scholarly journals Polynomials and functions with finite spectra on locally compact Abelian groups

1995 ◽  
Vol 51 (1) ◽  
pp. 33-42 ◽  
Author(s):  
B. Basit ◽  
A.J. Pryde
Keyword(s):  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Finite Spectrum ◽  
Beurling Algebra ◽  
Uniformly Continuous Function ◽  
Compact Abelian Groups ◽  
Indefinite Integral ◽  
Beurling Algebras ◽  
Primary Ideals ◽  
Uniformly Continuous

In this paper we define polynomials on a locally compact Abelian group G and prove the equivalence of our definition with that of Domar. We explore the properties of polynomials and determine their spectra. We also characterise the primary ideals of certain Beurling algebras on the group of integers Z. This allows us to classify those elements of that have finite spectrum. If ϕ is a uniformly continuous function with bounded differences then there is a Beurling algebra naturally associated with ϕ. We give a condition on the spectrum of ϕ relative to this algebra which ensures that ϕ is bounded. Finally we give spectral conditions on a bounded function on ℝ that ensure that its indefinite integral is bounded.

scholarly journals On a family of weighted convolution algebras

1990 ◽  
Vol 13 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Hans G. Feichtinger ◽  
A. Turan Gürkanli
Keyword(s):  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Asymptotic Estimates ◽  
Locally Compact ◽  
Invariance Properties ◽  
Beurling Algebra ◽  
Convolution Algebras ◽  
Approximate Identities ◽  
Beurling Algebras ◽  
Banach Ideals

Continuing a line of research initiated by Larsen, Liu and Wang [12], Martin and Yap [13], Gürkanli [15], and influenced by Reiter's presentation of Beurling and Segal algebras in Reiter [2,10] this paper presents the study of a family of Banach ideals of Beurling algebrasLw1(G),Ga locally compact Abelian group. These spaces are defined by weightedLp-conditions of their Fourier transforms. In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in section 3 and to show that two spaces of this type coincide if and only if their parameters are equal. In section 4 the existence of approximate identities in these algebras is established, from which, among other consequences, the bijection between the closed ideals of these algebras and those of the corresponding Beurling algebra is derived.

Translates of L∞ functions and of bounded measures

1964 ◽  
Vol 4 (4) ◽  
pp. 403-409 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Abelian Group ◽  
Continuous Functions ◽  
Group Element ◽  
Locally Compact Abelian Group ◽  
Space Of Continuous Functions ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Radon Measures ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

D. A. Edwards has shown [1] that if X is a locally compact Abelian group and f ∈ L∞, then the translate fa of f varies continuously with α if and only if f is (equal l.a.e. to) a bounded, uniformly continuous function. He remarks that this is a sort of dual to part of a result due to Plessner and Raikov which asserts that an element μ of the space Mb of bounded Radon measures on X belongs to L1 (i.e., is absolutely continuous relative to Haar measure) if and only its translates vary continuously with the group element, the relevant topology on Mb being that defined by the natural norm of Mb as the dual of the space of continuous functions vanishing at infinity. The proof he uses (ascribed to Reiter) applies equally well in both cases, and also to the case in which X is non-Abelian. A brief examination shows that in the latter case it is ultimately immaterial whether left- or right-translates are considered; since the extra complexities of this case are principally terminological, we shall direct no further attention to it.

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.

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.

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 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.

On the Extension of Uniformly Continuous Functions(1)

1975 ◽  
Vol 18 (1) ◽  
pp. 143-145 ◽  
Author(s):  
L. T. Gardner ◽  
P. Milnes
Keyword(s):  
Continuous Function ◽  
Uniform Space ◽  
Continuous Extension ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Totally Bounded ◽  
Special Cases ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

AbstractA theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.

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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