{
  "id": "2204.13907",
  "version": "v1",
  "published": "2022-04-29T07:00:47.000Z",
  "updated": "2022-04-29T07:00:47.000Z",
  "title": "Weak Convergence and Spectrality of Infinite Convolutions",
  "authors": [
    "Wenxia Li",
    "Jun Jie Miao",
    "Zhiqiang Wang"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CA",
    "math.FA",
    "math.PR"
  ],
  "abstract": "Let $\\{ A_k\\}_{k=1}^\\infty$ be a sequence of finite subsets of $\\mathbb{R}^d$ satisfying that $\\# A_k \\ge 2$ for all integers $k \\ge 1$. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution $$\\nu =\\delta_{A_1}*\\delta_{A_2} * \\cdots *\\delta_{A_n}*\\cdots, $$ where all sets $A_k \\subseteq \\mathbb{R}_+^d$ and $\\delta_A = \\frac{1}{\\# A} \\sum_{a \\in A} \\delta_a$. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in $\\mathbb{R}$ and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-29T07:00:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "42C30",
      "60B10"
    ],
    "keywords": [
      "infinite convolution",
      "weak convergence",
      "spectrality",
      "singular spectral measures",
      "necessary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}