Pub Date : 2024-07-26DOI: 10.1016/j.jat.2024.106083
Wojciech M. Kozlowski , Gianluca Vinti
We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.
{"title":"On approximation by rational functions in Musielak–Orlicz spaces","authors":"Wojciech M. Kozlowski , Gianluca Vinti","doi":"10.1016/j.jat.2024.106083","DOIUrl":"10.1016/j.jat.2024.106083","url":null,"abstract":"<div><p>We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106083"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000716/pdfft?md5=772962309c292532c60924a31afa1fa7&pid=1-s2.0-S0021904524000716-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141847123","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 : 2024-07-26DOI: 10.1016/j.jat.2024.106085
Deliang Chen
The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of as the closure of in different senses and the extension of continuous vector-valued functions are studied.
{"title":"The Machado–Bishop theorem in the uniform topology","authors":"Deliang Chen","doi":"10.1016/j.jat.2024.106085","DOIUrl":"10.1016/j.jat.2024.106085","url":null,"abstract":"<div><p>The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>×</mo><mi>J</mi><mo>,</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> as the closure of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow><mo>⊗</mo><mi>C</mi><mrow><mo>(</mo><mi>J</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> in different senses and the extension of continuous vector-valued functions are studied.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106085"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141848755","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.2024.106071
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":"303 ","pages":"Article 106071"},"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":"303 ","pages":"Article 106072"},"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":"303 ","pages":"Article 106075"},"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":"303 ","pages":"Article 106073"},"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":"303 ","pages":"Article 106074"},"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":"302 ","pages":"Article 106067"},"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}
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