Spaces defined by the Paley function

Асташкин, С. В.; Семенов, Э. М.

doi:10.1070/sm2013v204n07abeh004325

Cited by 9 publications

(4 citation statements)

References 18 publications

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“…a Haar chaos of order d and denote by H d ch the set of all such functions. 1 Observe that the latter series converges a.e. on [0, 1] and hence such a function x is welldefined.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

“…It is worth mentioning also that in the case 1 < p < ∞ Gamlen and Gaudet [8] have found the isomorphic types of subspaces spanned by subsequences of the Haar system. Specifically, they have showed that each subsequence {h n i } ⊂ {h n } spans a subspace in L p [0, 1], which is isomorphic either to ℓ p or to L p [0, 1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…, consisting of all p-multipliers and endowed with the 1 We follow here the common terminology related to chaoses in independent random variables (cf. [16, p. 147]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If X is a Banach function space, X ⊂ L 1 , then the Paley space P (X) consists of all functions x ∈ L 1 such that P x ∈ X and is equipped with the norm x P (X) := P x X [1]. In particular, P (L p ) = L p with equivalence of norms for every 1 < p < ∞.…”

Section: 3mentioning

confidence: 99%

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

Асташкин¹,

Терехин²

2020

Preprint

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Let f = ∞ k=0 c k h 2 k , where {h n } is the classical Haar system, c k ∈ C. Given a p ∈ (1, ∞), we find the sharp conditions, under which the sequence {f n } ∞ n=1 of dilations and translations of f is a basis in the space L p [0, 1], equivalent to {h n } ∞ n=1 . The results obtained depend substantially on whether p ≥ 2 or 1 < p < 2 and include as the endpoints of the L p -scale the spaces BM O d and H 1 d . The proofs are based on an appropriate splitting the set of positive integers N = ∪ ∞ d=1 N d so that the equivalence of {f n } ∞ n=1 to the Haar system in L p would be ensured by the fact that {f n } n∈N d is a basis in the subspace [h m , m ∈ N d ] Lp , equivalent to the Haar subsequence {h n } n∈N d for every d = 1, 2, .

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“…a Haar chaos of order d and denote by H d ch the set of all such functions. 1 Observe that the latter series converges a.e. on [0, 1] and hence such a function x is welldefined.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…, consisting of all p-multipliers and endowed with the 1 We follow here the common terminology related to chaoses in independent random variables (cf. [16, p. 147]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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