scholarly journals The spectral mapping property for p–multiplier operators on compact abelian groups

2005 ◽  
Vol 78 (3) ◽  
pp. 423-428 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Compact Set ◽  
Mapping Property ◽  
Spectral Mapping ◽  
Multiplier Operator ◽  
Algebraic Properties ◽  
Compact Abelian Groups ◽  
Multiplier Operators

AbstractLet G be a compact abelian group and 1< p < ∞. It is known that the spectrum σ (Tψ) of a Fourier p–multiplier operator Tψ acting in Lp(G), may fail to coincide with its natural spectrum ψ(Г) if p ≠ 2; here Γ is the dual group to G and the bar denotes closure in C. Criteria are presented, based on geometric, topological and/or algebraic properties of the compact set σ(Tψ), which are sufficient to ensure that the equality σ(Tψ) = ψ(Г)holds.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals Spectral mapping theorem for representations of measure algebras

1997 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
H. Seferoǧlu
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Homomorphism ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Unital Banach Algebra

Let G be a locally compact abelian group, M0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω: M0(G) → B be a continuous homomorphism of M0(G) into the unital Banach algebra B (possibly noncommutative) such that ω(L1(G)) is without order with respect to B in the sense that if for all b ∈ B, b.ω(L1(G)) = {0} implies b = 0. We prove that if sp(ω) is a synthesis set for L1(G) then the equality holds for each μ ∈ M0(G), where sp(ω) denotes the Arveson spectrum of ω, σB(.) the usual spectrum in B, the Fourier-Stieltjes transform of μ.

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

On Multipliers with Unconditionally Converging Fourier Series

1972 ◽  
Vol 24 (3) ◽  
pp. 477-484 ◽  
Author(s):  
Gregory F. Bachelis ◽  
Louis Pigno
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Complex Valued

Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.

Gap Series on Groups and Spheres

1966 ◽  
Vol 18 ◽  
pp. 389-398 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Gap Series ◽  
Image Position

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

scholarly journals Union results for thin sets

1990 ◽  
Vol 32 (2) ◽  
pp. 241-254
Author(s):  
Kathryn E. Hare
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Group Operation ◽  
Borel Measures ◽  
Thin Sets

Let G be a compact abelian group and let Γ be its (discrete) dual group. Denote by M(G) the space of complex regular Borel measures on G.Let E be a subset of Γ. Then:(i) E is called a Rajchman set if, for all μ ∈M(G) implies (ii) E is called a set of continuity if given ε > 0 there exists δ > 0 such that if and(iii) E is called a parallelepiped of dimension N if |E| = 2N and there are two-element sets . (The multiplication indicated here is the group operation.)

scholarly journals On vector-valued Hardy martingales and a generalized Jensen's inequality

2003 ◽  
Vol 2003 (9) ◽  
pp. 527-532
Author(s):  
Annela R. Kelly ◽  
Brian P. Kelly
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Analytic Functions ◽  
Jensen’S Inequality ◽  
Monotonicity Property ◽  
Dual Group ◽  
Analytic Vector ◽  
Jensen's Inequality ◽  
Vector Valued Functions ◽  
Vector Valued

We establish a generalized Jensen's inequality for analytic vector-valued functions on𝕋Nusing a monotonicity property of vector-valued Hardy martingales. We then discuss how this result extends to functions on a compact abelian groupG, which are analytic with respect to an order on the dual group. We also give a generalization of Helson and Lowdenslager's version of Jensen's inequality to certain operator-valued analytic functions.

scholarly journals On C.R. RAO’s theorem for locally compact abelian groups

2019 ◽  
Vol 484 (3) ◽  
pp. 273-276
Author(s):  
G. M. Feldman
Keyword(s):  
Abelian Group ◽  
Random Vector ◽  
Compact Abelian Group ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let x1, x2, x3 be independent random variables with values in a locally compact Abelian group X with nonvanish- ing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1x1 + a2x2 + a3x3, L2 = b1x1 + b2x2 + b3x3. It was proved that the distribution of the random vector (L1; L2) determines the distributions of the random variables xj up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao’s theorem for independent random variables with values in an a-adic solenoid.

scholarly journals Multipliers on spaces of functions on compact groups with p-summable Fourier transforms

1993 ◽  
Vol 47 (3) ◽  
pp. 435-442 ◽  
Author(s):  
Sanjiv Kumar Gupta ◽  
Shobha Madan ◽  
U.B. Tewari
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Compact Groups ◽  
Circle Group ◽  
Integrable Functions ◽  
Space Of Integrable Functions

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

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