{
  "id": "2209.09628",
  "version": "v1",
  "published": "2022-09-20T12:33:06.000Z",
  "updated": "2022-09-20T12:33:06.000Z",
  "title": "The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded by Certain Functions $f$",
  "authors": [
    "Hung Viet Chu"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $X$ be a Banach space and $(e_n)_{n=1}^\\infty$ be a basis. For a fixed function $f$ in a certain collection $\\mathcal{F}$ (closed under composition), we define and characterize ($f$, greedy) and ($f$, almost greedy) bases. These bases nontrivially extend the classical notion of greedy and almost greedy bases. We study relations among ($f$, (almost) greedy) bases as $f$ varies and show that while a basis is not almost greedy, it can still be ($f$, greedy) for some $f\\in \\mathcal{F}$. Furthermore, we prove that for all non-identity function $f\\in \\mathcal{F}$, we have the surprising equivalence $$\\mbox{($f$, greedy)}\\ \\Longleftrightarrow \\ \\mbox{($f$, almost greedy)}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-20T12:33:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A65",
      "46B15"
    ],
    "keywords": [
      "thresholding greedy algorithm",
      "approximations",
      "banach space",
      "bases nontrivially extend",
      "greedy bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}