On the Hausdorff-Young theorem for amalgams

Fournier, John J. F.

doi:10.1007/bf01323655

“…The proof of Theorem 3.1 in [7] applies, but we need to check that the results from [8], [9] and [10] are still valid in our setting. This requires the equivalence of the continuous and the discrete amalgam norms, which we showed in Propositions 19 and 21, together with uniform boundedness of translation along with the Hausdorff-Young theorem for these amalgam spaces.…”

Section: Functions That Are Square Integrable On a Neighbourhood Of Tmentioning

confidence: 99%

Wiener's theorem on hypergroups

Bloom¹,

Fournier²,

Leinert³

2015

Ann. Funct. Anal.

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The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f ∈ L 2 (T). This result has been extended to even exponents including p = ∞, but shown to fail for all other p ∈ (1, ∞] . All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents p ∈ [1, ∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 1 2 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

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“…and we can extract a sequence (f n ) from (f ι ) satisfying both (7) and (8), and (if necessary, passing to a subsequence thereof) converging pointwise a.e. to f. Using Fatou's lemma we obtain…”

Section: Remarkmentioning

confidence: 99%

Wiener's theorem for positive definite functions on hypergroups

Bloom¹,

Fournier²,

Leinert³

2014

Preprint

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The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f ∈ L 2 (T). This result has been extended to even exponents including p = ∞, but shown to fail for all other p ∈ (1, ∞] . All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents p ∈ [1, ∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 1 2 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

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“…In this paper, the methods of these authors are modified to prove the analogous result for other function spaces on R. For variants and applications of such results, see, for example, [4,9,11]. In particular, our results apply to amalgams of V and £9, as defined and studied in, for example, [2,3,5,6,7].…”

mentioning

confidence: 91%