Spectral synthesis on zero-dimensional locally compact Abelian groups

2019 ◽  
pp. 450-456
Author(s):  
Sergey S. Platonov
Keyword(s):  
Invariant Subspace ◽  
Linear Subspace ◽  
Spectral Synthesis ◽  
Linear Span ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Closed Linear Subspace ◽  
Compact Abelian Groups ◽  
Continuous Complex ◽  
Complex Valued

Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A closed linear subspace H⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H.

On multipliers of Fourier transforms

1972 ◽  
Vol 71 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

scholarly journals On a Generalized Wilson Functional Equation

2005 ◽  
Vol 12 (4) ◽  
pp. 595-606
Author(s):  
Roman Badora
Keyword(s):  
Functional Equation ◽  
Finite Group ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Locally Compact ◽  
Continuous Complex ◽  
A Finite Group ◽  
Complex Valued

Abstract We solve the functional equation where 𝐺 is a locally compact Abelian group, {𝑘0 = 𝑖𝑑𝐺, 𝑘1, . . . , 𝑘𝑁–1} is a finite group of continuous automorphisms of 𝐺, 𝑓 is a bounded continuous complex-valued function on 𝐺 and 𝑔 is an almost periodic function on 𝐺.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

Exponential Estimates for the Conjugate Function on Locally Compact Abelian Groups

1990 ◽  
Vol 33 (1) ◽  
pp. 34-44
Author(s):  
Nakhlé Habib Asmar
Keyword(s):  
Compact Support ◽  
Weak Type ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Exponential Estimates ◽  
Compact Abelian Groups ◽  
The Hilbert Transform ◽  
Almost All

AbstractLet G be a locally compact Abelian group, with character group X. Suppose that X contains a measurable order P. For the conjugate function of f is the function whose Fourier transform satisfies the identity for almost all χ in X where sgnp(χ) = - 1 , 0, 1, according as We prove that, when f is bounded with compact support, the conjugate function satisfies some weak type inequalities similar to those of the Hilbert transform of a bounded function with compact support in ℝ. As a consequence of these inequalities, we prove that possesses strong integrability properties, whenever f is bounded and G is compact. In particular, we show that, when G is compact and f is continuous on G, the function is integrable for all p > 0.

Locally compact groups with closed subgroups open and p-adic

1995 ◽  
Vol 118 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Karl H. Hofmann ◽  
Sidney A. Morris ◽  
Sheila Oates-Williams ◽  
V. N. Obraztsov
Keyword(s):  
Topological Group ◽  
Open Subgroup ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Character Group ◽  
Exceptional Groups ◽  
Open Set ◽  
Compact Abelian Groups

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

scholarly journals On weak spectral synthesis

1991 ◽  
Vol 43 (2) ◽  
pp. 279-282 ◽  
Author(s):  
K. Parthasarathy ◽  
Sujatha Varma
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Lie Group ◽  
Compact Abelian Group ◽  
Spectral Synthesis ◽  
Locally Compact Abelian Group ◽  
Fourier Algebra ◽  
Compact Lie Group ◽  
Locally Compact

Weak spectral synthesis fails in the group algebra and the generalised group algebra of any non compact locally compact abelian group and also in the Fourier algebra of any infinite compact Lie group.

scholarly journals Jackson's Theorem for locally compact abelian groups

1974 ◽  
Vol 10 (1) ◽  
pp. 59-66 ◽  
Author(s):  
Walter R. Bloom
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Integrable Function ◽  
Locally Compact Abelian Group ◽  
Classical Theorem ◽  
Extended Version ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Approximate Unit ◽  
Compact Abelian Groups

If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which‖f-tn‖p ≤(p; f; 1/n).In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

scholarly journals Positive linear operators and the approximation of continuous functions on locally compact abelian groups

1980 ◽  
Vol 30 (2) ◽  
pp. 180-186 ◽  
Author(s):  
Walter R. Bloom ◽  
Joseph F. Sussich
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Locally Compact Abelian Group ◽  
Periodic Functions ◽  
Locally Compact ◽  
Character Group ◽  
Spaces Of Continuous Functions ◽  
Compact Abelian Groups ◽  
Approximation Of Continuous Functions

AbstractIn 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2π-periodic functions and limn→rTnf = f uniformly for f = 1, cos and sin. then limn→rTnf = f uniformly for all f∈C. We extend this result to spaces of continuous functions defined on a locally compact abelian group G, with the test family {1, cos, sin} replaced by a set of generators of the character group of G.

scholarly journals Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Davood Alimohammadi ◽  
Sirous Moradi
Keyword(s):  
Compact Subset ◽  
Metric Space ◽  
Linear Subspace ◽  
Sufficient Condition ◽  
Compact Metric Space ◽  
Linear Subspaces ◽  
Lipschitz Algebra ◽  
Continuous Complex ◽  
Complex Valued

Let be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace of little Lipschitz algebra in this algebra and in particular showed that is dense in , whenever . Let be a compact subset of . We define new classes of Lipchitz algebras for and for , consisting of those continuous complex-valued functions on such that and , respectively. In this paper we obtain a sufficient condition for density of a linear subspace of extended little Lipschitz algebra in this algebra and in particular show that is dense in , whenever .

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.

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