3630cb524&pid=1-s2.0-S0021904524000418-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141329214","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号