{"title":"Norm convergence of subsequences of matrix transform means of Walsh–Fourier series","authors":"István Blahota","doi":"10.1007/s10998-024-00584-3","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\{a_{n}:\\ n\\in \\mathbb {P}\\}\\)</span> be an increasing sequence of positive integers. For every <span>\\(n\\in \\mathbb {P}\\)</span> let <span>\\(\\{t_{k,a_{n}}: 1\\le k\\le a_{n},\\ k\\in \\mathbb {P}\\}\\)</span> be a finite sequence of non-negative numbers such that </p><span>$$\\begin{aligned} \\sum _{k=1}^{a_{n}} t_{k,a_{n}}=1 \\end{aligned}$$</span><p>holds and </p><span>$$\\begin{aligned} \\lim _{n\\rightarrow \\infty }t_{k,a_{n}}=0 \\end{aligned}$$</span><p>is satisfied for any fixed <i>k</i>. Our main result (Theorem 6.5) is that we prove <span>\\(L_{1}\\)</span>-norm convergence </p><span>$$\\begin{aligned} \\sigma _{a_{n}}^{T}(f)\\rightarrow f \\end{aligned}$$</span><p>(for example, but not limited to <span>\\(a_{n}:=2^{n}\\)</span>, see Corollary 6.6 and Sect. 7) with weaker conditions than it was known before for matrix transform means and for some special means, namely Nörlund and weighted ones.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":"55 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Periodica Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10998-024-00584-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(\{a_{n}:\ n\in \mathbb {P}\}\) be an increasing sequence of positive integers. For every \(n\in \mathbb {P}\) let \(\{t_{k,a_{n}}: 1\le k\le a_{n},\ k\in \mathbb {P}\}\) be a finite sequence of non-negative numbers such that
is satisfied for any fixed k. Our main result (Theorem 6.5) is that we prove \(L_{1}\)-norm convergence
$$\begin{aligned} \sigma _{a_{n}}^{T}(f)\rightarrow f \end{aligned}$$
(for example, but not limited to \(a_{n}:=2^{n}\), see Corollary 6.6 and Sect. 7) with weaker conditions than it was known before for matrix transform means and for some special means, namely Nörlund and weighted ones.
期刊介绍:
Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica.
Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.