Set Of Spectral Synthesis Research Articles

A Characterization of Some Sets of Spectral Synthesis

I will describe joint work with the late Bob Warner. We knew that $$H^1(R)$$ gave better results for singular integrals than $$L^1(R)$$ ; our question was: Would the same be true for spectral synthesis?. We extend the Beurling–Pollard argument to give sufficient conditions for spectral synthesis in $$H^1(R)$$ . We motivate and construct a class of Q-scets which satisfy the boundary and union property of synthesis, and give examples of Q-sets. To some extent the technical parts of the argument extend to d-dimensional Euclidean spaces for $$d \ge 1$$ .

Représentations des algèbres de Jordan, variétés de Stiefel et synthèse spectrale

We study the Stiefel's manifolds which arise in the framework of the representations of the Euclidean Jordan and simple algebras. We prove that most of them are not sets of spectral synthesis by using the generalized Bessel functions associated with these representations. To cite this article: A.G. Kalliterakis, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Invariant subspaces of the maximal domain of the Fourier transform

Translation invariant subspaces of the maximal domain of the Fourier transform (the amalgam of $l^2$ with $L^1$) are characterised: it turns out that in this case all measurable subsets of the dual space are sets of spectral synthesis.

Compact operators, invariant subspaces, and spectral synthesis

Let L be a subspace lattice which contains a sequence { P n } of commuting projections such that for any subsequence { P n i }, VP n i = I and ΛP n i = 0. Then Alg( L)∩ K( H)=0. Suppose G is a compact abelian group and E ⊆ G is a closed set. There is a family L E of commutative subspace lattices for which Alg( L E )∩ K( H)≠0 precisely when E is a set of multiplicity in the sense of harmonic analysis. By showing that the graph of ⩽ is a set of uniqueness in 2 ∞ × 2 ∞ we obtain a “thin set” proof that Alg(2 ∞, ⩽, m 1 2 ) contains no nonzero compact operators. For an important class of sets K ⊆ G × G A min( K) = A max( K) iff K is a set of spectral synthesis in the group G X G. It follows that if E 1and E 2 are sets of spectral synthesis in G then Alg(P E 1 \\ ̄ bo Alg (P E 2 ) = Alg(P E 1 ⊗ P E 2 iff E 1 × E 2 is a set of spectral synthesis in G X G and if semigroups ∑ 1 and ∑ 2 are sets of spectral synthesis in G then Alg(∑ 1) \\ ̄ bo Alg(∑ 2) = Alg(∑ 1 × ∑ 2 iff ∑ 1 × ∑ 2 is a set of spectral synthesis in G × G. The operator algebra Alg( P 1⊗ P 2 ⊗ …) is synthetic iff for all n, Alg( P 1 ⊗ … ⊗ P n is synthetic. This implies that the operator algebras Alg(2 ∞, ⩽, m p ) are synthetic.

On Spectral Synthesis and Ergodicity in Spaces of Vector-Valued Functions

AbstractSpectral synthesis in L∞(ℝ, ℂN), N < 1, is considered. It is is proved that sets of spectral synthesis are necessarily sets of spectral resolution.These results are applied to investigate ergodic and mixing properties of some positive contractions on L1(G, ℂN).

On closed ideals in the motion group algebra

A right closed ideal is said to be right-primary if it is contained in precisely one right maximal closed ideal. A set E C d//where E = h(k(E)), is said to be a set of spectral synthesis if I = k(E) is the only right closed ideal such that h(1)= E. For the Euclidean motion group M(2) we have proved in [4] that 0 is not a set of spectral synthesis. That is, the one-sided Wiener's Tauberian Theorem does not hold for M(2). Our main purpose is to generalize this result by showing that for M(2) many subsets of ~ fail to be sets of spectral synthesis. In particular, every member of J// contains infinitely many right-primary ideals, whereas in the abelian case every closed primary ideal is maximal [3, p. 326]. Using the result in [4] and his methods in [-1, 2], Leptin (in a letter to the author) obtained the same results for generalized Ll-algebras (see Appendix).

A support preserving Hahn-Banach property to determine Helson-S sets

We imbed the space of pseudo-measuresA′(E) supported by a closed totally disconnected setE\( \subseteq \)R/2πZ into a space of distributions on an “imbedding” group. The basic technique is to find a sequence of measuresμm onE (non discrete measures generally) associated with eachT∈A′(E), so that, with an additional arithmetic condition, {μm} converges in a weaker than weak * topology to a measure μ, and μ=T. Using this framework we prove that a Helson set is a set of spectral synthesis if and only if certain of our distributions have a support preserving extension. We also introduce a uniqueness criterion, and show that the extension condition and uniqueness condition imply thatA′(E) is the space of measures supported byE.

(LF) spaces and distributions on compact groups and spectral synthesis onR/2?Z

LetE\( \subseteqq\)R/2πZ be closed with Lebesgue measure 0. We imbed the pseudo-measuresTsupported byE into a space of distributions on a specific compact connected group. The reason for this approach is to make use of the more tractable differentiable absolutely convergent Fourier series for the general problem to determine whenT is a measure. The specific results are outlined in the Introduction. Applications of the techniques presented here are used to obtain new criteria that a Helson set be a set of spectral synthesis in the author's forthcoming work (viz., “A Support Preserving Hahn-Banach Property and the Helson-S Set Problem” and “The Helson-S Set Problem and Discontinuous Homomorphisms on Metric Algebras”).