Sequences of dilations and translations equivalent to the Haar system in\n L^p-spaces

2020/05/10 by С. В. Асташкин, Astashkin, Sergey V., П. А. Терехин +1
Mathematics · #46E15 #46E20 #46E30 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2005.04648

openalex publication_date 2020/05/10 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Let f=\∑_k=0^\∞c_kh_2^k, where h_n is the classical Haar\nsystem, c_k\∈\ℂ. Given a p\∈ (1,\∞), we find the sharp\nconditions, under which the sequence f_{n n=1}^\∞ of dilations and\ntranslations of f is a basis in the space L^p[0,1], equivalent to\n h_{n n=1}^\∞. The results obtained depend substantially on whether\np\≥ 2 or 1<p<2 and include as the endpoints of the L_p-scale the spaces\nBMO_d and H_d¹. The proofs are based on an appropriate splitting the set\nof positive integers \ℕ=\∪_d=1^\∞ N_d so that the equivalence\nof f_{n n=1}^\∞ to the Haar system in L_p would be ensured by the\nfact that f_{n n\∈ Nd} is a basis in the subspace [h_m,m\∈\nN_d]_Lp, equivalent to the Haar subsequence h_{n n\∈ Nd} for every\nd=1,2,\….\n

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