Limiting Behavior of Infinite Products Scaled by Pisot Numbers View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2018-08-29

AUTHORS

Tian-You Hu, Ka-Sing Lau

ABSTRACT

For θ>1, the infinite product Γθ(x)=∏n=0∞cos(πθ-jx) is the Fourier transform of the Bernoulli convolution with scale θ-1. Its limiting behavior at infinity has been studied since the 1930’s, but is still not completely settled. In this note, we consider the limiting behavior of Γ(x)=Γθ1(x)Γθ2(λx), a question originally raised by Salem. For Pisot numbers θ1,θ2 that are exponentially commensurable, we show that the parameters λ such that Γ(x) does not tend to zero at infinity are countable, and in most cases they are dense in R. The explicit forms of such λ can also be identified. The conclusion is also true for Γ(x) with n products. More... »

PAGES

1-13

References to SciGraph publications

  • 1998-03. Asymptotic behavior of multiperiodic functions in JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
  • 2012-01. Convolutions of cantor measures without resonance in ISRAEL JOURNAL OF MATHEMATICS
  • 1993-03. Mean quadratic variations and Fourier asymptotics of self-similar measures in MONATSHEFTE FÜR MATHEMATIK
  • 2012-08. A property of Pisot numbers and Fourier transforms of self-similar measures in SCIENCE CHINA MATHEMATICS
  • 2000. Sixty Years of Bernoulli Convolutions in FRACTAL GEOMETRY AND STOCHASTICS II
  • 1973-03. PV-numbers and sets of multiplicity in PERIODICA MATHEMATICA HUNGARICA
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00041-018-9638-y

    DOI

    http://dx.doi.org/10.1007/s00041-018-9638-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1106415664


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0303", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Macromolecular and Materials Chemistry", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/03", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Chemical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Wisconsin\u2013Green Bay", 
              "id": "https://www.grid.ac/institutes/grid.267461.0", 
              "name": [
                "Department of Mathematics, University of Wisconsin-Green Bay, 54311, Green Bay, WI, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Hu", 
            "givenName": "Tian-You", 
            "id": "sg:person.012770157205.22", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012770157205.22"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Central China Normal University", 
              "id": "https://www.grid.ac/institutes/grid.411407.7", 
              "name": [
                "Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong", 
                "Department of Mathematics, University of Pittsburgh, 15217, Pittsburgh, PA, USA", 
                "Department of Mathematics, Central China Normal University, Wuhan, China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Lau", 
            "givenName": "Ka-Sing", 
            "id": "sg:person.0577752355.47", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0577752355.47"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf01311213", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1007176639", 
              "https://doi.org/10.1007/bf01311213"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01311213", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1007176639", 
              "https://doi.org/10.1007/bf01311213"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11856-011-0164-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1008614161", 
              "https://doi.org/10.1007/s11856-011-0164-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11425-012-4422-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1011642614", 
              "https://doi.org/10.1007/s11425-012-4422-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1090/s0002-9939-02-06398-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015923655"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1006/jfan.1993.1116", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1029592685"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1006/aima.1998.1773", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1038435077"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02018464", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039245448", 
              "https://doi.org/10.1007/bf02018464"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02018464", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039245448", 
              "https://doi.org/10.1007/bf02018464"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02475985", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041313012", 
              "https://doi.org/10.1007/bf02475985"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02475985", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041313012", 
              "https://doi.org/10.1007/bf02475985"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-0348-8380-1_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1049957895", 
              "https://doi.org/10.1007/978-3-0348-8380-1_2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-0348-8380-1_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1049957895", 
              "https://doi.org/10.1007/978-3-0348-8380-1_2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1080/10586458.1992.10504561", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1053597897"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1215/s0012-7094-44-01111-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064416344"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2307/2371053", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1069897780"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2307/2371641", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1069898355"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2018-08-29", 
        "datePublishedReg": "2018-08-29", 
        "description": "For \u03b8>1, the infinite product \u0393\u03b8(x)=\u220fn=0\u221ecos(\u03c0\u03b8-jx) is the Fourier transform of the Bernoulli convolution with scale \u03b8-1. Its limiting behavior at infinity has been studied since the 1930\u2019s, but is still not completely settled. In this note, we consider the limiting behavior of \u0393(x)=\u0393\u03b81(x)\u0393\u03b82(\u03bbx), a question originally raised by Salem. For Pisot numbers \u03b81,\u03b82 that are exponentially commensurable, we show that the parameters \u03bb such that \u0393(x) does not tend to zero at infinity are countable, and in most cases they are dense in R. The explicit forms of such \u03bb can also be identified. The conclusion is also true for \u0393(x) with n products.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/s00041-018-9638-y", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1042645", 
            "issn": [
              "1069-5869", 
              "1531-5851"
            ], 
            "name": "Journal of Fourier Analysis and Applications", 
            "type": "Periodical"
          }
        ], 
        "name": "Limiting Behavior of Infinite Products Scaled by Pisot Numbers", 
        "pagination": "1-13", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "b7c8feb85963eb4fa98c27fc7a33828798badf1c5d24b6ca4408b1beea62c36e"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00041-018-9638-y"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1106415664"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00041-018-9638-y", 
          "https://app.dimensions.ai/details/publication/pub.1106415664"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-10T23:20", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8693_00000494.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1007/s00041-018-9638-y"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00041-018-9638-y'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00041-018-9638-y'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00041-018-9638-y'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00041-018-9638-y'


     

    This table displays all metadata directly associated to this object as RDF triples.

    112 TRIPLES      21 PREDICATES      37 URIs      16 LITERALS      5 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00041-018-9638-y schema:about anzsrc-for:03
    2 anzsrc-for:0303
    3 schema:author Nee0110b757d34e179ea83cd065d701fd
    4 schema:citation sg:pub.10.1007/978-3-0348-8380-1_2
    5 sg:pub.10.1007/bf01311213
    6 sg:pub.10.1007/bf02018464
    7 sg:pub.10.1007/bf02475985
    8 sg:pub.10.1007/s11425-012-4422-y
    9 sg:pub.10.1007/s11856-011-0164-8
    10 https://doi.org/10.1006/aima.1998.1773
    11 https://doi.org/10.1006/jfan.1993.1116
    12 https://doi.org/10.1080/10586458.1992.10504561
    13 https://doi.org/10.1090/s0002-9939-02-06398-0
    14 https://doi.org/10.1215/s0012-7094-44-01111-7
    15 https://doi.org/10.2307/2371053
    16 https://doi.org/10.2307/2371641
    17 schema:datePublished 2018-08-29
    18 schema:datePublishedReg 2018-08-29
    19 schema:description For θ>1, the infinite product Γθ(x)=∏n=0∞cos(πθ-jx) is the Fourier transform of the Bernoulli convolution with scale θ-1. Its limiting behavior at infinity has been studied since the 1930’s, but is still not completely settled. In this note, we consider the limiting behavior of Γ(x)=Γθ1(x)Γθ2(λx), a question originally raised by Salem. For Pisot numbers θ1,θ2 that are exponentially commensurable, we show that the parameters λ such that Γ(x) does not tend to zero at infinity are countable, and in most cases they are dense in R. The explicit forms of such λ can also be identified. The conclusion is also true for Γ(x) with n products.
    20 schema:genre research_article
    21 schema:inLanguage en
    22 schema:isAccessibleForFree false
    23 schema:isPartOf sg:journal.1042645
    24 schema:name Limiting Behavior of Infinite Products Scaled by Pisot Numbers
    25 schema:pagination 1-13
    26 schema:productId N064167f8639d493a8fe026110ba7d4b5
    27 N392dca4ddd014ffba30043018f1c056a
    28 N87d8a76e4c33435689e81d17cb7bb9d1
    29 schema:sameAs https://app.dimensions.ai/details/publication/pub.1106415664
    30 https://doi.org/10.1007/s00041-018-9638-y
    31 schema:sdDatePublished 2019-04-10T23:20
    32 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    33 schema:sdPublisher Na2fc3a781118419484ed55e8c39c4bf7
    34 schema:url http://link.springer.com/10.1007/s00041-018-9638-y
    35 sgo:license sg:explorer/license/
    36 sgo:sdDataset articles
    37 rdf:type schema:ScholarlyArticle
    38 N064167f8639d493a8fe026110ba7d4b5 schema:name doi
    39 schema:value 10.1007/s00041-018-9638-y
    40 rdf:type schema:PropertyValue
    41 N392dca4ddd014ffba30043018f1c056a schema:name dimensions_id
    42 schema:value pub.1106415664
    43 rdf:type schema:PropertyValue
    44 N52ac10d7dae84d3f9097cd400a8e33fa rdf:first sg:person.0577752355.47
    45 rdf:rest rdf:nil
    46 N87d8a76e4c33435689e81d17cb7bb9d1 schema:name readcube_id
    47 schema:value b7c8feb85963eb4fa98c27fc7a33828798badf1c5d24b6ca4408b1beea62c36e
    48 rdf:type schema:PropertyValue
    49 Na2fc3a781118419484ed55e8c39c4bf7 schema:name Springer Nature - SN SciGraph project
    50 rdf:type schema:Organization
    51 Nee0110b757d34e179ea83cd065d701fd rdf:first sg:person.012770157205.22
    52 rdf:rest N52ac10d7dae84d3f9097cd400a8e33fa
    53 anzsrc-for:03 schema:inDefinedTermSet anzsrc-for:
    54 schema:name Chemical Sciences
    55 rdf:type schema:DefinedTerm
    56 anzsrc-for:0303 schema:inDefinedTermSet anzsrc-for:
    57 schema:name Macromolecular and Materials Chemistry
    58 rdf:type schema:DefinedTerm
    59 sg:journal.1042645 schema:issn 1069-5869
    60 1531-5851
    61 schema:name Journal of Fourier Analysis and Applications
    62 rdf:type schema:Periodical
    63 sg:person.012770157205.22 schema:affiliation https://www.grid.ac/institutes/grid.267461.0
    64 schema:familyName Hu
    65 schema:givenName Tian-You
    66 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012770157205.22
    67 rdf:type schema:Person
    68 sg:person.0577752355.47 schema:affiliation https://www.grid.ac/institutes/grid.411407.7
    69 schema:familyName Lau
    70 schema:givenName Ka-Sing
    71 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0577752355.47
    72 rdf:type schema:Person
    73 sg:pub.10.1007/978-3-0348-8380-1_2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1049957895
    74 https://doi.org/10.1007/978-3-0348-8380-1_2
    75 rdf:type schema:CreativeWork
    76 sg:pub.10.1007/bf01311213 schema:sameAs https://app.dimensions.ai/details/publication/pub.1007176639
    77 https://doi.org/10.1007/bf01311213
    78 rdf:type schema:CreativeWork
    79 sg:pub.10.1007/bf02018464 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039245448
    80 https://doi.org/10.1007/bf02018464
    81 rdf:type schema:CreativeWork
    82 sg:pub.10.1007/bf02475985 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041313012
    83 https://doi.org/10.1007/bf02475985
    84 rdf:type schema:CreativeWork
    85 sg:pub.10.1007/s11425-012-4422-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1011642614
    86 https://doi.org/10.1007/s11425-012-4422-y
    87 rdf:type schema:CreativeWork
    88 sg:pub.10.1007/s11856-011-0164-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1008614161
    89 https://doi.org/10.1007/s11856-011-0164-8
    90 rdf:type schema:CreativeWork
    91 https://doi.org/10.1006/aima.1998.1773 schema:sameAs https://app.dimensions.ai/details/publication/pub.1038435077
    92 rdf:type schema:CreativeWork
    93 https://doi.org/10.1006/jfan.1993.1116 schema:sameAs https://app.dimensions.ai/details/publication/pub.1029592685
    94 rdf:type schema:CreativeWork
    95 https://doi.org/10.1080/10586458.1992.10504561 schema:sameAs https://app.dimensions.ai/details/publication/pub.1053597897
    96 rdf:type schema:CreativeWork
    97 https://doi.org/10.1090/s0002-9939-02-06398-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1015923655
    98 rdf:type schema:CreativeWork
    99 https://doi.org/10.1215/s0012-7094-44-01111-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064416344
    100 rdf:type schema:CreativeWork
    101 https://doi.org/10.2307/2371053 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069897780
    102 rdf:type schema:CreativeWork
    103 https://doi.org/10.2307/2371641 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069898355
    104 rdf:type schema:CreativeWork
    105 https://www.grid.ac/institutes/grid.267461.0 schema:alternateName University of Wisconsin–Green Bay
    106 schema:name Department of Mathematics, University of Wisconsin-Green Bay, 54311, Green Bay, WI, USA
    107 rdf:type schema:Organization
    108 https://www.grid.ac/institutes/grid.411407.7 schema:alternateName Central China Normal University
    109 schema:name Department of Mathematics, Central China Normal University, Wuhan, China
    110 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
    111 Department of Mathematics, University of Pittsburgh, 15217, Pittsburgh, PA, USA
    112 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...