Interpolation and Sidon Sets for Compact Groups

Graham, Colin C.; Hare, Kathryn E.

doi:10.1007/978-1-4614-5392-5

Cited by 42 publications

(49 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The study of lacunary sets, such as Sidon sets and Λ(p) sets, constitutes an interesting theme in the theory of Fourier series on the circle group T. It has many applications in harmonic analysis and in the theory of Banach spaces, and various combinatorial and arithmetic properties of these sets have been studied extensively. These concepts have also been investigated in the context of more general compact abelian groups (with their discrete dual groups) and compact non-abelian groups; see [5], [9], [15] and the references cited therein. The study of these sets in the setting of discrete non-abelian groups was pioneered by Bozjeko [1], Figá-Talamanca [4] and Picardello [10].…”

Section: Introductionmentioning

confidence: 99%

A Non-abelian, Non-Sidon, Completely Bounded Set

Hare

Mohanty²

2020

Can. Math. Bull.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

A Non-abelian, Non-Sidon, Completely Bounded Set

Hare

Mohanty²

2020

Can. Math. Bull.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Motivated by this result, we proceed to look for all sequences sharing this property with (2 n ) n∈N . However, it turns out that this seemingly ergodic theoretical question was answered by harmonic analysts in the 1960's, and sequences with this property have been extensively studied under the name of interpolation sets [32,16,29,19,10].…”

mentioning

confidence: 99%

Interpolation sets and nilsequences

2020

Colloq. Math.

View full text Add to dashboard Cite

To give positive answer to a question of Frantzikinakis, we study a class of subsets of N, called interpolation sets, on which every bounded sequence can be extended to an almost periodic sequence on N. Strzelecki has proved that lacunary sets are interpolation sets. We prove that sets that are denser than all lacunary sets cannot be interpolation sets. We also extend the notion of interpolation sets to nilsequences and show that the analogue to Frantzikinakis' question for arbitrary sequences is false.

show abstract

“…Our basic references are [71,26,98] for topological spaces and for topologies on function spaces, [85,13,89] for the study of real-valued functions, [53] for topological groups and [52,32,86,44,10] for the study of the dual set of a topological group.…”

Section: Preliminary Results and Terminologymentioning

confidence: 99%

“…Now, one could analogously define central I 0 sets, but in that case, since central discrete measures must be supported on the centre of G [80], there would be groups for which not even all finite sets would be central I 0 sets (see [44,45]). As a consequence, there would be I 0 sets that are not central I 0 sets.…”

Section: Thus |P(f )mentioning

confidence: 99%

See 1 more Smart Citation

Interpolation and equicontinuity sets in topological groups and spaces of continuous functions

Ruiz¹

View full text Add to dashboard Cite

Index 105Bibliography 109 IntroductionClassical Hadamard sets (also called lacunary sets) have been studied since the beginning of the past century. Definition.A subset E = {n j } ∞ j=1 of N, with n 1 < n 2 < . . . is said to be a Hadamard set if there exists some q > 1 (called Hadamard ratio) such that n j+1 /n j ≥ q for all 1 ≤ j ≤ ∞. c j z n j has a radius of convergence equal to 1. Then f cannot be analytically continued across any portion of the arc |z| = 1.In the 1950's, some of these properties were more closely studied and finally became definitions. Since these properties were more functional analytic in nature, whereas the original lacunary property was arithmetic, it was no longer necessary to restrict attention to sets of integers.Therefore, lacunary sets have also been studied in the dual set of compact topological groups and in more general settings. Kahane [62] first used the term Sidon set in 1957 and the modern point of view of the notion appeared in Rudin's Book [86] in 1962. Let G = T, a Hadamard set E ⊆ G = Z is a special type of Sidon set.Given a compact abelian group G, one of the several characterisations of a Sidon set (see [66], for instance), stated that a subset E of G is a Sidon set if every bounded function on E is the restriction of the Fourier transform of a measure on G.Since the dual of a compact group is a discrete group and the Sidon sets are situated in the dual set, we can study the Sidon sets as subsets of discrete groups. In this sense, Picardello [76] extended in 1973 the usual definition of Sidon set in a group G to the discrete non-abelian case. 2 IntroductionA special kind of Sidon set is the I 0 sets. The concept of I 0 set in locally compact abelian groups was introduced by Hartman and Ryll-Nardzewski [48] in 1964, who considered the weak topology associated to a locally compact abelian (LCA, for short) group and introduced the notion of interpolation set or I 0 set. They defined that a subset A of a LCA group G is an I 0 set if every bounded function on A is the restriction of an almost periodic function on G (here, it is said that a complex-valued function f defined on G is almost periodic when it is the restriction of a continuous function defined on bG, the Bohr compactification of G).Therefore, an I 0 set is a subset A of G such that any bounded map on A can be interpolated by a continuous function on bG. As a consecuence, if A is a countably infinite I 0 set, then A bG is canonically homeomorphic to βω, the Stone-Čech compactification of ω. The main result given by Hartman and Ryll-Nardzewski is the following:For the particular case of discrete abelian groups, van Douwen achieved a remarkable progress by proving the existence of I 0 sets in very general situations. His main result can be formulated in the following way:Theorem. ([94, Th. 1.1.3]) Let G be a discrete Abelian group and let A be an infinite subset of G. Then, there is a subset B of A with |B| = |A| such that B is an I 0 set.In fact, van Douwen extended his result to the real line but left unresolv...

show abstract

Interpolation and Sidon Sets for Compact Groups

Cited by 42 publications

References 0 publications

A Non-abelian, Non-Sidon, Completely Bounded Set

A Non-abelian, Non-Sidon, Completely Bounded Set

Interpolation sets and nilsequences

Interpolation and equicontinuity sets in topological groups and spaces of continuous functions

Contact Info

Product

Resources

About