1,720,973 research outputs found
Schwartz correspondence for real motion groups in low dimensions
No full textFor a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space S(K\G/K) isomorphically onto the space S(Sigma_D), where Sigma_D is an embedded copy of the Gelfand spectrum in R^ell, canonically associated to a generating system D of G-invariant differential operators on G/K, and S(Sigma_D) consists of restrictions to Sigma_D of Schwartz functions on R^ell. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair (Mn,SOn) with n=3,4. The rather trivial case n=2 is included in previous work by the same authors
Gelfand transforms of polyradial functions on the Heisenberg group
No full textWe prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups
Gelfand pairs on the Heisenberg group and Schwartz functions
No full textLet H_n be the (2n + 1)-dimensional Heisenberg group and K a compact group of automorphisms of H_n such that (K \ltimes H_n,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on H_n and the space of Schwartz function on a closed subset of R^s homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on H_n
On the Schwartz correspondence for Gelfand pairs of polynomial growth
No full textLet (G;K) be a Gelfand pair, with G a Lie group of polynomial growth, and let Σ ⊂ Rl be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system D1; . . . ;Dl of G-invariant di erential operators on G=K and associating to a bounded spherical function φ the l-tuple of its eigenvalues under the action of the Dj 's. We say that property (S) holds for (G;K) if the spherical transform maps the bi-K-invariant Schwartz space S(KnG=K) isomorphically onto S(Σ), the space of restrictions to Σ of the Schwartz functions on Rl. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where G = K⋊N, with N nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs
Schwartz correspondence for the complex motion group on C-2
No full textIf (G, K) is a Gelfand pair, with G a Lie group of polynomial growth and K a compact subgroup of G, the Gelfand spectrum & sigma; of the bi-K-invariant algebra L1(K\G/K) admits natural embeddings into Rt spaces as a closed subset.For any such embedding, define S(& sigma;) as the space of restrictions to & sigma; of Schwartz functions on Rt. We call Schwartz correspondence for (G, K) the property that the spherical transform is an isomorphism of S(K\G/K) onto S(& sigma;).In all the cases studied so far, Schwartz correspondence has been proved to hold true. These include all pairs with G = K H and K abelian and a large number of pairs with G = K & alpha; H and H nilpotent.We prove Schwartz correspondence for the pair (U2 IX M2(C), U2), where M2(C) is the complex motion group and U2 = K acts on it by conjugation. Our proof goes through a detailed analysis of (M2(C), U2) as a strong Gelfand pair and reduction of the problem to Schwartz correspondence for each K-type & tau; & ISIN; K ⠂ with appropriate control of the estimates in terms of & tau;
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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