{
  "id": "1809.06541",
  "version": "v1",
  "published": "2018-09-18T05:43:48.000Z",
  "updated": "2018-09-18T05:43:48.000Z",
  "title": "Existence and exactness of exponential Riesz sequences and frames for fractal measures",
  "authors": [
    "Dorin Ervin Dutkay",
    "Shahram Emami",
    "Chun-Kit Lai"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_{\\epsilon}$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-18T05:43:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractal measures",
      "measures contain exponential riesz sequences",
      "exponential frame",
      "middle-third cantor measure",
      "no-overlap condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}