Gergely Kiss, Dávid Matolcsi, Máté Matolcsi, Gábor Somlai
{"title":"(Zp)d中的平铺和弱平铺。","authors":"Gergely Kiss, Dávid Matolcsi, Máté Matolcsi, Gábor Somlai","doi":"10.1007/s43670-023-00073-7","DOIUrl":null,"url":null,"abstract":"<p><p>We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary <i>p</i>-groups <math><msup><mrow><mo>(</mo><msub><mi>Z</mi><mi>p</mi></msub><mo>)</mo></mrow><mi>d</mi></msup></math>, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for <math><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math>, and prove some partial results for <math><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math>.</p>","PeriodicalId":74751,"journal":{"name":"Sampling theory, signal processing, and data analysis","volume":"22 1","pages":"1"},"PeriodicalIF":0.0000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10695897/pdf/","citationCount":"0","resultStr":"{\"title\":\"<ArticleTitle xmlns:ns0=\\\"http://www.w3.org/1998/Math/MathML\\\">Tiling and weak tiling in <ns0:math><ns0:msup><ns0:mrow><ns0:mo>(</ns0:mo><ns0:msub><ns0:mi>Z</ns0:mi><ns0:mi>p</ns0:mi></ns0:msub><ns0:mo>)</ns0:mo></ns0:mrow><ns0:mi>d</ns0:mi></ns0:msup></ns0:math>.\",\"authors\":\"Gergely Kiss, Dávid Matolcsi, Máté Matolcsi, Gábor Somlai\",\"doi\":\"10.1007/s43670-023-00073-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary <i>p</i>-groups <math><msup><mrow><mo>(</mo><msub><mi>Z</mi><mi>p</mi></msub><mo>)</mo></mrow><mi>d</mi></msup></math>, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for <math><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math>, and prove some partial results for <math><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math>.</p>\",\"PeriodicalId\":74751,\"journal\":{\"name\":\"Sampling theory, signal processing, and data analysis\",\"volume\":\"22 1\",\"pages\":\"1\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10695897/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Sampling theory, signal processing, and data analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s43670-023-00073-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/12/4 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Sampling theory, signal processing, and data analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s43670-023-00073-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/12/4 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for , and prove some partial results for .
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