Representation of Functions in Symmetric Spaces by Dilations and Translations
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IEF COMMUNICATIONS Representation of Functions in Symmetric Spaces by Dilations and Translations S. V. Astashkin and P. A. Terekhin Received February 24, 2019; in final form, June 19, 2019; accepted June 28, 2019
Abstract. Conditions under which the system of dilations and translations of a function f in a symmetric space X is a representing system in X are found. Previously a similar result was known 1 only for the spaces Lp , 1 p < ∞. In particular, each function f with 0 f (t) dt = 0 in a Lorentz space Λϕ generates an absolutely representing system of dilations and translations in this space if and only if the function ϕ(t) is submultiplicative. The key role in the proof is played by the notion of the multiplier space with respect to tensor product. Key words: system of dilations and translations, (absolutely) representing system, symmetric space, tensor product, multiplier space, frame, Lorentz space. DOI: 10.1134/S0016266320010050
1. Introduction. Let X be a separable symmetric space on I = [0, 1]. Under what conditions on a function f ∈ X is the system f (2k t − i) if t ∈ [ 2ik , i+1 ], 2k fk,i (t) = (1) i = 0, . . . , 2k − 1, k = 0, 1, . . . , 0 otherwise, of its dilations and translations representing in X? This means that, for each function x ∈ X, there exists a sequence {ξk,i } of coefficients such that k
x=
∞ 2 −1
ξk,i fk,i
(the convergence is in X).
(2)
k=0 i=0
For the spaces X = Lp , 1 p < ∞, this question was answered by Filippov and Oswald 1 in [1], where they proved that the obvious necessary condition 0 f (t) dt = 0 is simultaneously sufficient for the system of dilations and translations of a function f ∈ Lp to be a representing system in Lp . The key role in their proof was played by the fact that, for any function f ∈ Lp with 1 0 f (t) dt = 0, there exists a constant λ0 ∈ R such that 1 − λ0 f Lp < 1. In studying the possibility of representing functions by dilations and translations in any symmetric space, it has turned out that such a representation is not generally possible even in the power Lorentz space Λtα , 0 < α < 1. Another approach has proved more effective; it is based on an unexpected relationship between the problem under examination and the multiplier space of a symmetric space with respect to tensor product. In particular, we show that an analogue of the Filippov–Oswald theorem is valid for a Lorentz space Λϕ , provided that the function ϕ is submultiplicative, i.e., ϕ(st) Cϕ(s)ϕ(t) for some C > 0 and all s, t ∈ I. Recall that the Banach space X consisting of measurable real-valued functions x(t), t ∈ [0, 1], is said to be symmetric if (i) X is a function lattice, i.e., for any measurable function x(t) and any y ∈ X such that |x(t)| |y(t)|, t ∈ [0, 1], we have x ∈ X and xX yX ; c 2020 by Pleiades Publishing, Ltd. 0016–2663/19/5303–0174
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(ii) if functions x and y are equimeasurable, i.e., m{t ∈ [0, 1] : |x(t)| > τ } = m{t ∈ [0, 1] : |y(t)| > τ },
τ >0
(m is the Lebesgue measure), and y ∈ X, then x ∈ X and xX = yX . The spaces Lp , 1 p ∞,
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