A note on the closure of translations in $L^{p}$

Kinukawa, Masakiti

doi:10.2748/tmj/1178243449

Cited by 5 publications

(5 citation statements)

References 8 publications

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“…Boas [3] and Konyushkov [7] proved that the hypothesis of corollary implies (4) and ( 5) and, equivalent to A 0 (f) < °o by Theorem 2.…”

Section: Convergence Of V|c W | /3 Anc * Related Problems Stechkin Pr...mentioning

confidence: 88%

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On the convolution algebra of Beurling

Sunouchi

1967

Tohoku Math. J. (2)

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“…Boas [3] and Konyushkov [7] proved that the hypothesis of corollary implies (4) and ( 5) and, equivalent to A 0 (f) < °o by Theorem 2.…”

Section: Convergence Of V|c W | /3 Anc * Related Problems Stechkin Pr...mentioning

confidence: 88%

“…Kinukawa [6,7] also discussed the problem of spectral synthesis under the conditions (4) and (5). However it is sufficient under (4) or (5) and this turns to Ap(f) <°° and reduces to Beurling's idea.…”

Section: Convergence Of V|c W | /3 Anc * Related Problems Stechkin Pr...mentioning

confidence: 99%

On the convolution algebra of Beurling

Sunouchi

1967

Tohoku Math. J. (2)

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“…least for 1 < p < 2. This problem has been studied by Segal [22], Beurling [5], Pollard [20], Herz [7], Newman [18] and other authors (see [6,13,21]).…”

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confidence: 99%

Wiener's `closure of translates’problem and Piatetski-Shapiro's uniqueness phenomenon

Lev

Olevskii

2011

Ann. Math.

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N. Wiener characterized the cyclic vectors (with respect to translations) in ℓ p (Z) and L p (R), p = 1, 2, in terms of the zero set of the Fourier transform. He conjectured that a similar characterization should be true for 1 < p < 2. Our main result contradicts this conjecture.is non-zero almost everywhere.(ii) c is cyclic in ℓ 1 (Z) if and only if c(t) has no zeros.Part (i) is a consequence of the unitarity of the Fourier transform. Part (ii) is more delicate, the proof is based on the fact that the space ℓ 1 (Z) is a convolution algebra.In both cases the result can be stated as follows: c is a cyclic vector if and only if the set Z c := {t ∈ T : c(t) = 0} of the zeros of the Fourier transform (1) is "small" in a certain sense. Wiener conjectured (see [24], p. 93) that a similar result should be true for ℓ p spaces, at

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“…However, our proof of Theorems 3 and 4 adopted in this paper is quite elementary. That is, we shall discuss the problem along the line set by Beurling [1], Boas [2], Sunouchi [6], and Kinukawa [5].…”

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confidence: 99%

“…The one dimensional case of Theorem 1 is refered to Kinukawa [5]. aLp is defined by a class of F with β || F|| p < oo.…”

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confidence: 99%

Contractions of Fourier transforms in $R_{k}$

Kinukawa¹

1972

Tohoku Math. J. (2)

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Introduction.In the present paper, we shall characterize some functions, those which satisfy a Lipschitz condition, as Fourier transforms of a certain sub-class of L p (R k ), and we shall give a contraction theorem of L p -Fourier transforms.A complex valued function f(x ly x 2 , , x k ) on R k , the k-άim. Euclidean space, is denoted by f(x).When / has the following property (i) or (ii), we say / is (p)normalized: for any finiteJi+y interval I, where 1/p + 1/p' = 1; (ii) if p = 1, then / is continuous and lim^,.^ f(x) = 0. We denote the j-th difference of /(a?), with respect to heR k , by Δl(f(x)), that is, Δί(f(x)) -Σ(-l) i+m ( 3 )f(x + mh) . m=0 \m)

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A note on the closure of translations in $L^{p}$

Cited by 5 publications

References 8 publications

On the convolution algebra of Beurling

On the convolution algebra of Beurling

Wiener's `closure of translates’problem and Piatetski-Shapiro's uniqueness phenomenon

Contractions of Fourier transforms in $R_{k}$

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